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A Formal Definition of Dynamic Structure Discrete-Flow Components
Published in Gabriel A. Wainer, Pieter J. Mosterman, Discrete-Event Modeling and Simulation, 2018
We provide an intuitive definition of hyperreal numbers and the semantics of the addition and comparison operators. We revise first the notation for real intervals since we develop the definition of hyperreals based on the interval notation. The expression a ≥ t, t ∈ R defines the interval [t, +∞), and a > t defines the interval (t, + ∞). If we denote the left extreme of the interval (t, + ∞) by t + we can rewrite the interval (t, + ∞) as [t+, + ∞). To simplify the notation used in the next sections we define the infinitesimal number ε = t + −t. An alternative and canonical definition of infinitesimals in the realm of nonstandard analysis can be found by Goldblatt [9].
A parameter method for linear algebra and optimization with uncertainties
Published in Optimization, 2020
Nam Van Tran, Imme van den Berg
Nonstandard analysis distinguishes standard and nonstandard numbers, providing a convenient setting to deal with orders of magnitude of numbers. We adopt here the axiomatics IST of Nelson [28], which is an extension of common set theory ZFC. Introductions to IST are contained in e.g. [1,29] or [30]. Sets of IST are called internal. In this setting, nonstandard numbers are already present within , and standard real numbers are only Archimedean with respect to standard natural numbers. So different orders of magnitude already coexist within the real numbers. Limited numbers are real numbers bounded in absolute value by standard natural numbers. Real numbers larger in absolute value than limited numbers are called unlimited. Its reciprocals, together with 0, are called infinitesimal. Limited numbers which are not infinitesimal are called appreciable.
Nonstandard second-order formulation of the LWR model
Published in Transportmetrica B: Transport Dynamics, 2019
Even though it has been shown that all mathematical arguments in nonstandard analysis can be established in standard analysis, infinitesimal/infinite numbers and nonstandard analysis can be very useful for modeling purposes and have been widely used in physics and other areas. For examples, hyperreal numbers have led to simple representations of discontinuous functions, including the Dirac delta function. More importantly they enable the development of nonstandard analysis through intuitive and rigorous interpretations of derivatives and integrals with infinitesimal numbers, which were initiated by Leibniz (Davis 2005). In Hanqiao, St Mary, and Wattenberg (1986) and van den Berg (1998), nonstandard analysis was applied to solve the heat equation under initial conditions given by a Dirac delta function. In these studies, ε is defined as the limit of the time-step size, , and it was shown that ε can be simply replaced by in the discrete version of a differential equation and the continuous and discrete equations are equivalent for an infinitesimal .
Exploring public transport sustainability with neutrosophic logic
Published in Transportation Planning and Technology, 2019
Hence, a further generalization of FSs and IFSs was required. Recently a new theory developed by Florentin Smarandache (2007, 2010) has been outlined (Wang et al., 2005) which involves neutrosophic sets and neutrosophic logic. These are based on non-standard analysis pioneered by Abraham Robinson (1966). Non-standard analysis involves infinitesimals which are so small that there is no way to measure them; it has been defined as a branch of mathematical logic which introduces hyperreal numbers to allow for the existence of ‘genuine infinitesimals’, which are numbers that are less than 1/2, 1/3, 1/4, 1/5, … , but greater than zero (WolframMathWorld). Neutrosophic logic and sets were developed, according to Aggarwal et al. (2010) to represent a mathematical model of uncertainty, vagueness, ambiguity, imprecision, undefined, unknown, incompleteness, inconsistency, redundancy, contradiction present in data. The term neutrosophy means knowledge of neutral thought. This neutral represents the main distinction between fuzzy and intuitionistic fuzzy sets and logic (Majumdar, 2015). A unique feature of neutrosophy is that it can be used for modelling paradoxes. As Schumann and Smarandache (2007) remark, the paradox is the only proposition true and false in the same time in the same world, and indeterminate as well. Neutrosophic logic is a logic in which each proposition is estimated to have a degree of truth (T), a degree of indeterminacy (I) and a degree of falsity (F). A neutrosophic set is a set where each element of the universe has a degree of truth, indeterminacy and falsity respectively and which lies between ]−0, 1+[, the non-standard unit interval. The non-standard unit interval is an extension of the standard interval [0,1]. Unlike in intuitionistic fuzzy sets, where the incorporated uncertainty is dependent on the degree of belongingness and degree of non-belongingness, the uncertainty present in a neutrosophic set (the hesitancy/indeterminacy factor) is independent of the truth and falsity values. Thus, we do not assume that the incompleteness or indeterminacy degree is always given by 1 − (T + F).