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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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Cited by 10 (2 self)
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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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Cited by 5 (2 self)
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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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Cited by 4 (2 self)
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This paper is dedicated to the memory of Imre Lakatos
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.
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
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
INTERNAL TURING MACHINES
, 2004
"... Abstract. Using nonstandard analysis, we will extend the classical Turing machines into the internal Turing machines. The internal Turing machines have the capability to work with infinite (∗finite) number of bits while keeping the finite combinatoric structures of the classical Turing machines. We ..."
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Abstract. Using nonstandard analysis, we will extend the classical Turing machines into the internal Turing machines. The internal Turing machines have the capability to work with infinite (∗finite) number of bits while keeping the finite combinatoric structures of the classical Turing machines. We will show the following. The internal deterministic Turing machines can do in ∗polynomial time what a classical deterministic Turing machine can do in an arbitrary finite amount of time. Given an element of < ∗ M; ∗ x> ∈ Halt (more precisely, the ∗embedding of Halt), there is an internal deterministic Turing machine which will take < ∗ M; ∗ x> as input and halt in the ”yes ” state, and for < ∗ M; ∗ x>/ ∈ Halt, the internal deterministic Turing machine will halt in the ”no ” state. The language ∗ HALT can not be decided by the internal deterministic Turing machines. The internal deterministic Turing machines can be viewed as the asymptotic behavior of finite precision approximation to real number computations. It is possible to use the internal probabilistic Turing machines to simulate finite state quantum mechanics with infinite precision. This simulation suggests that no information can be transmitted instantaneously and at the same time, the Turing machine model can simulate instantaneous collapse of the wave function. The internal deterministic Turing machines are powerful, but if P = NP, then there are internal problems which the internal deterministic Turing machines can solve but not in ∗polynomial time. 1. Introduction. Nonstandard
NonStandard Semantics of Hybrid Systems Modelers ✩
"... Hybrid system modelers have become a corner stone of complex embedded system development. Embedded systems include not only control components and software, but also physical devices. In this area, Simulink is a de facto standard design framework, and Modelica a new player. However, such tools raise ..."
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Hybrid system modelers have become a corner stone of complex embedded system development. Embedded systems include not only control components and software, but also physical devices. In this area, Simulink is a de facto standard design framework, and Modelica a new player. However, such tools raise several issues related to the lack of reproducibility of simulations (sensitivity to simulation parameters and to the choice of a simulation engine). In this paper we propose using techniques from nonstandard analysis to define a semantic domain for hybrid systems. Nonstandard analysis is an extension of classical analysis in which infinitesimal (the ε and η in the celebrated generic sentence ∀ε∃η... of college maths) can be manipulated as first class citizens. This approach allows us to define both a denotational semantics, a constructive semantics, and a Kahn Process Network semantics for hybrid systems, thus establishing simulation engines on a sound but flexible mathematical foundation. These semantics offer a clear distinction between the concerns of
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