20 research outputs found

    Universe in a glass of iced-water. Exploration in off-the-wall physics

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    Various exploration in astrophysics has revealed many breakthroughs nowadays, not only with respect to James Webb Telescope, but also recent finding related to water and ice deposits in the Moon surface. Those new findings seem to bring us to new questions related to origin of Earth, Moon and the entire Universe

    Investigation, Development, and Evaluation of Performance Proving for Fault-tolerant Computers

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    A number of methodologies for verifying systems and computer based tools that assist users in verifying their systems were developed. These tools were applied to verify in part the SIFT ultrareliable aircraft computer. Topics covered included: STP theorem prover; design verification of SIFT; high level language code verification; assembly language level verification; numerical algorithm verification; verification of flight control programs; and verification of hardware logic

    Using Diagrammatic Reasoning for Theorem Proving in a Continuous Domain

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    Centre for Intelligent Systems and their ApplicationsThis project looks at using diagrammatic reasoning to prove mathematical theorems. The work is motivated by a need for theorem provers whose reasoning is readily intelligible to human beings. It should also have practical applications in mathematics teaching. We focus on the continuous domain of analysis - a geometric subject, but one which is taught using a dry algebraic formalism which many students find hard. The geometric nature of the domain makes it suitable for a diagram-based approach. However it is a difficult domain, and there are several problems, including handling alternating quantifiers, sequences and generalisation. We developed representations and reasoning methods to solve these. Our diagram logic isn't complete, but does cover a reasonable range of theorems. It utilises computers to extend diagrammatic reasoning in new directions – including using animation. This work is tested for soundness, and evaluated empirically for ease of use. We demonstrate that computerised diagrammatic theorem proving is not only possible in the domain of real analysis, but that students perform better using it than with an equivalent algebraic computer system

    A Unifying Field in Logics: Neutrosophic Logic.

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    The author makes an introduction to non-standard analysis, then extends the dialectics to “neutrosophy” – which became a new branch of philosophy. This new concept helps in generalizing the intuitionistic, paraconsistent, dialetheism, fuzzy logic to “neutrosophic logic” – which is the first logic that comprises paradoxes and distinguishes between relative and absolute truth. Similarly, the fuzzy set is generalized to “neutrosophic set”. Also, the classical and imprecise probabilities are generalized to “neutrosophic probability”

    Mathematical Explanation and Ontology: An Analysis of Applied Mathematics and Mathematical Proofs

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    The present work aims at providing an account of mathematical explanation in two different areas: scientific explanation and within mathematics. The research is addressed from two different perspectives: the one arising from an ontological concern about mathematical entities, and the other originating from a methodological choice: to study our chosen problems (mathematical explanation in science and in mathematics itself) in mathematical practice, that is to say, looking at the way mathematicians understand and perform their work in these diverse areas, including a case study for the context of intra-mathematical explanation. The central target is the analysis of the role that mathematical explanation plays in science and its relevance to the success or failure of scientific theories. The ontological question of whether the explanatory role of abstract objects, mathematical objects in particular, is enough to postulate their existence will be one of the issues to be addressed. Moreover, the possibility of a unified theory of explanation which can accommodate both external and internal mathematical explanation will also be considered. In order to go deeper into these issues, the research includes: (1) an analysis how the question of what is involved in internal mathematical explanation has been addressed in the literature, an analysis of the role of mathematical proof and the reasons why it makes sense to search for more explanatory proofs of already known results, and (2) an analysis of the relation between the use of mathematics in scientific explanation and the ontological commitment that arises from these explanatory tools in science. Part of the present work consists of an analysis of the explanatory role of mathematics through the study of cases reflecting this role. Case studies is one of the main sources of data in order to clarify the role mathematical entities play, among other methodological resources

    A UNIFYING FIELD IN LOGICS: NEUTROSOPHIC LOGIC. NEUTROSOPHY, NEUTROSOPHIC SET, NEUTROSOPHIC PROBABILITY AND STATISTICS

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    In 1960s Abraham Robinson has developed the non-standard analysis, a formalization of analysis and a branch of mathematical logic, which rigorously defines the infinitesimals

    A UNIFYING FIELD IN LOGICS: NEUTROSOPHIC LOGIC. NEUTROSOPHY, NEUTROSOPHIC SET, NEUTROSOPHIC PROBABILITY AND STATISTICS - 6th ed.

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    It was a surprise for me when in 1995 I received a manuscript from the mathematician, experimental writer and innovative painter Florentin Smarandache, especially because the treated subject was of philosophy - revealing paradoxes - and logics. He had generalized the fuzzy logic, and introduced two new concepts: a) “neutrosophy” – study of neutralities as an extension of dialectics; b) and its derivative “neutrosophic”, such as “neutrosophic logic”, “neutrosophic set”, “neutrosophic probability”, and “neutrosophic statistics” and thus opening new ways of research in four fields: philosophy, logics, set theory, and probability/statistics. It was known to me his setting up in 1980’s of a new literary and artistic avant-garde movement that he called “paradoxism”, because I received some books and papers dealing with it in order to review them for the German journal “Zentralblatt fur Mathematik”. It was an inspired connection he made between literature/arts and science, philosophy. We started a long correspondence with questions and answers. Because paradoxism supposes multiple value sentences and procedures in creation, antisense and non-sense, paradoxes and contradictions, and it’s tight with neutrosophic logic, I would like to make a small presentation

    Expert Conceptualizations of the Convergence of Taylor Series Yesterday, Today, and Tomorrow

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    Taylor series is a topic briefly covered in most university calculus sequences. In many cases it constitutes only one or two sections of a calculus textbook. With this limited exposure, what do calculus students really understand about the convergence of Taylor series? Do they think of Taylor series convergence as a sequence of converging polynomials? Do they think of convergence as a remainder going to zero? Do they think the Taylor series for sine really "equals" sine, or is it merely a good estimation for sine? Furthermore, how might experts respond to these questions?This study reported qualitative research methods which utilized multiple phases of data collection consisting of questionnaires and interviews from expert and novice (undergraduate student) participant groups. In addition, this study utilized multiple layers of analysis incorporating methods such as Strauss and Corbin's open coding and Sfard's discourse analysis. Using Tall and Vinner's notion of concept images, I analyzed and described the different ways in which both experts and novices conceptualized the convergence of Taylor series. In so doing, commonalities and differences amongst the expert and novice participant groups emerged.The main result from this study was found in the descriptions of thirteen different concept images that experts and novices employed concerning the convergence of Taylor series. Some of these images were used more than others by the different participant groups, and some images appeared to date back to the early years of calculus. Even though both groups employed a variety of images, on an individual level, experts were more prone to use a wider range of images that they efficiently and effectively employed as different situations prompted. The most notable difference between experts and novices was found in their graphical images of Taylor series convergence. Experts demonstrated little to no difficulties interpreting graphs of Taylor series, but the vast majority of novices were unable to correctly produce graphs related to Taylor series convergence. In several cases, novices appeared to be incorrectly applying previous knowledge of graphical properties of translating functions in an attempt to build their conceptions of Taylor polynomial graphs. This finding has implications for future research into the effects of graphical images, both dynamic and static, on student conceptions of the convergence of Taylor series

    A combination of nonstandard analysis and geometry theorem proving, with application to Newton's Principia

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