Academic literature on the topic 'Arithmetic average'

In this research, an efficient numerical method is developed to determine the steady-state waiting time distribution of a GI/G/1 queue by solving the discrete-time version of Lindley’s equation, when the queue is bounded on a finite interval. Then, by using Little’s Formula, we calculate the stationary distribution for the total number of customers in the queue. The derivations are based on the Wiener-Hopf factorization of random walks. The method is carried out using a successive approximation method, by improving the weighted average. Finally, to prove the effectiveness of our method, we apply the algorithm for Uniform, Geometric, and Gamma distributions, to find an approximation. An analytical interpretation is also presented to find the waiting time distribution for the Geom/Geom/1 queue, which is not based on a finite interval, as an example of the GI/G/1 queue. Moreover, compare to the other related methods it has been proven that our method is numerically stable, simple, and robust.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior<br>Este trabalho trata de duas das mais importantes desigualdades da MatemÃtica: a desigualdade entre as mÃdias geomÃtrica e aritmÃtica e a desigualdade de Cauchy-Schwarz. Apresentamos inicialmente diversas demonstraÃÃes para o caso n = 2, apÃs as quais seguem muitas demonstraÃÃes para o caso geral. Nessas demonstraÃÃes utilizamos Ãlgebra elementar, geometria euclidiana, construÃÃes geomÃtricas, geometria analÃtica, induÃÃo matemÃtica, convexidade de funÃÃes, multiplicadores de Lagrange entre outros assuntos. AlÃm disso foram selecionados vinte problemas que visam dar ao leitor uma melhor compreensÃo de como estas desigualdades podem ser aplicadas em diversos assuntos e de diversas formas, estimulando a criatividade dos alunos na resoluÃÃo de problemas.<br>This paper deals with two of the most important inequalities of Mathematics: the inequality between the geometric and arithmetic and Cauchy-Schwarz. Here several first statements for the case n = 2, after which many statements following for the general case. In these statements we use algebra elementary Euclidean geometry, geometric constructions, analytical geometry, mathematical induction, convexity of functions, Lagrange multipliers among other issues. Also selected were twenty problems that aim to give the reader a better understanding of how these inequalities can be applied in various subjects and in many ways, stimulating students' creativity in problem solving.

Orientador: Antônio Carlos Patrocinio<br>Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica<br>Made available in DSpace on 2018-08-25T12:22:05Z (GMT). No. of bitstreams: 1 Pereira_JaksonDaCruz_M.pdf: 1462332 bytes, checksum: 393b1f36be156bf1cdedb16da1cc3fcd (MD5) Previous issue date: 2014<br>Resumo: O presente trabalho se dedica ao estudo das médias aritmética, geométrica e harmônica. Inicialmente, definimos cada uma das médias e trabalhamos suas aplicações através da resolução de problemas. Posteriormente destacamos as desigualdades entre as médias e suas aplicações<br>Abstract: This dissertation is dedicated to the study of the arithmetic, geometric and harmonic means. Initially, we defined each of the means and applied its uses through problem resolution. Afterwards, we gave emphasis to the inequalities between the means and its uses<br>Mestrado<br>Matemática em Rede Nacional<br>Mestre em Matemática em Rede Nacional

Made available in DSpace on 2016-04-27T16:58:28Z (GMT). No. of bitstreams: 1 Jose Odair Ribeiro.pdf: 3472459 bytes, checksum: acdbee3b90dda93570dc1cdf69f5f2a3 (MD5) Previous issue date: 2007-10-10<br>Secretaria da Educação do Estado de São Paulo<br>The purpose of this study is to investigate the reading and interpretation of graphs and tables by specialist and non specialist teachers at Math, which act in Elementary School, with the objective of answering the following research question: What are the conceptions and competences that specialist and non specialist teachers at Math, which act in Elementary school have about basic concepts of Statistics? For such, a diagnostic research was developed with 40 teachers (20 polyvalent teachers G1 and 20 specialist at Math G2), coming from two schools of the public state network of the city of Mauá SP. The field work was constituted of two steps applying of the diagnostic tool, for both G1 and G2 and applying of semi-structured interviews in 10% of G1 and G2. The results obtained in each of these steps were analyzed, considering the object of the research reading and interpretation of graphs and tables as well as the four types of graphs used (column graph, line graph, sector graph and bar graph), a table of double entry and the concept of arithmetic average. The results showed the superiority of G2 over G1 and indicated that the conceptions and competences of teachers from both groups are still linked to a technicist view of Statistics, limited to a simple interpretation of basic concepts<br>Este estudo teve como objetivo investigar a leitura e a interpretação de gráficos e tabelas por professores especialistas e não especialistas em Matemática, que atuam no Ensino Fundamental, com a finalidade de responder à seguinte questão de pesquisa: Quais as concepções e competências que os professores especialistas e não especialistas em Matemática, que atuam no Ensino Fundamental têm sobre conceitos básicos de Estatística? Para tanto, foi desenvolvida uma pesquisa diagnóstica com 40 professores (20 polivalentes G1 e 20 especialistas em Matemática G2), advindos de duas escolas da rede pública estadual da cidade de Mauá SP. O trabalho de campo contemplou duas etapas aplicação do instrumento diagnóstico, tanto no G1 como no G2 e aplicação de entrevistas semi-estruturada em 10% de G1 e G2. Os resultados obtidos em cada uma das etapas foram analisados, considerando o objeto da pesquisa leitura e interpretação de gráficos e tabelas bem como os quatro tipos de gráficos usados (gráfico de colunas, gráfico de linhas, gráfico de setores e gráfico de barras), uma tabela de dupla entrada e o conceito de média aritmética. Os resultados evidenciaram a superioridade do G2 sobre o G1 e indicaram que as concepções e competências dos professores de ambos os grupos, ainda, se encontram vinculadas a uma visão tecnicista da Estatística, limitada a uma interpretação simples dos conceitos básicos

Made available in DSpace on 2016-04-27T16:59:02Z (GMT). No. of bitstreams: 1 Fabio Muniz do Amaral.pdf: 2248512 bytes, checksum: 42a09ee45471a17734b25bc5d5ed22d8 (MD5) Previous issue date: 2010-05-18<br>The purpose of this study is to validates a didactic sequence verifying the forms the teachers of elementary Education understand and work the meanings of the center and spread measures, mode, arithmetic and mean average, according to the levels of operation of Aline Robert's knowledge (1998). To reach this objective we elaborated a didactic sequence that contemplates activities in order to contribute in the (re)construction of the meanings of the necessary knowledge so that the teachers have better conditions of applying them in classroom. We discussed the characteristics of the mode, Arithmetic and Medium average, besides the properties that each of those measures presents, using several registrations us to explore all possibilities. Our theoretical references are those proposed by Robert (1998) and the Theory of Semiotic Representation Registers (DUVAL, 2003) to analyze the activities to be used with the teachers. As methodology we used assumptions Didactic Engineering. The analysis of the resolutions and discussions with teachers throughout the implementation process of the sequence revealed their willingness and desire to learn as well as their participation in all the process of (re)construction of knowledge. In this sense we see that the explanation of the meanings of the measures of central tendency for teachers provided greater security for the performance of their work in the classroom, so we infer that the sequence contributed to a better training to the students<br>O presente trabalho teve como objetivo validar uma sequência didática, verificando de que forma as professoras do Ensino Fundamental de uma Escola de Educação Básica entendem e trabalham os significados das medidas de tendência central: Moda, Média aritmética e Mediana, segundo os níveis de funcionamento dos conhecimentos de Aline Robert (1998). Para atingir este objetivo, foi elaborada uma sequência didática que contemplou atividades para contribuir na (re)construção dos significados dos conhecimentos necessários para que as professoras tenham melhores condições de aplicá-los em sala de aula. As características de Moda, Média aritmética e Mediana foram discutidas, além das propriedades que cada uma dessas medidas apresentam. Vários registros foram utilizados para explorar todas as possibilidades de aprendizagem. Como referencial teórico, os níveis propostos por Robert (1998) e a Teoria dos Registros de Representação Semiótica (DUVAL, 2003) para analisar e serem trabalhados em conjunto com as professoras que participaram do estudo. Como metodologia usamos pressupostos da Engenharia Didática. A análise das resoluções e as discussões das professoras durante todo o processo de aplicação da sequência revelaram a disposição e vontade de aprender, além da participação de todas no processo de (re)construção desses conhecimentos. Nesse sentido, percebeu-se que o esclarecimento dos significados das medidas de tendência central proporcionou às professoras maior segurança no desenvolvimento de seu trabalho em sala de aula, portanto, inferiu-se que a sequência contribuiu para que as professoras possam proporcionar melhor formação a seus alunos

Doutoramento em Matemática<br>Nesta tese de Doutoramento desenvolve-se principalmente uma abordagem algébrica à teoria de sistemas de controlo não lineares. No entanto, outros tópicos são também estudados. Os tópicos tratados são os seguidamente enunciados: fórmulas para sistemas de controlo sobre álgebras de Lie livres, estabilidade de um sistema de corpos rolantes, algoritmos para aritmética digital, e equações integrais de Fredholm não lineares. No primeiro e principal tópico estudam-se representações para as soluções de sistemas de controlo lineares no controlo. As suas trajetórias são representadas pelas chamadas séries de Chen. Estuda-se a representação formal destas séries através da introdução de várias álgebras não associativas e técnicas específicas de álgebras de Lie livres. Sistemas de coordenadas para estes sistemas são estudados, nomeadamente, coordenadas de primeiro tipo e de segundo tipo. Apresenta-se uma demonstração alternativa para as coordenadas de segundo tipo e obtêm-se expressões explícitas para as coordenadas de primeiro tipo. Estas últimas estão intimamente ligadas ao logaritmo da série de Chen que, por sua vez, tem fortes relações com uma fórmula designada na literatura por “continuous Baker-Campbell- Hausdorff formula”. São ainda apresentadas aplicações à teoria de funções simétricas não comutativas. É, por fim, caracterizado o mapa de monodromia de um campo de vectores não linear e periódico no tempo em relação a uma truncatura do logaritmo de Chen. No segundo tópico é estudada a estabilizabilidade de um sistema de quaisquer dois corpos que rolem um sobre o outro sem deslizar ou torcer. Constroem-se controlos fechados e dependentes do tempo que tornam a origem do sistema de dois corpos num sistema localmente assimptoticamente estável. Vários exemplos e algumas implementações em Maple°c são discutidos. No terceiro tópico, em apêndice, constroem-se algoritmos para calcular o valor de várias funções fundamentais na aritmética digital, sendo possível a sua implementação em microprocessadores. São também obtidos os seus domínios de convergência. No último tópico, também em apêndice, demonstra-se a existência e unicidade de solução para uma classe de equações integrais não lineares com atraso. O atraso tem um carácter funcional, mostrando-se ainda a diferenciabilidade no sentido de Fréchet da solução em relação à função de atraso.<br>In this PhD thesis several subjects are studied regarding the following topics: formulas for nonlinear control systems on free Lie algebras, stabilizability of nonlinear control systems, digital arithmetic algorithms, and nonlinear Fredholm integral equations with delay. The first and principal topic is mainly related with a problem known as the continuous Baker-Campbell-Hausdorff exponents. We propose a calculus to deal with formal nonautonomous ordinary differential equations evolving on the algebra of formal series defined on an alphabet. We introduce and connect several (non)associative algebras as Lie, shuffle, zinbiel, pre-zinbiel, chronological (pre-Lie), pre-chronological, dendriform, D-I, and I-D. Most of those notions were also introduced into the universal enveloping algebra of a free Lie algebra. We study Chen series and iterated integrals by relating them with nonlinear control systems linear in control. At the heart of all the theory of Chen series resides a zinbiel and shuffle homomorphism that allows us to construct a purely formal representation of Chen series on algebras of words. It is also given a pre-zinbiel representation of the chronological exponential, introduced by A.Agrachev and R.Gamkrelidze on the context of a tool to deal with nonlinear nonautonomous ordinary differential equations over a manifold, the so-called chronological calculus. An extensive description of that calculus is made, collecting some fragmented results on several publications. It is a fundamental tool of study along the thesis. We also present an alternative demonstration of the result of H.Sussmann about coordinates of second kind using the mentioned tools. This simple and comprehensive proof shows that coordinates of second kind are exactly the image of elements of the dual basis of a Hall basis, under the above discussed homomorphism. We obtain explicit expressions for the logarithm of Chen series and the respective coordinates of first kind, by defining several operations on a forest of leaf-labelled trees. It is the same as saying that we have an explicit formula for the functional coefficients of the Lie brackets on a continuous Baker-Campbell-Hausdorff-Dynkin formula when a Hall basis is used. We apply those formulas to relate some noncommutative symmetric functions, and we also connect the monodromy map of a time-periodic nonlinear vector field with a truncation of the Chen logarithm. On the second topic, we study any system of two bodies rolling one over the other without twisting or slipping. By using the Chen logarithm expressions, the monodromy map of a flow and Lyapunov functions, we construct time-variant controls that turn the origin of a control system linear in control into a locally asymptotically stable equilibrium point. Stabilizers for control systems whose vector fields generate a nilpotent Lie algebra with degree of nilpotency · 3 are also given. Some examples are presented and Maple°c were implemented. The third topic, on appendix, concerns the construction of efficient algorithms for Digital Arithmetic, potentially for the implementation in microprocessors. The algorithms are intended for the computation of several functions as the division, square root, sines, cosines, exponential, logarithm, etc. By using redundant number representations and methods of Lyapunov stability for discrete dynamical systems, we obtain several algorithms (that can be glued together into an algorithm for parallel execution) having the same core and selection scheme in each iteration. We also prove their domains of convergence and discuss possible extensions. The last topic, also on appendix, studies the set of solutions of a class of nonlinear Fredholm integral equations with general delay. The delay is of functional character modelled by a continuous lag function. We ensure existence and uniqueness of a continuous (positive) solution of such equation. Moreover, under additional conditions, it is obtained the Fr´echet differentiability of the solution with respect to the lag function.

In this thesis we mainly focused on the usage of the Conditional Value-at-Risk (CVaR) in risk management and on the pricing of the arithmetic average basket and Asian options in the Black-Scholes framework via a new log-normal sum approximation method. Firstly, we worked on the linearization procedure of the CVaR proposed by Rockafellar and Uryasev. We constructed an optimization problem with the objective of maximizing the expected return under a CVaR constraint. Due to possible intermediate payments we assumed, we had to deal with a re-investment problem which turned the originally one-period problem into a multiperiod one. For solving this multi-period problem, we used the linearization procedure of CVaR and developed an iterative scheme based on linear optimization. Our numerical results obtained from the solution of this problem uncovered some surprising weaknesses of the use of Value-at-Risk (VaR) and CVaR as a risk measure. In the next step, we extended the problem by including the liabilities and the quantile hedging to obtain a reasonable problem construction for managing the liquidity risk. In this problem construction the objective of the investor was assumed to be the maximization of the probability of liquid assets minus liabilities bigger than a threshold level, which is a type of quantile hedging. Since the quantile hedging is not a perfect hedge, a non-zero probability of having a liability value higher than the asset value exists. To control the amount of the probable deficient amount we used a CVaR constraint. In the Black-Scholes framework, the solution of this problem necessitates to deal with the sum of the log-normal distributions. It is known that sum of the log-normal distributions has no closed-form representation. We introduced a new, simple and highly efficient method to approximate the sum of the log-normal distributions using shifted log-normal distributions. The method is based on a limiting approximation of the arithmetic mean by the geometric mean. Using our new approximation method we reduced the quantile hedging problem to a simpler optimization problem. Our new log-normal sum approximation method could also be used to price some options in the Black-Scholes model. With the help of our approximation method we derived closed-form approximation formulas for the prices of the basket and Asian options based on the arithmetic averages. Using our approximation methodology combined with the new analytical pricing formulas for the arithmetic average options, we obtained a very efficient performance for Monte Carlo pricing in a control variate setting. Our numerical results show that our control variate method outperforms the well-known methods from the literature in some cases.