13 research outputs found

    A Proof of a Generalization of Niven\u27s Theorem Using Algebraic Number Theory

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    Niven’s theorem states that the sine, cosine, and tangent functions are rational for only a few rational multiples of π. Specifically, for angles θ that are rational multiples of π, the only rational values of sin(θ) and cos(θ) are 0, ±½, and ±1. For tangent, the only rational values are 0 and ±1. We present a proof of this fact, along with a generalization, using the structure of ideals in imaginary quadratic rings. We first show that the theorem holds for the tangent function using elementary properties of Gaussian integers, before extending the approach to other imaginary quadratic rings. We then show for which rational multiples of π the squares of the sine, cosine, and tangent functions are rational, providing a generalized form of Niven’s theorem. We end with a discussion of a few related combinatorial identities

    Selected introductory concepts from combinatorial mathematics, 1967

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    This paper is concerned with the development of that part of combinatorial mathematics that deals with existence-type problems. This development is accomplished through the framework of modern algebra. Beginning with such elementary tonics as sets, permutations, and combinations the paper goes on to the principle of Inclusion and exclusion, recurrence relations, the elegant Theorem of Ramsey, and an introduction to systems of distinct representatives. In addition to the treatment of combinatorial mathematics as a mathematical system in itself, a few of the multitudinous applications of this theory are presented. These include applications and relationships to the theory of numbers, matrices, group and field theory, and conbiaatorlal-type problems which occur in every day life

    Time granularity in simulation models within a multi-agent system

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    The understanding of how processes in natural phenomena interact at different scales of time has been a great challenge for humans. How information is transferred across scales is fundamental if one tries to scale up from finer to coarse levels of granularity. Computer simulation has been a powerful tool to determine the appropriate amount of detail one has to impose when developing simulation models of such phenomena. However, it has proved difficult to represent change at many scales of time and subject to cyclical processes. This issue has received little attention in traditional AI work on temporal reasoning but it becomes important in more complex domains, such as ecological modelling. Traditionally, models of ecosystems have been developed using imperative languages. Very few of those temporal logic theories have been used for the specification of simulation models in ecology. The aggregation of processes working at different scales of time is difficult (sometimes impossible) to do reliably. The reason is because these processes influence each other, and their functionality does not always scale to other levels. Thus the problems to tackle are representing cyclical and interacting processes at many scales and providing a framework to make the integration of such processes more reliable. We propose a framework for temporal modelling which allows modellers to represent cyclical and interacting processes at many scales. This theory combines both aspects by means of modular temporal classes and an underlying special temporal unification algorithm. To allow integration of different models they are developed as agents with a degree of autonomy in a multi-agent system architecture. This Ecoagency framework is evaluated on ecological modelling problems and it is compared to a formal language for describing ecological systems

    AN INVESTIGATION ABOUT HIGH SCHOOL MATHEMATICS TEACHERS' BELIEFS ABOUT TEACHING GEOMETRY

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    There continues to exist a dilemma about how, why and when geometry should be taught. The aim of this study was to examine high school mathematics teachers' beliefs about geometry and its teaching with respect to its role in the curriculum, the uses of manipulatives and dynamic geometry software in the classroom, and the role of proofs. In this study belief is taken as subjective knowledge (Furinghetti and Pehkonen, 2002). Data were collected from 520 teachers using questionnaires that included both statements that required responses on a Likert scale and open-ended questions. Also an intervention case study was conducted with one teacher. A three factor solution emerged from the analysis that revealed a disposition towards activities, a disposition towards an appreciation of geometry and its applications and a disposition towards abstraction. These results enabled classification of teachers into one of eight groups depending on whether their scores were positive or negative on the three factors. Knowing the teacher typology allows for appropriate professional development activities to be undertaken. This was done in the case study where techniques for scaffolding proofs were used as an intervention for a teacher who had a positive disposition towards activities and appreciation of geometry and its applications but a negative disposition towards abstraction. The intervention enabled the teacher successfully to teach her students how to understand and construct proofs. The open-ended responses on the questionnaire were analysed to obtain a better understanding of the teachers' beliefs. Four themes, the formal, intuitive, utilitarian and the mathematical, emerged from the analysis, which support the modal arguments given by Gonzalez and Herbst (2006). The findings reveal a disconnect between some high school teachers' beliefs about why geometry is important to study and the current position of the Standards Movement; and between whether geometry should be taught as part of an integrated curriculum or as a one-year course

    Elastodynamics of Failure in a Continuum

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    A general treatment of the elastodynamics of failure in a prestressed elastic continuum is given, with particular emphasis on the geophysical aspects of the problem. The principal purpose of the study is to provide a physical model of the earthquake phenomenon, which yields an explicit description of the radiation field in terms of source parameters. The Green's tensor solution to the equations of motion in a medium with moving boundaries is developed. Using this representation theorem, and its specialization to the scalar case by means of potentials, it is shown that material failure in a continuum can be treated equivalently as a boundary value problem or as an initial value problem. The initial value representation is shown to be preferable for geophysical purposes, and the general solution for a growing and propagating rupture zone is given. The energy balance of the phenomenon is discussed with particular emphasis on the physical source of the radiated energy. It is also argued that the flow of energy is the controlling factor for the propagation and growth of a failure zone. Failure should then be viewed as a generalized phase change of the medium. The theory is applied to the simple case of a growing and propagating spherical failure zone. The model is investigated in detail both analytically and numerically. The analysis is performed in the frequency domain and the radiation fields are given in the form of multipolar expansions. The necessary theorems for the manipulation of such expansions for seismological purposes are proved, and their use discussed on the basis of simple examples. The more realistic ellipsoidal failure zone is investigated. The static problem of an arbitrary ellipsoidal inclusion under homogeneous stress of arbitrary orientation is solved. It is then shown how the analytical solution can be combined with numerical techniques to yield more realistic models. The conclusion is that this general approach yields a very flexible model which can be adapted to a wide variety of physical circumstances. In spite of the simplicity of the model, the predicted radiation field is rather complex; it is discussed as a function of source parameters, and scaling laws are derived which ease the interpretation of observed spectra. Preliminary results in the time domain are also shown. It is concluded that the model can be compared favorably both with the observations, and with results obtained from purely numerical models

    Understanding Inquiry, an Inquiry into Understanding: a conception of Inquiry Based Learning in Mathematics

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    IBL (Inquiry Based Learning) is a group of educational approaches centered on the student and aiming at developing higher-level thinking, as well as an adequate set of Knowledge, Skills, and Attitudes (KSA). IBL is at the center of recent educational research and practice, and is expanding quickly outside of schools: in this research we propose such forms of instruction as Guided Self-Study, Guided Problem Solving, Inquiry Based Homeschooling, IB e-learning, and particularly a mixed (Inquiry-Expository) form of lecturing, named IBLecturing. The research comprises a thorough review of previous research in IBL; it clarifies what is and what is not Inquiry Based Learning, and the distinctions between its various forms: Inquiry Learning, Discovery Learning, Case Study, Problem Based Learning, Project Based Learning, Experiential Learning, etc. There is a continuum between Pure Inquiry and Pure Expository approaches, and the extreme forms are very infrequently encountered. A new cognitive taxonomy adapted to the needs of higher-level thinking and its promotion in the study of mathematics is also presented. This research comprises an illustration of the modeling by an expert (teacher, trainer, etc.) of the heuristics and of the cognitive and metacognitive strategies employed by mathematicians for solving problems and building proofs. A challenging problem has been administered to a group of gifted students from secondary school, in order to get more information about the possibility of implementing Guided Problem Solving. Various opportunities for further research are indicated, for example applying the recent advances of cognitive psychology on the role of Working Memory (WM) in higher-level thinking

    The Machinery of Freedom: Guide to a Radical Capitalism

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    This book argues the case for a society organized by private property, individual rights, and voluntary co-operation, with little or no government. David Friedman\u27s standpoint, known as \u27anarcho-capitalism\u27, has attracted a growing following as a desirable social ideal since the first edition of The Machinery of Freedom appeared in 1971. This new edition is thoroughly revised and includes much new material, exploring fresh applications of the author\u27s libertarian principles. Among topics covered: how the U.S. would benefit from unrestricted immigration; why prohibition of drugs is inconsistent with a free society; why the welfare state mainly takes from the poor to help the not-so-poor; how police protection, law courts, and new laws could all be provided privately; what life was really like under the anarchist legal system of medieval Iceland; why non-intervention is the best foreign policy; why no simple moral rules can generate acceptable social policies -- and why these policies must be derived in part from the new discipline of economic analysis of law.https://digitalcommons.law.scu.edu/monographs/1006/thumbnail.jp

    Bowdoin Alumnus Volume 32 (1957-1958)

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    https://digitalcommons.bowdoin.edu/alumni-magazines/1030/thumbnail.jp

    Bowdoin Orient v.88, no.1-21 (1958-1959)

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    https://digitalcommons.bowdoin.edu/bowdoinorient-1950s/1009/thumbnail.jp

    Bowdoin Orient v.96, no.1-24 (1966-1967)

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    https://digitalcommons.bowdoin.edu/bowdoinorient-1960s/1007/thumbnail.jp
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