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  1. Dwight R. Bean (1976). Effective Coloration. Journal of Symbolic Logic 41 (2):469-480.
    We are concerned here with recursive function theory analogs of certain problems in chromatic graph theory. The motivating question for our work is: Does there exist a recursive (countably infinite) planar graph with no recursive 4-coloring? We obtain the following results: There is a 3-colorable, recursive planar graph which, for all k, has no recursive k-coloring; every decidable graph of genus p ≥ 0 has a recursive 2(χ(p) - 1)-coloring, where χ(p) is the least number of colors which will suffice (...)
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  2. Vieri Benci, Leon Horsten & Sylvia Wenmackers (2013). Non-Archimedean Probability. Milan Journal of Mathematics 81 (1):121-151.
    We propose an alternative approach to probability theory closely related to the framework of numerosity theory: non-Archimedean probability (NAP). In our approach, unlike in classical probability theory, all subsets of an infinite sample space are measurable and only the empty set gets assigned probability zero (in other words: the probability functions are regular). We use a non-Archimedean field as the range of the probability function. As a result, the property of countable additivity in Kolmogorov’s axiomatization of probability is replaced by (...)
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  3. Arthur Walter Burks & Jesse B. Wright, Sequence Generators, Graphs, and Formal Languages.
    A sequence generator is a finite graph, more general than, but akin to, the usual state diagram associated with a finite automaton. The nodes of a sequence generator represent complete states, and each node is labeled with an input and an output state. An element of the behavior of a sequence generator is obtained by taking the input and output states along an infinite path of the graph.Sequence generators may be associated with formulas of the monadic predicate calculus, in which (...)
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  4. Kathleen M. Clark (2014). History of Mathematics in Mathematics Teacher Education. In Michael R. Matthews (ed.), International Handbook of Research in History, Philosophy and Science Teaching. Springer. 755-791.
    The purpose of this chapter is to provide a broad view of the state of the field of history of mathematics in education, with an emphasis on mathematics teacher education. First, an overview of arguments that advocate for the use of history in mathematics education and descriptions of the role that history of mathematics has played in mathematics teacher education in the United States and elsewhere is given. Next, the chapter details several examples of empirical studies that were conducted with (...)
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  5. Elio Conte (2012). On a Simple Derivation of the Effect of Repeated Measurements on Quantum Unstable Systems by Using the Regularized Incomplete Beta-Function. Advanced Studies in Theoretical Physics 6 (25):1207-1213.
    a simple derivation of the effect induced from repeated measurements on quantum unstable systems is obtained by using the regularized incomplete beta - function .
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  6. Stanislas Dehaene, Véronique Izard, Pierre Pica & Elizabeth Spelke (2009). Response to Comment on "Log or Linear? Distinct Intuitions on the Number Scale in Western and Amazonian Indigene Cultures". Science 323 (5910):38.
    The performance of the Mundurucu on the number-space task may exemplify a general competence for drawing analogies between space and other linear dimensions, but Mundurucu participants spontaneously chose number when other dimensions were available. Response placement may not reflect the subjective scale for numbers, but Cantlon et al.'s proposal of a linear scale with scalar variability requires additional hypotheses that are problematic.
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  7. Orkia Derkaoui (2014). Safe Bounds in Semidefinite Programming by Using Interval Arithmetic. American Journal of Operations Research 4:293-300.
    Efficient solvers for optimization problems are based on linear and semidefinite relaxations that use floating point arithmetic. However, due to the rounding errors, relaxation thus may overestimate, or worst, underestimate the very global optima. The purpose of this article is to introduce an efficient and safe procedure to rigorously bound the global optima of semidefinite program. This work shows how, using interval arithmetic, rigorous error bounds for the optimal value can be computed by carefully post processing the output of a (...)
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  8. Anatolij Dvurečenskij & Jiří Janda (2013). On Bilinear Forms From the Point of View of Generalized Effect Algebras. Foundations of Physics 43 (9):1136-1152.
    We study positive bilinear forms on a Hilbert space which are not necessarily bounded nor induced by some positive operator. We show when different families of bilinear forms can be described as a generalized effect algebra. In addition, we present families which are or are not monotone downwards (Dedekind upwards) σ-complete generalized effect algebras.
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  9. Gérard G. Emch (2002). Mathematical Topics Between Classical and Quantum Mechanics. Studies in History and Philosophy of Science Part B 33 (1):148-150.
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  10. Eva-Maria Engelen, Christian Fleischhack, C. Giovanni Galizia & Katharina Landfester (eds.) (2010). Heureka: Evidenzkriterien in den Wissenschaften. Ein Kompendium für den interdisziplinären Gerauch. Spektrum Springer.
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  11. William Feller (1968). An Introduction to Probability Theory and its Applications Vol. I. John Wiley & Sons.
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  12. James Franklin (2006). Review of N. Wildberger, Divine Proportions: Rational Trigonometry to Universal. [REVIEW] Mathematical Intelligencer 28 (3):73-74.
    Reviews Wildberger's account of his rational trigonometry project, which argues for a simpler way of doing trigonometry that avoids irrationals.
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  13. James Franklin (1996). Proof in Mathematics: An Introduction. Quakers Hill Press.
    Why do students take the instruction "prove" in examinations to mean "go to the next question"? Because they have not been shown the simple techniques of how to do it. Mathematicians meanwhile generate a mystique of proof, as if it requires an inborn and unteachable genius. True, creating research-level proofs does require talent; but reading and understanding the proof that the square of an even number is even is within the capacity of most mortals.
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  14. James Franklin (1988). Homomorphisms Between Verma Modules in Characteristic P. Journal of Algebra 112:58-85.
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  15. Michael N. Fried (2014). History of Mathematics in Mathematics Education. In Michael R. Matthews (ed.), International Handbook of Research in History, Philosophy and Science Teaching. Springer. 669-703.
    This paper surveys central justifications and approaches adopted by educators interested in incorporating history of mathematics into mathematics teaching and learning. This interest itself has historical roots and different historical manifestations; these roots are examined as well in the paper. The paper also asks what it means for history of mathematics to be treated as genuine historical knowledge rather than a tool for teaching other kinds of mathematical knowledge. If, however, history of mathematics is not subordinated to the ideas and (...)
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  16. Eduard Glas (2014). A Role for Quasi-Empiricism in Mathematics Education. In Michael R. Matthews (ed.), International Handbook of Research in History, Philosophy and Science Teaching. Springer. 731-753.
    Although there are quite a few directions in modern philosophy of mathematics that invoke some essential role for (quasi-)empirical material, this chapter will be devoted exclusively to what may be considered the seminal tradition. This enabled me to present the subject as one coherent whole and to forestall the discussion getting scattered in a diversity of directions without doing justice to any one of them. -/- Quasi-empiricism in this tradition is the view that the logic of mathematical inquiry is based, (...)
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  17. K. Gödel (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik 38 (1):173--198.
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  18. Judith V. Grabiner (2014). The Role of Mathematics in Liberal Arts Education. In Michael R. Matthews (ed.), International Handbook of Research in History, Philosophy and Science Teaching. Springer. 793-836.
    The history of the continuous inclusion of mathematics in liberal education in the West, from ancient times through the modern period, is sketched in the first two sections of this chapter. Next, the heart of this essay (Sects. 3, 4, 5, 6, and 7) delineates the central role mathematics has played throughout the history of Western civilization: not just a tool for science and technology, mathematics continually illuminates, interacts with, and sometimes challenges fields like art, music, literature, and philosophy – (...)
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  19. S. Heikkilä, A Fixed Point Theorem and its Use in Kripke's Theory of Truth.
    Chain iteration methods are used to prove a fixed point theorem for set mappings. As an application, a general rule is constructed for finding in Kripke's theory of truth, based on Strong Kleene evaluation scheme for treating sentences lacking a truth-value, language interpretations containing their truth predicates.
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  20. Harold T. Hodes (1988). Book Review. The Lambda-Calculus. H. P. Barendregt(. [REVIEW] Philosophical Review 97 (1):132-7.
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  21. Harold T. Hodes (1982). Jumping to a Uniform Upper Bound. Proceedings of the American Mathematical Society 85 (4):600-602.
    A uniform upper bound on a class of Turing degrees is the Turing degree of a function which parametrizes the collection of all functions whose degree is in the given class. I prove that if a is a uniform upper bound on an ideal of degrees then a is the jump of a degree c with this additional property: there is a uniform bound b<a so that b V c < a.
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  22. Véronique Izard, Pierre Pica, Elizabeth S. Spelke & Stanislas Dehaene (2008). Exact Equality and Successor Function: Two Key Concepts on the Path Towards Understanding Exact Numbers. Philosophical Psychology 21 (4):491 – 505.
    Humans possess two nonverbal systems capable of representing numbers, both limited in their representational power: the first one represents numbers in an approximate fashion, and the second one conveys information about small numbers only. Conception of exact large numbers has therefore been thought to arise from the manipulation of exact numerical symbols. Here, we focus on two fundamental properties of the exact numbers as prerequisites to the concept of EXACT NUMBERS : the fact that all numbers can be generated by (...)
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  23. Uffe Thomas Jankvist (2014). On the Use of Primary Sources in the Teaching and Learning of Mathematics. In Michael R. Matthews (ed.), International Handbook of Research in History, Philosophy and Science Teaching. Springer. 873-908.
    In this chapter, an attempt is made to illustrate why the study of primary original sources is, as often stated, rewarding and worth the effort, despite being extremely demanding for both teachers and students. This is done by discussing various reasons for as well as different approaches to using primary original sources in the teaching and learning of mathematics. A selection of these reasons and approaches will be illustrated through a number of examples from the literature on using original sources (...)
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  24. Tinne Hoff Kjeldsen (2014). The Role of History and Philosophy in University Mathematics Education. In Michael R. Matthews (ed.), International Handbook of Research in History, Philosophy and Science Teaching. Springer. 837-871.
    University level mathematics is organised differently in different universities. In this paper we consider mathematics programmes leading to a graduate degree in mathematics. We briefly introduce a multiple perspective approach to the history of mathematics from its practices, reflections about uses of history and the research direction in philosophy of mathematics denoted ‘Philosophy of Mathematical Practice’. We link history and philosophy of mathematical practices to recent ideas in mathematics education in order to identify different roles history and philosophy can play (...)
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  25. V. G. Makhanʹkov (1989). Soliton Phenomenology. Kluwer Academic.
    Solitons, i.e. solitary localized waves with particle-like behaviour, and multi-solitons occur virtually everywhere. There is a good reason for that in that there is a solid, albeit somewhat heuristic argument which says that for wave-like phenomena the 'soliton approximation' is the next one after the linear one. It is also not too difficult via some searching in the voluminous literature - many hundreds of papers on solitons each year - to write down a long list of equations which admit soliton (...)
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  26. Richard Montague (1961). Fraenkel's Addition to the Axioms of Zermelo. In Bar-Hillel, Yehoshua & [From Old Catalog] (eds.), Journal of Symbolic Logic. Jerusalem, Magnes Press, Hebrew University;. 662-662.
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  27. Catherine Rowett (2013). Philosophy's Numerical Turn: Why the Pythagoreans' Interest in Numbers is Truly Awesome. In Dirk Obbink & David Sider (eds.), Doctrine and Doxography: Studies on Heraclitus and Pythagoras. De Gruyter. 3-32.
    Philosophers are generally somewhat wary of the hints of number mysticism in the reports about the beliefs and doctrines of the so-called Pythagoreans. It's not clear how much Pythagoras himself (as opposed to his later followers) indulged in speculation about numbers, or in more serious mathematics. But the Pythagoreans whom Aristotle discusses in the Metaphysics had some elaborate stories to tell about how the universe could be explained in terms of numbers—not just its physics but perhaps morality too. Was this (...)
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  28. Stuart Rowlands (2014). Philosophy and the Secondary School Mathematics Classroom. In Michael R. Matthews (ed.), International Handbook of Research in History, Philosophy and Science Teaching. Springer. 705-730.
    Although much has been written about the philosophy for children programme in the academic literature (and the press), there is very little on philosophy and the mathematics classroom, and the little there is has tended to treat mathematics within the context of this programme. By contrast, however, this chapter proposes a radically different perspective whereby the mathematics teacher is the dominant authority and directs the discourse from the front of the class. The aim of this perspective is to move the (...)
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  29. Yaroslav Sergeyev (2010). Counting Systems and the First Hilbert Problem. Nonlinear Analysis Series A 72 (3-4):1701-1708.
    The First Hilbert problem is studied in this paper by applying two instruments: a new methodology distinguishing between mathematical objects and mathematical languages used to describe these objects; and a new numeral system allowing one to express different infinite numbers and to use these numbers for measuring infinite sets. Several counting systems are taken into consideration. It is emphasized in the paper that different mathematical languages can describe mathematical objects (in particular, sets and the number of their elements) with different (...)
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  30. Yaroslav Sergeyev (2010). Lagrange Lecture: Methodology of Numerical Computations with Infinities and Infinitesimals. Rendiconti Del Seminario Matematico dell'Università E Del Politecnico di Torino 68 (2):95–113.
    A recently developed computational methodology for executing numerical calculations with infinities and infinitesimals is described in this paper. The approach developed has a pronounced applied character and is based on the principle “The part is less than the whole” introduced by the ancient Greeks. This principle is applied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The point of view on infinities and infinitesimals (and in general, on Mathematics) presented in this paper (...)
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  31. Yaroslav Sergeyev (2009). Numerical Computations and Mathematical Modelling with Infinite and Infinitesimal Numbers. Journal of Applied Mathematics and Computing 29:177-195.
    Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not related to the non-standard analysis) is used to work with finite, infinite, and infinitesimal numbers numerically. This can be done on a new kind of a computer – the Infinity Computer – able to work with all these types of numbers. The new computational tools both give possibilities to (...)
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  32. Yaroslav Sergeyev (2009). Numerical Point of View on Calculus for Functions Assuming Finite, Infinite, and Infinitesimal Values Over Finite, Infinite, and Infinitesimal Domains. Nonlinear Analysis Series A 71 (12):e1688-e1707.
    The goal of this paper consists of developing a new (more physical and numerical in comparison with standard and non-standard analysis approaches) point of view on Calculus with functions assuming infinite and infinitesimal values. It uses recently introduced infinite and infinitesimal numbers being in accordance with the principle ‘The part is less than the whole’ observed in the physical world around us. These numbers have a strong practical advantage with respect to traditional approaches: they are representable at a new kind (...)
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  33. Yaroslav Sergeyev (2009). Evaluating the Exact Infinitesimal Values of Area of Sierpinski's Carpet and Volume of Menger's Sponge. Chaos, Solitons and Fractals 42: 3042–3046.
    Very often traditional approaches studying dynamics of self-similarity processes are not able to give their quantitative characteristics at infinity and, as a consequence, use limits to overcome this difficulty. For example, it is well know that the limit area of Sierpinski’s carpet and volume of Menger’s sponge are equal to zero. It is shown in this paper that recently introduced infinite and infinitesimal numbers allow us to use exact expressions instead of limits and to calculate exact infinitesimal values of areas (...)
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  34. Yaroslav D. Sergeyev (2008). A New Applied Approach for Executing Computations with Infinite and Infinitesimal Quantities. Informatica 19 (4):567-596.
    A new computational methodology for executing calculations with infinite and infinitesimal quantities is described in this paper. It is based on the principle ‘The part is less than the whole’ introduced by Ancient Greeks and applied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). It is shown that it becomes possible to write down finite, infinite, and infinitesimal numbers by a finite number of symbols as particular cases of a unique framework. The (...)
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  35. Mark Sharlow, Generalizing the Algebra of Physical Quantities.
    In this paper, I define and study an abstract algebraic structure, the dimensive algebra, which embodies the most general features of the algebra of dimensional physical quantities. I prove some elementary results about dimensive algebras and suggest some directions for future work.
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  36. Michael A. Sherbon, Nature's Information and Harmonic Proportion.
    The history of science is polarized by debates over Plato and Aristotle’s holism versus the atomism of Democritus and others. This includes the complementarity of continuous and discrete, one and the many, waves and particles, and analog or digital views of reality. The three-fold method of the Pythagorean paradigm of unity, duality, and harmony enables the calculation of fundamental physical constants required by the forces of nature in the formation of matter; thereby demonstrating Plato’s archetypal viewpoint.
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  37. Michael A. Sherbon (2014). Fundamental Nature of the Fine-Structure Constant. International Journal of Physical Research 2 (1):1-9.
    Arnold Sommerfeld introduced the fine-structure constant that determines the strength of the electromagnetic interaction. Following Sommerfeld, Wolfgang Pauli left several clues to calculating the fine-structure constant with his research on Johannes Kepler's view of nature and Pythagorean geometry. The Laplace limit of Kepler's equation in classical mechanics, the Bohr-Sommerfeld model of the hydrogen atom and Julian Schwinger's research enable a calculation of the electron magnetic moment anomaly. Considerations of fundamental lengths such as the charge radius of the proton and mass (...)
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  38. L. B. Sultanova (2012). Historical Dynamics of Implicit and Intuitive Elements of Mathematical Knowledge. Liberal Arts in Russia 1 (1):30--35.
    The article deals with historical dynamics of implicit and intuitive elements of mathematical knowledge. The author describes historical dynamics of implicit and intuitive elements and discloses a historical and evolutionary mechanism of building up mathematical knowledge. Each requirement to increase the level of theoretical rigor in mathematics is historically realized as a three-stage process. The first stage considers some general conditions of valid mathematical knowledge recognized by the mathematical community. The second one reveals the level of theoretical rigor increasing, while (...)
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  39. Kazuya Tsuboi (2014). Consideration on the Flow Velocity in the Experimental Analysis of the Flame Displacement Speed Using DNS Data of Turbulent Premixed Flames with Different Lewis Numbers. Open Journal of Fluid Dynamics 4 (3):278-287.
    The flame displacement speed is one of the major characteristics in turbulent premixed flames. The flame displacement speed is experimentally obtained from the displacement normal to the flame surface, while it is numerically evaluated by the transport equation of the flame surface. The flame displacement speeds obtained both experimentally and numerically cannot be compared directly because their definitions are different. In this study, two kinds of experimental flame displacement speeds—involving the mean inflow velocity and the local flow velocity—were simulated using (...)
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  40. Kazuya Tsuboi (2014). DNS Analysis on the Indirect Relationship Between the Local Burning Velocity and the Flame Displacement Speed of Turbulent Premixed Flames. Open Journal of Fluid Dynamics 4 (3):288-297.
    The local burning velocity and the flame displacement speed are the dominant properties in the mechanism of turbulent premixed combustion. The flame displacement speed and the local burning velocity have been investigated separately, because the flame displacement speed can be used for the discussion of flame-turbulence interactions and the local burning velocity can be used for the discussion of the inner structure of turbulent premixed flames. In this study, to establish the basis for the discussion on the effects of turbulence (...)
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  41. Kevin Williams & Lynnclaire Dennis, The Science of Life Discovered From Lynnclaire Dennis' Near-Death Experience. Afterlife.
    Elsevier, the world's leading provider of science and health information, published an academic/scientific textbook about a new mathematical discovery discovered in a near-death experience (NDE) that matches the dynamics of living and life-like (social) systems and has applications in general systems theory, universal systems modelling, human clinical molecular genetics modelling, medical informatics, astrobiology, education and other areas of study. This article is about Lynnclaire Dennis and how she brought back perhaps the greatest scientific discovery ever from a NDE. The Mereon (...)
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  42. Richard Zach, Matthias Baaz & Norbert Preining (2007). First-Order Gödel Logics. Annals of Pure and Applied Logic 147 (1):23-47.
    First-order Gödel logics are a family of finite- or infinite-valued logics where the sets of truth values V are closed subsets of [0,1] containing both 0 and 1. Different such sets V in general determine different Gödel logics GV (sets of those formulas which evaluate to 1 in every interpretation into V). It is shown that GV is axiomatizable iff V is finite, V is uncountable with 0 isolated in V, or every neighborhood of 0 in V is uncountable. Complete (...)
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