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Text in Afrikaans / A study of the philosophical and historical foundations of infinity highlights the problematic development of infinity. Aristotle distinguished between potential and actual infinity, but rejected the latter. Indeed, the interpretation of actual infinity leads to contradictions as seen in the paradoxes of Zeno. It is difficult for a human being to understand actual infinity. Our logical schemes are adapted to finite objects and events. Research shows that students focus primarily on infinity as a dynamic or neverending process. Individuals may have contradictory intuitive thoughts at different times without being aware of cognitive conflict. The intuitive thoughts of students about both the actual (at once) infinite and potential (successive) infinity are very complex. The problematic nature of actual infinity and the contradictory intuitive cognition should be the starting point in the teaching of the concept infinity. / Educational Studies / M.Ed. (Mathematic Education)

自2009年美國提出重返亞洲(Pivot to Asia)的主張後，其對於東亞地區的注意力明顯提高，對區域間的盟友如菲律賓、日本，也加強了各式援助與安全保證。與此同時，隨著中國自改革開放後實力提升，對於向外發展的需求也日益提升。在此過程中，海權的發展，對其而言至關重要，然而中國所面對的是對其發展抱持疑慮的周邊環境。儘管中國強調和平發展原則，然他國仍對其擴展影響力感到不安，其中美國亦將中國認為是潛在的挑戰者，故而採取支持周邊國家的方式限制中國，使中國在近年的海上爭議中與他國呈現僵持的局面。 本研究主要以Robert Jervis 的認知理論為研究途徑，探討中美在互動中是否產生特定的機制，及對於中國海權發展影響。本研究發現，由認知的角度切入，中美兩國在互動過程中，確有機制存在，而主要起始點為兩國對於中國身分認知是否相符，若是肯定的答案，則兩國對於中國海權的認知將產生理性相符的結果。然而當兩國認知存在差距時，將會導向以非理性相符的態度看待中國海權，中國認為其是良性發展，但現實中卻是造成他國不安；美國認為限制中國符合自身利益，忽略中國和平發展的意圖，形成兩國皆將主觀認知加之於客觀不相容的現實的情況。當兩者的非理性相符認知碰撞的情況下，以本研究的釣魚台爭端為例，將使中國在向外海權發展中，尤其是對海洋領土的主張受到阻礙。 / Since the claiming of “Pivot to Asia”, the U.S. enhanced its support and security insurance toward alliance in East Asia as well. Meanwhile, with the growing of its strength, China’s demand of reaching outward also keeps rising. During this process, the developing of sea power plays an important role. However, the environment which China facing is not as supportive as it expected. Despite its emphasis on peaceful development, there is still anxiety in other countries, especially the U.S. which considers China as possible challenger, thus supporting countries in order to restrict China, leading to stalemate in recent dispute on sea issue. Using theory of perception of Robert Jervis as main approach, in order to investigate the mechanism between Sino-U.S. interaction, also its influence on China’s sea power . We find out that there truly is mechanism during the process , and the main point is difference of cognition on China’s identity. If the cognition of the two countries are corresponded, the result of perception on China’s sea power rational cognitive consistency. When perception gap occurs, it will lead to the result of irrational cognitive consistency, which China considers its development as benign, causing the feeling of threatening in reality. In the other hand, the U.S. believes that setting restraint on China brings benefits, thus ignoring China’s true intention, which form the situation of adding subjective perception on contradictive reality. As the Diaoyu Island dispute in the thesis, China’s sea power meets obstacle when the irrational cognitive consistency collapsed.

O conceito de número de rotação originalmente definido para homeomorfismos do círculo S1 que preservam orientação pode ser generalizado para todo homeomorfismo h do anel fechado S1×[0; 1] isotópico à identidade, onde obtemos o chamado conjunto de rotação. Neste trabalho estudamos o caso em que o conjunto de rotação de h se reduz somente a um número irracional ? (neste caso dizemos que h é uma pseudo-rotação irracional), obtendo que para qualquer inteiro positivo n, existe um arco simples ? que une uma componente do bordo do anel à outra, de tal modo que ? é disjunto de seus n primeiros iterados por h: Este resultado é um análogo do Teorema de Kwapisz concernente a difeomorfismos do toro bidimensional [14]. Posteriormente e utilizando o primeiro resultado, provamos que a rotação rígida de ângulo pode ser aproximada por um homeomorfismo conjugado a h. Finalmente, mostramos que ser uma pseudo-rotação irracional é uma propriedade necessária para que um homeomorfismo tenha a propriedade de interseção de curvas e não tenha pontos periódicos. / The concept of rotation number originally defined for orientation preserving homeomorphisms of the circle S1 can be generalized for any homeomorphism h of closed annulus S1×[0; 1] which is isotopic to the identity. In this setting we obtain the so called rotation set. In this work we study the case when the rotation set of h is reduced to a single irrational number ? (we say that h is an irrational pseudo-rotation), and we prove that for any positive integer n, there exists a simple arc ? joining one of the boundary components of annulus to the other, such that ? is disjoint from its n first iterates under h: This result is an analogue of a theorem of Kwapisz dealing with diffeomorphisms of the two-torus [14]. Subsequently and applying the first result, we prove that a rigid rotation of angle can be approximated by a homeomorphism that is conjugate to h: Finally, we prove that to be an irrational pseudo-rotation is a necessary property in order that a homeomorphism has the curves intersection property and no periodic points.

Ringtones are a common distracting sound in modern workspaces. In an earlierexperiment, ringtones increasing in volume (looming) produced greater attentional capture effectin the context of serial short-term memory, than ringtones with sudden onsets that decreased involume (receding). To determine whether this effect occurred merely because the loudest part ofthe looming ringtone coincided with the most sensitive part of the serial short-term memory task,this study repeated the sound conditions of the first experiment, but altered their timing. In thisstudy, the onset of the ringtones were brought forward in time such that the loudest part of thelooming ringtone now coincided with the part of the serial short-term memory task wherein theonset of the looming ringtone occurred in the first experiment. The looming ringtone againproduced more disruption than the receding ringtone, which failed to disrupt performance relativeto the quiet control condition. The presence of a masking sound eliminated the looming ringtoneeffect, as in the previous study. The results here support previous work demonstrating that thelooming sounds give rise to attentional capture and that this reflects an evolutionary adaptation tounconsciously react to approaching sounds/objects.

In this thesis we present and solve selected problems arising from digital geometry and combinatorics on words. We consider digital straight lines and, equivalently, upper mechanical words with positive irrational slopes a<1 and intercept 0. We formulate a continued fraction (CF) based description of their run-hierarchical structure. Paper I gives a theoretical basis for the CF-description of digital lines. We define for each irrational positive slope less than 1 a sequence of digitization parameters which fully specifies the run-hierarchical construction. In Paper II we use the digitization parameters in order to get a description of runs using only integers. We show that the CF-elements of the slopes contain the complete information about the run-hierarchical structure of the line. The index jump function introduced by the author indicates for each positive integer k the index of the CF-element which determines the shape of the digitization runs on level k. In Paper III we present the results for upper mechanical words and compare our CF-based formula with two well-known methods, one of which was formulated by Johann III Bernoulli and proven by Markov, while the second one is known as the standard sequences method. Due to the special treatment of some CF-elements equal to 1 (essential 1's in Paper IV), our method is currently the only one which reflects the run-hierarchical structure of upper mechanical words by analogy to digital lines. In Paper IV we define two equivalence relations on the set of all digital lines with positive irrational slopes a<1. One of them groups into classes all the lines with the same run length on all digitization levels, the second one groups the lines according to the run construction in terms of long and short runs on all levels. We analyse the equivalence classes with respect to minimal and maximal elements. In Paper V we take another look at the equivalence relation defined by run construction, this time independently of the context, which makes the results more general. In Paper VI we define a run-construction encoding operator, by analogy with the well-known run-length encoding operator. We formulate and present a proof of a fixed-point theorem for Sturmian words. We show that in each equivalence class under the relation based on run length on all digitization levels (as defined in Paper IV), there exists exactly one fixed point of the run-construction encoding operator.

Limit theorems in the sense of weak convergence of probability measures for the Lerch zeta-function with algebraic irrational parameter are obtained. A theorem of mentioned type on the complex plane, a joint limit theorem for a collection of Lerch zeta-functions on the complex plane as well as a limit theorem in the space of analytic functions are proved. The theorems obtained characterize the asymptotic behaviour of the Lerch zeta-function and can be applied in the investigation of the universality of that function. / Yra gautos Lercho dzeta funkcijos su algebriniu iracionaliuoju parametru ribinės teoremos silpno tikimybinių matų konvergavimo prasme. Yra įrodyta minėto tipo teorema kompleksinėje plokštumoje, jungtinė ribinė teorema Lercho dzeta funkcijų rinkiniui kompleksinėje plokštumoje ir teorema analizinių funkcijų erdvėje. Įrodytos teoremos charakterizuoja Lercho dzeta funkcijų asimptotinį elgesį ir gali būti taikomos šios funkcijos universalumui tirti.

Yra gautos Lercho dzeta funkcijos su algebriniu iracionaliuoju parametru ribinės teoremos silpno tikimybinių matų konvergavimo prasme. Yra įrodyta minėto tipo teorema kompleksinėje plokštumoje, jungtinė ribinė teorema Lercho dzeta funkcijų rinkiniui kompleksinėje plokštumoje ir teorema analizinių funkcijų erdvėje. Įrodytos teoremos charakterizuoja Lercho dzeta funkcijų asimptotinį elgesį ir gali būti taikomos šios funkcijos universalumui tirti. / Limit theorems in the sense of weak convergence of probability measures for the Lerch zeta-function with algebraic irrational parameter are obtained. A theorem of mentioned type on the complex plane, a joint limit theorem for a collection of Lerch zeta-functions on the complex plane as well as a limit theorem in the space of analytic functions are proved. The theorems obtained characterize the asymptotic behaviour of the Lerch zeta-function and can be applied in the investigation of the universality of that function.

Text in Afrikaans / A study of the philosophical and historical foundations of infinity highlights the problematic development of infinity. Aristotle distinguished between potential and actual infinity, but rejected the latter. Indeed, the interpretation of actual infinity leads to contradictions as seen in the paradoxes of Zeno. It is difficult for a human being to understand actual infinity. Our logical schemes are adapted to finite objects and events. Research shows that students focus primarily on infinity as a dynamic or neverending process. Individuals may have contradictory intuitive thoughts at different times without being aware of cognitive conflict. The intuitive thoughts of students about both the actual (at once) infinite and potential (successive) infinity are very complex. The problematic nature of actual infinity and the contradictory intuitive cognition should be the starting point in the teaching of the concept infinity. / Educational Studies / M.Ed. (Mathematic Education)

Submitted by Emanuel Varela Cardoso (emanuel.varela@ufcg.edu.br) on 2018-11-09T18:09:57Z No. of bitstreams: 1 ANA CLÁUDIA GUEDES DOS SANTOS – DISSERTAÇÃO (PPGMat) 2013.pdf: 24981615 bytes, checksum: d442e8df3b32727e30684e3cbd516a9b (MD5) / Made available in DSpace on 2018-11-09T18:09:57Z (GMT). No. of bitstreams: 1 ANA CLÁUDIA GUEDES DOS SANTOS – DISSERTAÇÃO (PPGMat) 2013.pdf: 24981615 bytes, checksum: d442e8df3b32727e30684e3cbd516a9b (MD5) Previous issue date: 2013-08 / Capes / Este trabalho tem como proposta pedagógica apresentar aos alunos o conceito de segmentos comensuráveis e de segmentos incomensuráveis, mostrando a importância desses conceitos para o estudo dos números racionais e irracionais. Veremos um processo de verificação da comensurabilidade de dois segmentos, doravante P.V.C.D.S, que é um processo geométrico de verificação de comensurabilidade de dois segmentos. A partir do P.V.C.D.S, apresentamos a demonstração clássica de que p2 é irracional, com uma abordagem geométrica, mostrando que o segmento do lado de um quadrado de medida 1 e o segmento de sua diagonal são incomensuráveis. Ainda apresentamos um estudo sobre expressões decimais, no qual será apresentado um teorema que nos permite verificar se uma fração irredutível possui representação decimal finita ou infinita e periódica. Também apresentamos outro teorema que nos permite transformar expressões decimais finitas e infinitas e periódicas na sua forma de fração. Por fim, apresentaremos algumas sugestões de atividades, que englobam todo conteúdo do presente TCC. Essas atividades foram aplicadas a uma turma de 1 ano do Ensino Médio de uma escola pública, e as respostas dos alunos estão anexadas ao trabalho. / This work have pedagogical proposed to introduce the concept of commensurable segments and incommensurable segments, showing the importance of these concepts for the study of rational and irrational numbers. We will stabelish a verification process to detect the mensurability of two segments, which is a geometric process. We present the classic demonstration that root of 2 is irrational with a geometric approach, showing that the segment of the side of a square measuring its diagonal are immeasurable. We still will present a study on decimal expressions, and prove a theorem that allows to check that an irreducible fraction has decimal representation finite or infinite and periodic. We also present another theorem that allows us to turn decimal expressions finite or infinite and periodic on its fraction form. Finally we present some suggestions for activities that include all content of the TCC. These activities have been applied to a class of 1st year of high school at a public school, and the students’ answers are attached to the work.