Submitted by Angela Maria de Oliveira (amolivei@uepg.br) on 2018-11-29T18:07:06Z No. of bitstreams: 2 license_rdf: 811 bytes, checksum: e39d27027a6cc9cb039ad269a5db8e34 (MD5) Maikon Luiz.pdf: 959643 bytes, checksum: aaf472f5b8a9a29532793d01234788a9 (MD5) / Made available in DSpace on 2018-11-29T18:07:06Z (GMT). No. of bitstreams: 2 license_rdf: 811 bytes, checksum: e39d27027a6cc9cb039ad269a5db8e34 (MD5) Maikon Luiz.pdf: 959643 bytes, checksum: aaf472f5b8a9a29532793d01234788a9 (MD5) Previous issue date: 2018-10-19 / Neste trabalho,estudamos os números e os polinomios de Bernoulli,bem como algumas de suas aplicações mais importantes em Teoria dos Números. Com base em uma caracterização ao simples, os polinômios de Bernoulli são introduzidos e, posteriormente, os números de Bernoulli. As séries de Fourier dos polinomios de Bernoulli são utilizadas na demonstração da equação funcional da função teta. Esta equação, por sua vez, é utilizada na demonstração da celebre equação funcional da função zeta, que tem importância central na teoria da distribuição dos números primos. Além das conexões com a funções especiais zeta e teta, discutimos também, em detalhe,conexões entre os números e os polinomios de Bernoulli com a função gama. Essas relações são então exploradas para produzir belas fórmulas para certos valores da função zeta, entre outras aplicações. / In this work we study Bernoulli numbers and Bernoulli polynomials, as well as some of its most important applications in Number Theory. Based on a simple characterization, the Bernoulli polynomials are introduced and, later, the Bernoulli numbers. The Fourier series of the Bernoulli polynomials are used to demonstrate the functional equation of the theta function. This equation, in turn, is used in the proof of the famous functional equation of the zeta function, which is central to the theory of prime number distribution. In addition to the connections with the special functions zeta and theta, we also discuss, in detail, connections between the Bernoulli numbers and Bernoulli polynomials with the gamma function. These relations are then explored to produce beautiful formulas for certain values of the zeta function,among other applications.

臺灣婦女生育率下降快速，近年來屢創新低，堪稱全球生育率最低的國家，總生育率自民國89年1.68、降為民國98年1.03，民國99年甚至降至0.90以下，提升生育率成為政府施政的重要課題。因為資料限制，生育率大多以總生育率(Total Fertility Rate)表示，而非較能反映婦女一生生育總數的世代完成生育率(Completed Cohort Fertility Rate)。這兩者間存有不少差異，以生育率下降的臺灣為例，總生育率會因生育時機遞延而低估世代生育率，以總生育率詮釋生育率可能有瑕疵。有鑒於此，本文以比較「世代」及「年代」兩者的差異，以生育率及死亡率為研究對象，探討較適宜描述臺灣特性的模型。 由於世代生育率會有資料不足的問題，本文使用外推法(Extrapolation)補足年齡較高（如35歲以上）的婦女生育率，並以四種模型估計年代生育率與世代生育率，包括Gamma模型、Gompertz模型、主成份分析(Principle Component Analysis)與單一年齡組個別估計法，希望找出適合預測臺灣世代完成生育率的模型。除了台灣資料，也用日本、法國與美國的世代生育率資料，比較各國世代生育率模型的異同。另外，本文也以世代及年代兩種觀點，類似生育率的探討方式，比較常用死亡率模型的優劣。 不論是生育率或是死亡率資料，配適模型結果皆以世代資料可得到較好的估計結果，生育率以單一年齡組個別估計法為最佳的模型，死亡率則以Gamma模型、主成份分析、單一年齡組個別估計法為較佳的模型。 / Taiwan’s fertility rates have been declining radically in recent years, much faster than most countries in the world. For example, the total fertility rate (TFR) is 1.68 in 2000, 1.03 in 2009, and even reduces to 0.90 in 2010. Therefore, one of the top priorities for Taiwan government policies is to enhance the willingness of having children. Due to the data availability, the TFR is used more often, although the completed cohort fertility rate (CFR) is a more reasonable measurement. However, previous studies showed that the TFR is likely to be influenced by the deferring (i.e., tempo effect) of childbearing and produces misleading results. In order to measure the effect of deferring childbearing, this study focuses on exploring the difference of measures in the view of cohort and period (especially the CFR vs. TFR) and evaluates which fertility and mortality model is more appropriate for Taiwan. Because there are fewer complete cohort fertility data, we use extrapolation to make up the higher age-group fertility data (such as aged 35 and above). We consider four fertility models in this study, including Gamma model, Gompertz model, principal component analysis, and individual group estimation. We use the data from Taiwan, Japan, France and United State data to evaluate these fertility models. The results indicate that the parametric models (Gamma and Gompertz) have the worst performance, probably due to the rapid change of fertility behaviors. In addition, similar to evaluating the fertility models, we compare the performance of frequently used mortality models using the cohort and period mortality data. The result shows that using cohort data to estimate fertility and mortality is better than period data. Also individual group estimation is the best model to fit fertility; the better models to fit mortality are Gamma model, principle component analysis and individual group estimation.

台灣地區的人口隨著生育率及死亡率的雙重下降而呈現快速老化，其中生育率的降低影響尤為顯著。民國50年時，台灣平均每位婦女生育5.58個小孩，到了民國70年卻只生育1.67個小孩，去年（民國90年）生育率更創歷年新低，只有1.4。死亡率的下降可由平均壽命的延長看出，民國75年時男性為70.97歲，女性為75.88歲；到了民國90年，男性延長到72.75歲，女性延長到78.49歲。由於生育率的變化幅度高於死亡率，對人口結構的影響較大，因此本文分成兩個部份，主要在研究台灣地區15至49歲婦女生育率的變化趨勢，再將研究結果用於台灣地區未來人口總數及其結構的預測。 　　本研究第一部分是生育率的研究，引進Gamma函數、Gompertz函數、Lee-Carter法三種模型及單一年齡組個別估計法，以民國40年至84年（西元1951年至1995年）的資料為基礎，民國85年至89年（西元1996年至2000年）資料為檢測樣本，比較模型的優劣，尋求較適合台灣地區生育率的模型，再以最合適的模型預測民國91年至140年（西元2002年至2051年）的生育率。第二部分是人口推估，採用人口變動要素合成方法（Cohort Component Projection Method）推估台灣地區未來50年的人口總數及其結構，其中生育率採用上述最適合台灣地區的模型、死亡率則引進國外知名的Lee-Carter法及SOA法（Society of Actuaries），探討人口結構，並與人力規劃處的結果比較之。 / Both the fertility rate and mortality rate have been experiencing dramatic decreases in recent years. As a result, the population aging has become one of the major concerns in Taiwan area, and the proportion of the elderly (age 65 and over) increases promptly from 2.6% in 1965 to 8.8% in 2001. The decrease of fertility rate is especially significant. For example, the total fertility rate was 5.58 in 1961, and then decreases dramatically to 1.67 in 1981 (1.4 in 2001), a reduction of almost 70% within 20 years. 　　The goal of this paper is to study the population aging in Taiwan area, in particular, the fertility pattern. The first part of this paper is to explore the fertility models and decide which model is the most suitable based on age-fertility fertility rates in Taiwan. The models considered are Gamma function, Gompertz function, Lee-Carter method and individual group estimation. We use the data from 1951 to 1995 as pilot data and 1996 to 2000 as test data to judge which model fit well. The second part of this study is to project the Taiwan population for the next 50 years, i.e. 2002-2051. The projection method used is Cohort Component Projection method, assuming the population in Taiwan area is closed. We also compare our projection result to that by Council for Economic Planning and Development, the Executive Yuan of the Republic of China.