On [summation of 1 raised to the power 1] equivalence relations over the natural numbers

"Vegeu el resum a l'inici del document del fitxer adjunt"

Table of the logarithms of the natural numbers from 1 to 108000 /

Mode of access: Internet.

Numbers and Languages

The thesis presents results obtained during the authors PhD-studies. First systems of language equations of a simple form consisting of just two equations are proved to be computationally universal. These are systems over unary alphabet, that are seen as systems of equations over natural numbers. The systems contain only an equation X+A=B and an equation X+X+C=X+X+D, where A, B, C and D are eventually periodic constants. It is proved that for every recursive set S there exists natural numbers p and d, and eventually periodic sets A, B, C and D such that a number n is in S if and only if np+d is in the unique solution of the abovementioned system of two equations, so all recursive sets can be represented in an encoded form. It is also proved that all recursive sets cannot be represented as they are, so the encoding is really needed. Furthermore, it is proved that the family of languages generated by Boolean grammars is closed under injective gsm-mappings and inverse gsm-mappings. The arguments apply also for the families of unambiguous Boolean languages, conjunctive languages and unambiguous languages. Finally, characterizations for morphisims preserving subfamilies of context-free languages are presented. It is shown that the families of deterministic and LL context-free languages are closed under codes if and only if they are of bounded deciphering delay. These families are also closed under non-codes, if they map every letter into a submonoid generated by a single word. The family of unambiguous context-free languages is closed under all codes and under the same non-codes as the families of deterministic and LL context-free languages.

A aprendizagem da divisão: um olhar sobre os procedimentos usados pelos alunos

Este artigo apresenta e discute alguns aspetos sobre a aprendizagem da divisão com números naturais, focando-se nos procedimentos usados por alunos de uma turma do 3.º ano na resolução de tarefas de divisão. Os resultados apresentados fazem parte de uma investigação mais abrangente que teve como finalidade a compreensão do modo como os alunos aprofundam a aprendizagem da multiplicação numa perspetiva de desenvolvimento do sentido do número. A investigação realizada seguiu uma metodologia de design research, na modalidade de experiência de ensino. A análise das produções escritas dos alunos e de episódios de sala de aula relativos às discussões coletivas sobre as resoluções das tarefas propostas mostra que os alunos usam uma diversidade de procedimentos e que estes evoluem significativamente ao longo da experiência de ensino. Esta evolução parece ser suportada pelas características das tarefas, os seus contextos e números, assim como pela articulação, desde logo estabelecida, entre a divisão e a multiplicação. Além disso, o recurso ao modelo retangular parece, também, ter contribuído para a progressão para procedimentos multiplicativos, baseados na decomposição de um dos fatores. Os resultados do estudo permitem ainda perceber que a evolução dos procedimentos usados pelos alunos e a sua diversidade não são alheias ao ambiente de sala de aula construído.

A multiplicação e a divisão de números naturais: uma análise de manuais escolares de 1.º ciclo

Dissertação apresentada à Escola Superior de Educação de Lisboa para obtenção de grau de mestre em Educação Matemática na Educação Pré-escolar e nos 1.º e 2.º ciclos do Ensino Básico

Desenvolvimento da fluência no cálculo mental através do uso de estratégias de cálculo: um estudo no 4.º e no 6.º ano de escolaridade

Relatório de estágio de mestrado em Ensino do 1º e 2º Ciclo do Ensino Básico

Vom Finanz- zum Wissenschaftsbetrug : Methode, den Irrungen in der medizinischen Literatur beizukommen [From financial to scientific fraud : Methods to detect discrepancies in the medical literature].

Fraud is as old as Mankind. There are an enormous number of historical documents which show the interaction between truth and untruth; therefore it is not really surprising that the prevalence of publication discrepancies is increasing. More surprising is that new cases especially in the medical field generate such a huge astonishment. In financial mathematics a statistical tool for detection of fraud is known which uses the knowledge of Newcomb and Benford regarding the distribution of natural numbers. This distribution is not equal and lower numbers are more likely to be detected compared to higher ones. In this investigation all numbers contained in the blinded abstracts of the 2009 annual meeting of the Swiss Society of Anesthesia and Resuscitation (SGAR) were recorded and analyzed regarding the distribution. A manipulated abstract was also included in the investigation. The χ(2)-test was used to determine statistical differences between expected and observed counts of numbers. There was also a faked abstract integrated in the investigation. A p<0.05 was considered significant. The distribution of the 1,800 numbers in the 77 submitted abstracts followed Benford's law. The manipulated abstract was detected by statistical means (difference in expected versus observed p<0.05). Statistics cannot prove whether the content is true or not but can give some serious hints to look into the details in such conspicuous material. These are the first results of a test for the distribution of numbers presented in medical research.

A non reductionist logicism with explicit definitions

This paper introduces and examines the logicist construction of Peano Arithmetic that can be performed into Leśniewski’s logical calculus of names called Ontology. Against neo-Fregeans, it is argued that a logicist program cannot be based on implicit definitions of the mathematical concepts. Using only explicit definitions, the construction to be presented here constitutes a real reduction of arithmetic to Leśniewski’s logic with the addition of an axiom of infinity. I argue however that such a program is not reductionist, for it only provides what I will call a picture of arithmetic, that is to say a specific interpretation of arithmetic in which purely logical entities play the role of natural numbers. The reduction does not show that arithmetic is simply a part of logic. The process is not of ontological significance, for numbers are not shown to be logical entities. This neo-logicist program nevertheless shows the existence of a purely analytical route to the knowledge of arithmetical laws.

The problem of the graphic artist

'The problem of the graphic artist' is a small example of applying elementary mathematics (divisibility of natural numbers) to a real problem which we ourselves have actually experienced. It deals with the possibilities for partitioning a sheet of paper into strips. In this contribution we report on a teaching unit in grade 6 as well as on informal tests with students in school and university. Finally we analyse this example methodologically, summarise our observations with pupils and students, and draw some didactical conclusions.

Uma abordagem no ensino de frações baseadas em atividades para o 6º ano do ensino fundamantal

The objective of this work if constitutes in creation a proposal for activities, in the discipline of mathematics, for the 6th year of Elementary School, that stimulates the students the develop the learning of the content of fractions, from the awareness of the insufficiency of the natural numbers for solve several problems. Thus, we prepared a set with twelve activities, starting by the comparison between measures, presenting afterward some of the meanings of fractions and ending with the operations between fractions. For so much, use has been made of materials available for use in the classroom, of forma ludic, for resolution of challenges proposed. Through these activities, it becomes possible students to recognize the necessity of using fractions for solve a amount larger of problems

Aplicação dos registros de representação semiótica no ensino-aprendizagem da matemática: um estudo com alunos do sexto ano do ednsino fundamental

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

O jogo de escopa adaptado para o uso em sala de aula

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

A note on the theory SID<ω of stratified induction

We introduce and analyse a theory of finitely stratified general inductive definitions over the natural numbers, inline image, and establish its proof theoretic ordinal, inline image. The definition of inline image bears some similarities with Leivant's ramified theories for finitary inductive definitions.

El significado aritmológico de la Tríada y sus Proyecciones Filosófico-Religiosas

Bertrand Russell dio una solución lógica general a la definición de los números caracterizando al número tres como la clase que es correspondiente a todas las clases biunívocas con los grupos de tres. Para Frege la definición del número era una de las cruces de la comprensión de la aritmética. Hegel, por su parte, bajo el impulso de la triplicidad kantiana de los juicios sintéticos, teoréticos y prácticos y la influencia de la concepción trinitaria cristiana advertía que la contraposición de los opuestos al no ser contradictoria permitía el desarrollo deviniente. La aritmología pitagórica sobre los números naturales destacó la definición del tres caracterizada por su naturaleza de mediedad. Retomando estas bases de pura inteligibilidad y las especulaciones gnósticas sobre la tríada mostraremos sus proyecciones filosófico-religiosas.

Genetic Algorithm for Finding the Key’s Length and Cryptanalysis of the Permutation Cipher

In this article we discuss a possibility to use genetic algorithms in cryptanalysis. We developed and described the genetic algorithm for finding the secret key of a block permutation cipher. In this case key is a permutation of some first natural numbers. Our algorithm finds the exact key’s length and the key with controlled accuracy. Evaluation of conducted experiment’s results shows that the almost automatic cryptanalysis is possible.