CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Este trabalho relata a experiÃncia de aplicaÃÃo em sala de aula de uma sequÃncia de atividades de trigonometria com destaque para a construÃÃo, estudo e utilizaÃÃo de um ciclo trigonomÃtrico ou prancha trigonomÃtrica numa turma de 2 ano TÃcnico em ComÃrcio (em nÃvel mÃdio) de uma escola profissional da rede estadual de ensino do estado do CearÃ. As atividades aplicadas constam de estudo de semelhanÃa de triÃngulos, das razÃes trigonomÃtricas no triÃngulo retÃngulo e das funÃÃes circulares seno, cosseno e tangente facilitadas por sua vez, pelo manuseio da prancha trigonomÃtrica. A avaliaÃÃo dos resultados do trabalho se dà atravÃs do acompanhamento das facilidades/dificuldades encontradas pelos alunos na confecÃÃo dos materiais e na resoluÃÃo das atividades propostas. / This paper reports the experience of the application in the classroom of a sequence of activities trigonometry with emphasis on the construction, study and use of a trigonometric trigonometric cycle or board in a class of 2nd year in Technical Trading (medium level) a professional state school education in the state of CearÃ. The activities listed in the applied study of similar triangles, the trigonometric ratios in right triangle and circular functions sine, cosine and tangent facilitated in turn by the management of trigonometric board. The evaluation of the results of the work is through the monitoring of the facilities/difficulties encountered by students in the preparation of materials and the resolution of the proposed activities.

Изучаются коэффициенты тригонометрического полинома при односторонних ограничениях на отрезках [0, π] и [0, 2π]. Для нечетных тригонометрических полиномов степени не выше двух получены точные верхние и нижние границы первого коэффициента при линейном одностороннем ограничении. Для полиномов более высоких степеней определено поведение максимальных и минимальных возможных значений при одностороннем линейном ограничении. Проведено исследование поведения максимальных и минимальных значений первых коэффициентов синус-полиномов при двусторонних линейных ограничениях. / We study coefficients of a trigonometric polynomial under a one-sided constraint on the intervals [0, π] и [0, 2π]. We obtain sharp upper and lower estimates of the first coefficient of an odd trigonometric polynomial of degree at most two under a linear constraint. For polynomials of higher degree, we find the behavior of maximal and minimal coefficients of a sine-polynomial under two-sided linear constraints.

Essa dissertação analisará uma abordagem investigativa de ensino de funções trigonométricas que prioriza a compreensão da relação entre movimentos circulares em diferentes velocidades com a formação gráfica gerada por esses movimentos. Com o auxílio do software Geogebra, diferentes movimentos foram criados, o que proporcionou a investigação gráfica por parte dos alunos. A atividade foi realizada no laboratório de informática onde, constantemente, houve investigação por parte dos alunos e intervenções significativas por parte do professor. Escolheu-se para essa pesquisa uma análise qualitativa embasada no processo descritivo das ações ocorridas em sala de aula. Para conhecer as características dessa abordagem, foi utilizado um estudo de casos. Após a atividade, os alunos conseguiram interpretar os principais movimentos gerados na circunferência e traduzi-los na sua forma gráfica. A análise mostra que os alunos não somente conseguiram desenvolver significados aos movimentos circulares, como também interpretaram corretamente situações cotidianas estabelecidas pelo professor ao fim do trabalho / This dissertation will analyze an investigative approach to the teaching of trigonometric functions that prioritizes the understanding of the relation between circular movements at different speeds with the graphical formation generated by these movements. With the help of the software Geogebra, different movements were created, which provided the graphic investigation by the students. The activity was carried out in the computer lab where, constantly, there was investigation by the students and significant interventions by the teacher. For this research, a qualitative analysis based on the descriptive process of the actions taken in the classroom was chosen. To know the characteristics of this approach, a case study was used. After the activity, the students were able to interpret the main movements generated on the circumference and translate them into their graphic form. The analysis shows that the students not only managed to develop meanings to the circular movements, but also correctly interpreted daily situations established by the teacher at the end of the work

Cette thèse est constituée de trois chapitres. Le premièr chapitre porte sur l’examen desconditions de validité du principe d’équipartition de l’énergie totale de la solution de l’équationdes ondes associée au laplacien de Dunkl trigonométrique. Enfin, nous établissons lecomportement asymptotique de l’équipartition dans le cas général. Les résultats de cettepartie ont fait l’objet de la publication [8]. Le deuxième chapitre, publié avec J.Ph. Ankeret M. Sifi [6], montre que les fonctions d’Opdam dans le cas de rang 1 satisfont à uneformule produit. Cela nous a permis de définir une structure de convolution du genre hypergroupe.En particulier, on montre que cette convolution satisfait l’analogue du phénomènede Kunze-Stein. Le dernier chapitre est consacrée à l’étude des propriétés dispersives et estimationsde Strichartz pour la solution de l’équation de Schrödinger associée au laplaciende Dunkl trigonométrique unidimensionnel [7]. Cette étude commence par des estimationsoptimales du noyau de la chaleur et de Schrödinger. À l’aide de ces résultats, ainsi que lesoutils d’analyse harmonique dévellopée dans le chapitre 2, on montre des éstimées de typeStrichartz qui permettent de trouver des conditions d’admissibilité pour des équations deSchrödinger semi-linéaires. / This thesis consists of three chapters. The first one is concerned with energy properties of the wave equation associated with the trigonometric Dunkl Laplacian. We establish the conservation of the total energy, the strict equipartition of energy under suitable assumptions and the asymptotic equipartition in the general case. These results were published in [8]. The second chapter, in collaboration with J.Ph. Anker and M. Sifi [6], shows that Opdam’s functions in the rank one case satisfy a product formula. We then define and study a convolution structure related to Opdam’s functions. In particular, we prove that this convolution fulfills a Kunze-Stein type phenomena. The last chapter deals with dispersive and Strichartz estimates for the linear Schrödinger equation associated with the one dimensional trigonometric Dunkl Laplacian [7]. We establish sharp estimates for the heat kernel in complex time, and therefore for the Schrödinger kernel. We then use these estimates together with tools from chapter 2 to deduce dispersive and Strichartz inequalities for the linear Schrödinger equation and apply them to well–posedness in the nonlinear case.

I give an overview of the basic theory of Hilbert spaces necessary to understand the convergence of the Fourier series for square integrable functions. I state the necessary theorems and definitions to understand the formulations of the problem in a Hilbert space framework, and then I give some applications of the theory along the way.

Multi-Polynomial Higher Order Neural Network Group Models (MPHONNG) program developed by the author will be studied in this thesis. The thesis also investigates the use of MPHONNG for financial data and rainfall data simulation and prediction. The MPHONNG is combined with characteristics of Polynomial function, Trigonometric polynomial function and Sigmoid polynomial function. The models are constructed with three layers Multi-Polynomial Higher Order Neural Network and the weights of the models are derived directly from the coefficents of the Polynomial form, Trignometric polynomial form and Sigmoid polynomial form. To the best of the authors knowledge, it is the first attempt to use MPHONNG for financial data and rainfall data simulation and prediction. Results proved satisfactory, and confirmed that MPHONNG is capable of handling high frequency, high order nonlinear and discontinuous data. / Master of Science (Hons)

This thesis investigates a new method for financial data simulation using novel neural network models developed by the author. Using two improved models for financial data simulation and prediction, the trigonometric polynomial higher order neural network group models have been developed. The theoretical principles of these improved models are presented and demonstrated in the thesis. It is the first attempt to use trigonometric polynomial high order neural network group models for financial data simulation. We could not find any references to using trigonometric polynomial high order neural network group models for financial data simulation in the extensive literature search conducted for this thesis, including a thorough Internet search on this topic. The author has developed a computer program, called 'THONG'. The program, running on X-windows, is based on the new neural network models developed, and also uses group models. This program allows users to apply in practice his new analysis and prediction method. The 'THONG' program is a user-friendly GUI system. All the steps of the operation in this system are easily controlled using a mouse. Both system operation and system mode can be viewed during the processing of data. THONG models have proven capable of handling high frequency, high order nonlinear and discontinuous data. The results of processing the experimental data using the THONG financial simulator are presented in the thesis. These results confirm that the THONG group models converge without difficulty, and are considerably more accurate than traditional neural network models. / Doctor of Philosophy (PhD)

The function $sin x$ as one of the six trigonometric functions is fundamental in nearly every branch of mathematics, and its applications. In this thesis, we study an integral equation related to that of $sin x$: $mbox{~for~}xin[-frac{hat{pi}_{p}}{2},~frac{hat{pi}_{p}}{2}] mbox{~and~} p>1$ $$x=int_0^{S_{p}(x)}(1-|t|^{p})^{-frac{1}{p}}dt.$$ Here $hat{pi}_{p}=frac{2pi}{psin(frac{pi}{p})}=2int_0^1(1-t^{p})^{-frac{1}{p}}dt.$ We find that the function $S_{p}(x)$ is well defined. Its properties are also similar to those of $sin x$ : differentiation, identities, periodicity, asymptotic expansions, $cdots$, etc. For example, we have $$|S_{p}(x)|^{p}+|S'_{p}(x)|^{p}=1mbox{~~and~~}frac{d}{dx}(|S'_{p}(x)|^{p-2}S'_{p}(x))=-(p-1)|S_{p}(x)|^{p-2}S_{p}(x).$$ We call $S_{p}(x)$ the generalized sine function. Similarly, we define the generalized cosine function $C_{p}(x)$ by $|x|=int_{C_{p}(x)}^{1}(1- t^{p})^{-frac{1}{p}}dt$ for $xin[-frac{hat{pi}_{p}}{2}$,~$frac{hat{pi}_{p}}{2}]$ and derive its properties. Thus we obtain two sets of trigonometric functions: egin{itemize} item[(i)]$~S_{p}(x),~ S'_{p}(x),~ T_{p}(x)=frac{S_{p}(x)}{S'_{p}(x)},~RT_{p}(x)=frac{S'_{p}(x)}{S_{p}(x)},~ SE_{p}(x)=frac{1}{S'_{p}(x)},~ RS_{p}(x)=frac{1}{S_{p}(x)}~;$ item[(ii)]$~C_{p}(x),~ C'_{p}(x),~RCT_{p}(x)=-frac{C'_{p}(x)}{C_{p}(x)},~ CT_{p}(x)=-frac{C_{p}(x)}{C'_{p}(x)},~RC_{p}(x)=frac{1}{C_{p}(x)},~ CS_{p}(x)=-frac{1}{C'_{p}(x)}mbox{~¡C~}$ end{itemize}These two sets of functions have similar differentiation formulas, identities and periodic properties as the classical trigonometric functions. They coincide when $p=2$. Their graphs and asymptotic expansions are also interesting. Through this study, we understand more about the theoretical framework of trigonometric functions.

In this paper we consider the exact $D$-optimal design problem for linear trigonometric regression models with or without intercept on a partial circle. In a recent papper Dette, Melas and Pepelyshev (2001) found explicit solutions of approximate $D$-optimal designs for trigonometric regression models with intercept on a partial circle. The exact optimal designs are determined by means of moment sets of trigonometric functions. It is shown that the structure of the optimal designs depends on both the length of the design interval and the number of the design points.

Essa dissertação analisará uma abordagem investigativa de ensino de funções trigonométricas que prioriza a compreensão da relação entre movimentos circulares em diferentes velocidades com a formação gráfica gerada por esses movimentos. Com o auxílio do software Geogebra, diferentes movimentos foram criados, o que proporcionou a investigação gráfica por parte dos alunos. A atividade foi realizada no laboratório de informática onde, constantemente, houve investigação por parte dos alunos e intervenções significativas por parte do professor. Escolheu-se para essa pesquisa uma análise qualitativa embasada no processo descritivo das ações ocorridas em sala de aula. Para conhecer as características dessa abordagem, foi utilizado um estudo de casos. Após a atividade, os alunos conseguiram interpretar os principais movimentos gerados na circunferência e traduzi-los na sua forma gráfica. A análise mostra que os alunos não somente conseguiram desenvolver significados aos movimentos circulares, como também interpretaram corretamente situações cotidianas estabelecidas pelo professor ao fim do trabalho / This dissertation will analyze an investigative approach to the teaching of trigonometric functions that prioritizes the understanding of the relation between circular movements at different speeds with the graphical formation generated by these movements. With the help of the software Geogebra, different movements were created, which provided the graphic investigation by the students. The activity was carried out in the computer lab where, constantly, there was investigation by the students and significant interventions by the teacher. For this research, a qualitative analysis based on the descriptive process of the actions taken in the classroom was chosen. To know the characteristics of this approach, a case study was used. After the activity, the students were able to interpret the main movements generated on the circumference and translate them into their graphic form. The analysis shows that the students not only managed to develop meanings to the circular movements, but also correctly interpreted daily situations established by the teacher at the end of the work