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Shannon’s sampling theorem is one of the most important results of modern signal theory. It describes the reconstruction of any band-limited signal from a finite number of its samples. On the other hand, although less well known, there is the discrete sampling theorem, proved by Cooley while he was working on the development of an algorithm to speed up the calculations of the discrete Fourier transform. Cooley showed that a sampled signal can be resampled by selecting a smaller number of samples, which reduces computational cost. Then it is possible to reconstruct the original sampled signal using a reverse process. In principle, the two theorems are not related. However, in this paper we will show that in the context of Non-Standard Mathematical Analysis (NSA) and Hyperreal Numerical System ∗R, the two theorems are equivalent. The difference between them becomes a matter of scale. With the scale changes that the hyperreal number system allows, the discrete variables and functions become continuous, and Shannon’s sampling theorem emerges from the discrete sampling theorem. Resumen El teorema del muestreo de Shannon es uno de los resultados más importantes de la moderna teoría de señales. Este describe la reconstrucción de toda señal de banda limitada desde un número finito de sus muestras. Por otra parte, aunque menos conocido, se tiene el teorema del muestreo discreto, demostrado por Cooley mientras trabajaba en la elaboración de un algoritmo para acelerar los cálculos de la transformada discreta de Fourier. Cooley demostró que una señal muestreada se puede volver a muestrearla mediante la selección de un número menor de muestras, lo cual reduce el costo computacional. Luego, es posible reconstruir la señal muestreada original mediante un proceso inverso. En principio, los dos teoremas no están relacionados. Sin embargo, en este artíclo demostraremos que, en el contexto del Análisis Matemático No Estándar (ANS) y el Sistema Numérico Hiperreal ∗R, los dos teoremas son equivalentes. La diferencia entre ellos se vuelve un asunto de escala. Con los cambios de escala que permite realizar el sistema numérico hiperreal, las variables y funciones discretas se vuelven continuas, y el teorema del muestreo de Shannon emerge del teorema del muestreo discreto.

In general, the state of a system in which a physical quantity such as mass is distributed over space can be modeled by a function that represents the density distribution. The purpose of this paper is to introduce special functions that can be applied when in the system to be modeled, where the quantity is distributed over isolated points. For that matter, the expanded real numbers are introduced as an ordered subring of the hyperreal number field that does not contain any infinitesimals, and hyperreal delta functions are defined as special functions from the real numbers to the expanded real numbers satisfying the condition that (i) the support is a singleton, and (ii) the integral over the reals is a nonzero real number. These newly defined hyperreal delta functions, and tensor products thereof, then provide a general tool, applicable for the mathematical modeling of physical systems in which infinitely high densities occur.

Cilj diplomskega dela je z uvedbo infinitezemalnih števil predstaviti drugačen pristop h klasični analizi. Infinitezimalna števila uvedemo s pomočjo petih aksiomov. Največji poudarek je na principu prenosa, ki povezuje trditve, ki veljajo nad hiperrealnimi števili, s trditvami, ki veljajo nad realnimi števili. Skozi diplomsko delo dokazujemo, da so definicije iz nestandardne analize ekvivalentne tistim iz klasične analize, nato pa nove definicije uporabimo za dokaze izrekov iz klasične analize. Zaradi obširnosti klasične analize, se v diplomskem delu omejimo na koncepte zveznosti, limite, odvoda in kompaktnosti. In the thesis we introduce infinitesimals to present a new approach to classical analysis. We do so by introducing a set of five axioms. The emphasis is on the transfer axiom which helps us transfer theorems from nonstandard analysis to classical analysis. Throughout the thesis we show that the definitions in non-standard analysis are equivalent to those from classical analysis, and use non-standard definitions to prove theorems from classical analysis. Due to the large reach of classical analysis, we limit ourselves to concepts of continuity, limits, differentiation and compactness.

We examine an argument of Reeder suggesting that the nilpotent infinitesimals in Paolo Giordano's ring extension of the real numbers $^{\bullet}\mathbb{R}$ are smaller than any infinitesimal hyperreal number from Abraham Robinson's nonstandard analysis $^\ast\mathbb{R}$. Our approach consists in the study of two canonical order-preserving homomorphisms taking values in ${^{\bullet}\mathbb{R}}$ and in ${^\ast\mathbb{R}}$, respectively, and whose domain is Henle's extension of the real numbers in the framework of "non-nonstandard" analysis. In particular, we will show that there exists a nonzero element in Henle's ring that is "too small" to be registered as nonzero in Paolo Giordano's ring, while it is seen as a nonzero infinitesimal in ${^\ast\mathbb{R}}$. This result suggests that some hyperreal infinitesimals are smaller than the nilpotent infinitesimals. We argue that the apparent contradiction with the conclusions by Reeder is only due to the presence of nilpotent elements in ${^{\bullet}\mathbb{R}}$.

There are many natural phenomena that can best be described by the use of infinitesimal and infinite numbers (see e.g. [ 1 , 5 , 13 , 23 ]. However, until now, the Non-standard techniques have been applied to theoretical models. In this paper we investigate the possibility to implement such models in numerical simulations. First we define the field of Euclidean numbers which is a particular field of hyperreal numbers. Then, we introduce a set of families of Euclidean numbers, that we have called altogether algorithmic numbers, some of which are inspired by the IEEE 754 standard for floating point numbers. In particular, we suggest three formats which are relevant from the hardware implementation point of view: the Polynomial Algorithmic Numbers, the Bounded Algorithmic Numbers and the Truncated Algorithmic Numbers. In the second part of the paper, we show a few applications of such numbers.

This research discusses the validity of the properties of real numbers set to hyperreal numbers set, i.e. algebraic, ordered, and completeness properties, by using a finitely additive measure. This finitely additive measure is a map from the power set of natural numbers set onto set {0,1}. The subset of natural numbers set has measure zero if it’s finite and one if it’s infinite. The set of hyperreal numbers is constructed from equivalence classes of the set of all sequences of real numbers by using a relation involving the finitely additive measure, that is, two sequences of real numbers are said to be related if and only if those two sequences are the same almost everywhere. In the hyperreal numbers set, there exist infinitesimal numbers besides 0. Infinitesimal number is a number which is less than any positive real number and greater than any negative real number. So, in hyperreal number set, there are some smallest positive numbers. The results show that the hyperreal numbers set is an ordered and complete field.

Treating divergent series properly has been an ongoing issue in mathematics. However, many of the problems in divergent series stem from the fact that divergent series were discovered prior to having a number system which could handle them. The infinities that resulted from divergent series led to contradictions within the real number system, but these contradictions are largely alleviated with the hyperreal number system. Hyperreal numbers provide a framework for dealing with divergent series in a more comprehensive and tractable way., Comment: 9 pages

Cardinality arguments against regular probability measures aim to show that no matter which ordered field H we select as the measures for probability, we can find some event space F of sufficiently large cardinality such that there can be no regular probability measure from F into H. In particular, taking H to be hyperreal numbers won’t help to guarantee that probability measures can always be regular. I argue that such cardinality arguments fail, since they rely on the wrong conception of the role of numbers as measures of probability. With the proper conception of their role we can see that for any event space F, of any cardinality, there are regular hyperreal-valued probability measures.

In this article, we propose to introduce the differential of a function through a non-classical way, lying on hyperreals and infinite microscopes. This approach is based on the developments of nonstandard analysis, wants to be more intuitive than the classical one and tries to emphasize the functional and geometric aspects of the differential. In the second part of the work, we analyze the results of an experiment made with undergraduate students who had been taught calculus by a non standard way for nearly two years.

International audience; In update logic as studied by Baltag, Moss, Solecki and van Benthem, little attention is paid to the interpretation of an action by an agent, which is just assumed to depend on the situation. This is actually a complex issue that nevertheless complies to some logical dynamics. In this paper, we tackle this topic. We also deal with actions that change propositional facts of the situation. In parallel, we propose a formalism to accurately represent an agent's epistemic state based on hyperreal numbers. In that respect, we use infinitesimals to express what would surprise the agents (and by how much) by contradicting their beliefs. We also use a subjective probability to model the notion of belief. It turns out that our probabilistic update mechanism satisfies the AGM postulates of belief revision.

The method of fluxions was originally given by Newton among others in order to determine the tangent to a curve. In this note, we will formulate this method by the light of some modern mathematical tools: using the concept of limit, but also with hyperreal numbers and their standard parts and with dual numbers; another way is the use of virtual microscopes both in the contexts of classical and non standard analysis.

The infinitely small and the infinitely large are essential in calculus. They have appeared throughout its history in various guises: infinitesimals, indivisibles, differentials, evanescent quantities, moments, infinitely large and infinitely small magnitudes, infinite sums, power series, limits, and hyperreal numbers. And they have been fundamental at both the technical and conceptual levels – as underlying tools of the subject and as its foundational underpinnings. We will consider examples of these aspects of the infinitely small and large as they unfolded in the history of calculus from the 17th through the 20th centuries. We will also present `didactic observations' at relevant places in the historical account.

This paper first describes the construction and use of the hyperreals in the theorem-prover Isabelle within the framework of higher-order logic (HOL). The theory, which includes infinitesimals and infinite numbers, is based on the hyperreal number system developed by Abraham Robinson in his nonstandard analysis (NSA). The construction of the hyperreal number system has been carried out strictly through the use of definitions to ensure that the foundations of NSA in Isabelle are sound. Mechanizing the construction has required that various number systems including the rationals and the reals be built up first. Moreover, to construct the hyperreals from the reals has required developing a theory of filters and ultrafilters and proving Zorn's lemma, an equivalent form of the axiom of choice.This paper also describes the use of the new types of numbers and new relations on them to formalize familiar concepts from analysis. The current work provides both standard and nonstandard definitions for the various notions, and proves their equivalence in each case. To achieve this aim, systematic methods, through which sets and functions are extended to the hyperreals, are developed in the framework. The merits of the nonstandard approach with respect to the practice of analysis and mechanical theorem-proving are highlighted throughout the exposition.

I mplicit functions arise throughout mathematical analysis. They are especially prominent in applications to economics. From the beginning of mathematical economics one must work with implicit functions, as in analysis of a consumer and his indifference curves, or of a firm and its isoquants. Consequently, in teaching calculus to business students we come early to the challenge of explaining differentiation of implicit functions. The aim of this note is to give a systematic approach to computing the successive derivatives of an implicit function of one real variable, an approach that we think brings students naturally to the main results. The idea is that smooth curves looked at closely enough are straight, so that analysis problems are locally problems of linear algebra. Our method invokes that intuition: we look at curves through (virtual) microscopes—but we pay for this by having to deal with hyperreal numbers. Although Leibniz and Newton, for instance, already worked with ‘‘infinitesimals’’, the rigorous treatment called ‘‘nonstandard analysis’’ was introduced only in 1961 by A. Robinson [4]. An especially simple presentation for didactical purposes was given by Keisler [3]; we shall mostly adopt his definitions and notations. We recall that the hyperreal numbers extend the real ones with the same algebraic rules; technically, the set *R of the hyperreal numbers is a non-archimedean ordered field in which the real line R is embedded. Moreover, *R contains at least one infinitesimal (and then it must contain infinitely many). An infinitesimal is a number e such that its absolute value is less thanevery real number butwhich is unequal to 0; its reciprocal 1e is infinite, that is, is a number whose absolute value is greater than every real number. Clearly, non-zero infinitesimals and infinite numbers are not real. A hyperreal number x, which is not infinite, is of course said to be finite; for any such x there exists one and only one real number r, which is infinitely close to x, that is, such that the difference x r is an infinitesimal: this r is called the standard part of x and is denoted by r = st(x). Formally, st is a ring homomorphism from the set of finite hyperreal numbers to R; and its kernel is the set of infinitesimals. Moreover, a function of one or several real variables may have a natural extension to the hyperreal numbers, with the same definition and the same properties as the original one. Indeed, if a real-valued function is defined by a system of formulas, its extension can be obtained by applying the same formulas to the hyperreal system. In this article, we adopt the same notation for a real function and for its natural extension. The concept of (virtual) microscope is well known (see, for example, [1, 2, 5]). For a point P(a, b) in the hyperreal plane *R and a positive infinite hyperreal number H, a microscope pointed on P and with H as power magnifies the distances from P by a factor H; more explicitly, it is a map, denoted by MPH ; defined on *R as follows: MPH : x; y ð Þ 7! X ;Y ð Þ with X 1⁄4 H x a ð Þ and Y 1⁄4 H y b ð Þ

Chapter 2 discusses different systems of numbers, emphasizing on reasons making them so important. It starts with the system of natural numbers and presents its existence as a natural consequence from the axiom of infinity. Next, the systems of integers and rational numbers are constructed over the system of natural numbers. The least upper bound property and its lack in the system of rational numbers are emphasized. The system of real numbers is introduced as an ordered field with least upper bound property, which improves the system of rational numbers. For completeness of the whole picture on numbers, brief information about the systems of extended real, complex and hyperreal numbers is also given. The chapter ends with a discussion of cardinality, emphasizing on finite, countable and continuum cardinalities.

Kirchhoff's laws fail to hold in general for infinite electrical networks. Standard calculus is simply incapable of resolving this paradox because it cannot provide the infinitesimals and more generally the hyperreal currents and voltages that such networks often require. However, nonstandard analysis can do precisely this. The idea of a nonstandard electrical network is introduced in this paper and is used to reestablish Kirchhoff's laws for a fairly broad class of infinite electrical networks. The second section herein presents a fairly brief tutorial on infinitesimals, hyperreal numbers, and the key ideas of nonstandard analysis needed for a comprehension of this paper.

A construction of the real number system based on almost homomorphisms of the integers Z was proposed by Schanuel, Arthan, and others. We combine such a construction with the ultrapower or limit ultrapower construction, to construct the hyperreals out of integers. In fact, any hyperreal field, whose universe is a set, can be obtained by such a one-step construction directly out of integers. Even the maximal (i.e., On-saturated) hyperreal number system described by Kanovei and Reeken (2004) and independently by Ehrlich (2012) can be obtained in this fashion, albeit not in NBG. In NBG, it can be obtained via a one-step construction by means of a definable ultrapower (modulo a suitable definable class ultrafilter)., 17 pages, 1 figure

A nonstandard universe is constructed from a superstructure in a Boolean-valued model of set theory. This provides a new framework of nonstandard analysis with which methods of forcing are incorporated naturally. Various new principles in this framework are provided together with the following applications: (1) An example of an ℵ1-saturated Boolean ultrapower of the real number field which is not Scott complete is constructed. (2) Infinitesimal analysis based on the generic extension of the hyperreal numbers is provided, and the hull completeness theorem and the Loeb measure construction are extended to objects in the generic extension of the internal universe. (3) The reduction theory of the Boolean-valued complex numbers are developed as a prototype of the applications to the topological reduction theory of Boolean sheaves or operator algebras.

The infinitely small and the infinitely large – in one form or another – are essential in calculus. In fact, they are among the distinguishing features of calculus compared to some other branches of mathematics, for example algebra. They have appeared throughout the history of calculus in various guises: infinitesimals, indivisibles, differentials, evanescent quantities, moments, infinitely large and infinitely small magnitudes, infinite sums, power series, limits, and hyperreal numbers. And they have been fundamental at both the technical and conceptual levels – as underlying tools of the subject and as its foundational underpinnings. We will consider examples of these aspects of the infinitely small and large as they unfolded in the history of calculus from the seventeenth through the twentieth centuries. This will, in fact, entail discussing central issues in the development of calculus.

In ordinary qualitative reasoning (QR), qualitative behavior of the dynamical systems is predicted by assignment of qualitative values such as \{+, 0, $-$\} into model variables based on proper transition rules. Unfortunately, due to ambiguities in these ordinarily introduced qualitative values, their arithmetic and transition rules cause predictions to be redundant, sometimes even inacurate. In this paper, we present a new method of qualitative reasoning which, besides using hyperreal numbers, takes into account their $\varepsilon$-{\bf H} ranking in describing both qualitative values and qualitative derivatives of variables and also employs a convergence filter to investigate the infinitesimal asymptotic behavior of qualitative variables. We applied this qualitative reasoning method to envision a temporally hierarchical complex system. The result shows that this method provides a more detailed and natural qualitative solution than previous methods like Kuipers's time abstraction in envisioning temporally hierarchical complex system.

Il libro presenta, con dimostrazioni dettagliate, le costruzioni classiche dei numeri reali di Méray-Cantor (1872) e di Dedekind (1872) e le confronta con la definizione assiomatica di Hilbert, quest'ultima proposta sia nella forma originale (1900) che in una riscrittura moderna. Sono illustrate e comparate anche le costruzioni basate sugli allineamenti di cifre, e le definizioni assiomatiche basate sulle proprietà di "separazione". Gli autori mostrano come le note difficoltà didattiche nell'introduzione dei numeri reali derivino spesso da implicite mutue contaminazioni fra le varie impostazioni e suggeriscono come queste possano essere eliminate.

In a nonstandard universe, the κ-saturation property states that any family of fewer than κ internal sets with the finite intersection property has a nonempty intersection. An ordered field F is said to have the λ-Bolzano-Weierstrass property iff F has cofinality λ and every bounded λ-sequence in F has a convergent λ-subsequence. We show that if κ < λ are uncountable regular cardinals and βα < λ whenever α < κ and β < λ then there is a κ-saturated nonstandard universe in which the hyperreal numbers have the λ-Bolzano-Weierstrass property. The result also applies to certain fragments of set theory and second order arithmetic.

The field of real numbers being extended as a larger commutative field, we investigate the possibility of defining a scalar product for the distributions of finite discrete support. Then we focus on the most simple possible extension (which is an ordered field), we provide explicit formulas for this scalar product, and we exhibit a structure of positive definite inner-product space. In a one-dimensional application to the Schroedinger equation, the distributions supported by the origin are embedded into a bra-ket vector space, where the "singular" potential describing point interaction is defined in a natural way. A contact with the hyperreal numbers that arise in nonstandard analysis is possible but not essential, our extensions of $\bf R$ and $\bf C$ being obtained by a quite elementary method., 27 pages

The hyperreal numbers of nonstandard analysis are characterized in purely algebraic terms as homomorphic images of a suitable class of rings of functions.

In this paper we present the main features of Nonstandard Analysis without any use of Logic or any deep tool from set theory. The hyperreal numbers are introduced by three axioms of algebraic type whose consistency is proved by algebraic means. Probably this way of introducing Nonstandard Analysis, is not the most interesting and elegant. Moreover, part of the richness and the power of Nonstandard Analysis (as the Transfer and the Saturation Principles) are lost. Nevertheless, most of its elementary applications can be carried out without any problem. In order to show this, the last four sections of this paper are devoted to simple applications to Elementary Calculus.

After briefly considering the ancient Greek and nineteenth-century history of incommensurables (magnitudes that do not have a common aliquot part) and incomparables (magnitudes such that the larger can never be surpassed by any finite number of additions of the smaller to itself), this paper undertakes two tasks. The first task is to consider whether the numerical accommodation of incommensurables by means of the extension of the ordered field of rational numbers to the field of reals is `similar' or analogous to the numerical accommodation of incomparables by means of the extension of the ordered field of reals to the field of hyperreals. The second task is to evaluate several contemporary attempts to use concepts and techniques of the nonstandard mathematics of hyperreals to address classical, Zenonian puzzles concerning continuous magnitudes. The result of both these undertakings is, in a certain sense, `deflationary'.

Members of *ℝ are called hyperreal numbers, while members of ℝ are real and sometimes called standard. *ℚ consists of hyperrationals, *ℤ of hyperintegers, and *ℕ of hypernaturals. That *ℚ consists precisely of quotients m/n of hyperintegers m, n ∈ *ℤ follows by transfer of the sentence $$ \forall x \in \mathbb{R}\left[ {x \in \mathbb{Q} \leftrightarrow \exists y,z \in \mathbb{Z}\left( {z \ne 0 \wedge x = y/z} \right)} \right]. $$ It is now time to examine the basic arithmetical and algebraic structure of *ℝ, particularly in its relation to the structure of ℝ.

Publisher Summary This chapter focuses on the statistical applications of information energy. Applications to some related areas, such as solving logic problems, connections with hyperreal numbers, noiseless coding, etc., are also made. The chapter also analyzes the notion of information energy to the continuous case and examines some of its properties. The difference between the continuous and the discrete case is worth emphasizing—that is, the information energy in the continuous case is not a limit of the information energy in the discrete case. These differences between the discrete and continuous distribution warns that the results holding for the discrete case cannot be extended directly to the continuous case. They need independent verification. Furthermore, the chapter discusses the information energy of a point process. After studying the properties relative to the rate of change of information energy, it is proved that given the class of stationary point processes, the information energy is minimum for the Poisson process.

After a rigorous introduction to hyperreal numbers, we give in terms of non standard analysis, (1) a Lagrangian statement of classical physics, and (2) a statement of formal quantum scattering. © 1988 Plenum Publishing Corporation.

An exposition of the Keisler's elementary treatment of non-standard methods in elementary calculus.