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Foundations Of Nonstandard Analysis  A Gentle Introduction to Nonstandard Extemsions
 In Nonstandard analysis (Edinburgh
"... this paper is to describe the essential features of the resulting frameworks without getting bogged down in technicalities of formal logic and without becoming dependent on an explicit construction of a specific field ..."
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this paper is to describe the essential features of the resulting frameworks without getting bogged down in technicalities of formal logic and without becoming dependent on an explicit construction of a specific field
Modelling of Complex Software Systems: a Reasoned Overview
"... This paper is devoted to the presentation of the key concepts on which a mathematical theory of complex (industrial) systems can be based. We especially show how this formal framework can capture the realness of modern information technologies. We also present some new modelling problems that are na ..."
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This paper is devoted to the presentation of the key concepts on which a mathematical theory of complex (industrial) systems can be based. We especially show how this formal framework can capture the realness of modern information technologies. We also present some new modelling problems that are naturally emerging in the specific context of complex software systems.
Towards a functional formalism for modelling complex industrial systems
 ComPlexUs, special Issue : Complex Systems  European Conference 2005
, 2006
"... This paper is dedicated to the memory of Imre Lakatos ..."
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This paper is dedicated to the memory of Imre Lakatos
A Rigorous Real Time Feynman Path Integral and Propagator
 J. Phys. A: Math. Gen
"... This work partially fulfills the author’s Ph.D. thesis requirements ..."
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This work partially fulfills the author’s Ph.D. thesis requirements
A DEFENCE OF MATHEMATICAL PLURALISM
, 2004
"... We approach the philosophy of mathematics via a discussion of the differences between classical mathematics and constructive mathematics, arguing that each is a valid activity within its own context. ..."
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We approach the philosophy of mathematics via a discussion of the differences between classical mathematics and constructive mathematics, arguing that each is a valid activity within its own context.
Conceptions of the Continuum
"... Abstract: A number of conceptions of the continuum are examined from the perspective of conceptual structuralism, a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. This puts into question ..."
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Abstract: A number of conceptions of the continuum are examined from the perspective of conceptual structuralism, a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. This puts into question the idea from current set theory that the continuum is somehow a uniquely determined concept. Key words: the continuum, structuralism, conceptual structuralism, basic structural conceptions, Euclidean geometry, Hilbertian geometry, the real number system, settheoretical conceptions, phenomenological conceptions, foundational conceptions, physical conceptions. 1. What is the continuum? On the face of it, there are several distinct forms of the continuum as a mathematical concept: in geometry, as a straight line, in analysis as the real number system (characterized in one of several ways), and in set theory as the power set of the natural numbers and, alternatively, as the set of all infinite sequences of zeros and ones. Since it is common to refer to the continuum, in what sense are these all instances of the same concept? When one speaks of the continuum in current settheoretical
Neutrosophic logics on NonArchimedean Structures
 Critical Review, Creighton University, USA
"... We present a general way that allows to construct systematically analytic calculi for a large family of nonArchimedean manyvalued logics: hyperrationalvalued, hyperrealvalued, and padic valued logics characterized by a special format of semantics with an appropriate rejection of Archimedes ’ ax ..."
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We present a general way that allows to construct systematically analytic calculi for a large family of nonArchimedean manyvalued logics: hyperrationalvalued, hyperrealvalued, and padic valued logics characterized by a special format of semantics with an appropriate rejection of Archimedes ’ axiom. These logics are built as different extensions of standard manyvalued logics (namely, Lukasiewicz’s, Gödel’s, Product, and Post’s logics). The informal sense of Archimedes ’ axiom is that anything can be measured by a ruler. Also logical multiplevalidity without Archimedes ’ axiom consists in that the set of truth values is infinite and it is not wellfounded and wellordered. We consider two cases of nonArchimedean multivalued logics: the first with manyvalidity in the interval [0, 1] of hypernumbers and the second with manyvalidity in the ring Zp of padic integers. On the base of nonArchimedean valued logics, we construct nonArchimedean valued interval neutrosophic logics by which we can describe neutrality phenomena.
Extending du BoisReymond’s Infinitesimal and Infinitary Calculus Theory Part 5 Nonreversisble arithmetic and limits
"... An algebra for comparing functions at infinity with infinireals, comprising of infinitesimals and infinities, is developed: where the unknown relation is solved for. Generally, we consider positive monotonic functions f and g, arbitrarily small or large, with relation z: f z g. In general we requir ..."
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An algebra for comparing functions at infinity with infinireals, comprising of infinitesimals and infinities, is developed: where the unknown relation is solved for. Generally, we consider positive monotonic functions f and g, arbitrarily small or large, with relation z: f z g. In general we require f, g, f − g and fg to be ultimately monotonic. 1.
algorithm proposed
, 2005
"... The goals of this paper are to show the following. First, Grover’s algorithm can be viewed as a digital approximation to the analog quantum ..."
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The goals of this paper are to show the following. First, Grover’s algorithm can be viewed as a digital approximation to the analog quantum
Elliptic Differential Equations and their Discretizations By
"... A study of elliptic differential equations is carried out, from the point of view of interconnecting the discrete with the analytical. Approximate maximum principles and barrier postulates, acting on functions with hyperfinite domains, are introduced. The methods are specially adapted for proofs of ..."
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A study of elliptic differential equations is carried out, from the point of view of interconnecting the discrete with the analytical. Approximate maximum principles and barrier postulates, acting on functions with hyperfinite domains, are introduced. The methods are specially adapted for proofs of convergence of discretizations for linear elliptic PDE’s. The wellknown Brouwer degree theory is extended to hyperfinite dimensional spaces, with the purpose of applying it to show convergence of discretizations in nonlinear elliptic problems. Acknowledgements ii I would like to express my gratitude to all the people who have contributed to this work, with their advice, teachings, friendship or help. I am indebted to Professor H. Jerome Keisler for teaching me the fascinating world of nonstandard analysis, and its logic foundations, and his advice and encouragement in the research leading to this thesis. I thank Professors Paul Rabinowitz and Sigurd Angenent for their excellent lectures in partial differential equations. To Prof. Paul Milewski, I thank for showing me the beauty of applied mathematics. I also thank Prof. Kenneth Kunen for his lectures in logic. A very special thanks to Maria João. Her joy in life, her constant encouragement and love is a grant of life and hope. Our common journey through life and Mathematics has been a most pleasant one. I thank my beloved Joana for being a great daughter. To my parents, I thank their love, support and encouragement. They have taught me that true wealth results from pursue of knowledge and the mastering of an art. I also thank theirs, and my wife’s, patience in handling all official affairs in Lisbon, during the time I have stayed abroad. I thankfully acknowledge the support of the following organizations: Junta