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Let a1, a2, … be a numerical sequence. As the main task of the paper, we consider the classical problem of computing the sum ∑n=1∞an when the series is either conditionally convergent or divergent. We demonstrate that the concept of grossone, recently proposed by Sergeyev, can be useful in both computing this sum and studying properties of summation methods. We also consider the problem of choosing the upper limit in the sum if we wish to replace the infinity sign ∞ with a grossone-based quantity. Finally, we discuss some properties of prime numbers in the grossone universe and make an attempt of analyzing the celebrated Euler’s product formula.

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... (Notice that the noncontradictoriness of the ①-based computational methodology has been studied in depth in [15][16][17].) From the practical point of view, this methodology has given rise both to a new supercomputer patented in several countries (see [18]) and called Infinity Computer and to a variety of applications starting from optimization (see [12,[19][20][21][22][23][24]) and going through infinite series (see [13,[25][26][27][28]), fractals and cellular automata (see [25,[29][30][31][32]), hyperbolic geometry and percolation (see [33,34]), the first Hilbert problem and Turing machines (see [13,35,36]), infinite decision making processes and probability (see [13,[37][38][39]), numerical differentiation and ordinary differential equations (see [40][41][42][43]), etc. ...

... The next lemma proves that L in (28) is nonsingular under the assumptions in Lemma 4.1. (28). Then, we have ...

... Proof The first two relations follow immediately from (27), Table 3 and recalling that C k is nonsingular. Moreover, since β k = Ap k 2 / r k 2 , note that in (28) we have ...

We consider an iterative computation of negative curvature directions, in large-scale unconstrained optimization frameworks, needed for ensuring the convergence toward stationary points which satisfy second-order necessary optimality conditions. We show that to the latter purpose, we can fruitfully couple the conjugate gradient (CG) method with a recently introduced approach involving the use of the numeral called Grossone. In particular, recalling that in principle the CG method is well posed only when solving positive definite linear systems, our proposal exploits the use of grossone to enhance the performance of the CG, allowing the computation of negative curvature directions in the indefinite case, too. Our overall method could be used to significantly generalize the theory in state-of-the-art literature. Moreover, it straightforwardly allows the solution of Newton’s equation in optimization frameworks, even in nonconvex problems. We remark that our iterative procedure to compute a negative curvature direction does not require the storage of any matrix, simply needing to store a couple of vectors. This definitely represents an advance with respect to current results in the literature.

... In particular, metamathematical investigations on the new theory and its consistency can be found in [66]. The methodology described here has been successfully applied in several areas of Mathematics and Computer Science: single and multiple criteria optimization (see [20,[32][33][34]43,116]), cellular automata (see [29][30][31]), Euclidean and hyperbolic geometry (see [68,69]), percolation (see [56,57,110]), fractals (see [15,87,89,97,102,110]), infinite series and the Riemann zeta function (see [91,96,99,101,114]), the first Hilbert problem, Turing machines, and supertasks (see [81,93,103,104]), numerical differentiation and numerical solution of ordinary differential equations (see [2,74,95,98,105]), etc. Some of these applications will be discussed in the following pages. ...

... Notice that some traditional summation techniques can be interpreted as certain weighted averages on sums having x addends (see [114,115] for a detailed discussion). Hereinafter we consider what happens in the x-based framework in situations where certain summation techniques lead to negative sums of divergent series that contain infinitely many positive integers. ...

... and, therefore, the following theorem holds (a similar result obtained using a different reasoning can be found in [114]). Let us first briefly present Euler's proof and then comment upon it. ...

... The version of grossone based on Axioms 2,3 resembles the version studied by L.H. Kauffman in [9]. We refer to [10] for detailed discussion of the Axioms 1-3 and their relation to the original Axioms of Ya. Sergeyev. ...

... Unlike the sum (1), the sum (2) is uniquely defined and any rearrangement of terms does not change the answer. More precisely, the following result has been proved in [10]. ...

... As shown in [10] the concept of ① could be very useful for comparing methods of summation of divergent series. In particular, the use of grossone allowes an introduction of probability measures like the discrete uniform or binomial distributions on the set {1, 2, . . . ...

In this presentation, several areas of mathematics are considered where the concept of grossone developed by Ya. Sergeyev in his small book [1] and a series of papers [2, 3, 4, 5, 6, 7, 8], can be very useful. Let us start with discussing the axioms of grossone and suggest some minor variations to the axioms of Ya. Sergeyev. The version of the grossone, which will be used in this work, will allow us to consider limits of conditionally convergent and divergent sequences.

... .} and generalizing the ordinary discrete uniform and binomial distributions. Both of these extensions have been recently discussed in Calude and Dumitrescu (2020) and mentioned in Zhigljavsky (2012); both extensions use the notion of grossone. The grossone, introduced in Sergeyev (2013) and denoted by 1 , is a model of infinity which, as shown in Sergeyev (2009), Sergeyev (2017 and many other publications can be very useful in solving diverse problems of computational mathematics and optimization; in such applications, 1 is used as numerical infinity. ...

... The grossone, introduced in Sergeyev (2013) and denoted by 1 , is a model of infinity which, as shown in Sergeyev (2009), Sergeyev (2017 and many other publications can be very useful in solving diverse problems of computational mathematics and optimization; in such applications, 1 is used as numerical infinity. Grossone can also be useful as a theoretical model of infinity, see, e.g., (Zhigljavsky 2012;Sergeyev 2017). Some historical, philosophical and logical aspects of grossone have been considered in Lolli (2012), Lolli (2015), Hansson (2020). ...

We study properties of two probability distributions defined on the infinite set \(\{0,1,2, \ldots \}\) and generalizing the ordinary discrete uniform and binomial distributions. Both extensions use the grossone-model of infinity. The first of the two distributions we study is uniform and assigns masses \(1/\textcircled {1}\) to all points in the set \( \{0,1,\ldots ,\textcircled {1}-1\}\), where \(\textcircled {1}\) denotes the grossone. For this distribution, we study the problem of decomposing a random variable \(\xi \) with this distribution as a sum \(\xi {\mathop {=}\limits ^\mathrm{d}} \xi _1 + \cdots + \xi _m\), where \(\xi _1 , \ldots , \xi _m\) are independent non-degenerate random variables. Then, we develop an approximation for the probability mass function of the binomial distribution Bin\((\textcircled {1},p)\) with \(p=c/\textcircled {1}^{\alpha }\) with \(1/2<\alpha \le 1\). The accuracy of this approximation is assessed using a numerical study.

... Drawing inspiration from "the physical world around us" 1 he developed a numerical computational system based on two fundamental atoms , the ordinary natural number 1 ∈ N for finite quantities, and a new infinite unit ①called grossone for infinite and infinitesimal quantities: the reader can see [49,52,57,58,60,62] for detailed introduction surveys, the book [47] written in a popular way or [29,34] for more technical insights. This new numerical system has already been applied in many different research areas of pure and applied mathematics, and also of several experimental sciences: for instance, in optimization and numerical differentiation (see [15,16,53,69] ), in Euclidean and hyperbolic geometry (see [31,32] ), fractals (see [9,10,48,50,55,59,66] ), cellular automata (see [12,13] and in the context of [7] under investigation), numerical solution of ordinary differential equations (see [1,33,56,65] ), infinite series (see [51,54,61,68] ), percolation (see [21,22,66] ), Turing machines and supertasks (see [43,63,64] ), etc. Instead, as far as we know, in the present paper the new system is used for the first time in connection with complex numbers and variables, and also with doubly-infinite series. ...

... From here onwards we will tacitly assume that the reader is familiar with Sergeyev's numerical system, knows how to develop computations and how to use the basic notions and properties. Numerous examples and extensive discussions on the new framework can be easily found in [47][48][49][50][51][52]54,55,[57][58][59]68] and in the references therein. It is very important to also remark that all the proofs present in the paper are purely computational. ...

The Z-transform is an important mathematical tool to model sample-data control systems or other discrete-data systems. Since the Z-transform is defined as sum of an infinite number of addends, it is very interesting to look at it from non-classical points of view through one of the many current theories that today provide a wide range of different infinities and infinitesimals. In this paper, therefore, we choose to adopt a new simple applied approach recently proposed by Y.D. Sergeyev that allows one to execute easily numerical computations with various sizes of infinite and infinitesimal numbers. Using this new approach, we obtain a very different type of Z-transform of a complex sequence (or better, a family of infinitely many Z-transforms attached to the same sequence) whose existence is guaranteed almost everywhere on C, unlike what happens in traditional analysis in which the bilateral Z-transform often does not exist anywhere.

... In particular, metamathematical investigations on the new theory and its non-contradictory can be found in [17]. The ①-based methodology has been successfully applied in several areas of Mathematics and Computer Science: single and multiple criteria optimization (see [21,22,23,24,25]), cellular automata (see [26,27]), Euclidean and hyperbolic geometry (see [28,29]), percolation (see [30]), fractals (see [31,32,33,34,35]), infinite series and the Riemann zeta function (see [36,37,38,39,40]), the first Hilbert problem, Turing machines, and supertasks (see [41,42,20,43]), numerical differentiation and numerical solution of ordinary differential equations (see [44,45,46,47,48]), etc. In this paper, divergent series and Ramanujan summation are studied. ...

... Notice that some traditional summation techniques can be interpreted as certain weighted averages on sums having ① addends (see [40,50] for a detailed discussion). Let us see now what happens in the ①-based framework in situations where certain summation techniques produce results where to divergent series containing infinitely many positive integers negative results are assigned. ...

A computational methodology called Grossone Infinity Computing introduced with the intention to allow one to work with infinities and infinitesimals numerically has been applied recently to a number of problems in numerical mathematics (optimization, numerical differentiation, numerical algorithms for solving ODEs, etc.). The possibility to use a specially developed computational device called the Infinity Computer (patented in USA and EU) for working with infinite and infinitesimal numbers numerically gives an additional advantage to this approach in comparison with traditional methodologies studying infinities and infinitesimals only symbolically. The grossone methodology uses the Euclid’s Common Notion no. 5 ‘The whole is greater than the part’ and applies it to finite, infinite, and infinitesimal quantities and to finite and infinite sets and processes. It does not contradict Cantor’s and non-standard analysis views on infinity and can be considered as an applied development of their ideas. In this paper we consider infinite series and a particular attention is dedicated to divergent series with alternate signs. The Riemann series theorem states that conditionally convergent series can be rearranged in such a way that they either diverge or converge to an arbitrary real number. It is shown here that Riemann’s result is a consequence of the fact that symbol ∞ used traditionally does not allow us to express quantitatively the number of addends in the series, in other words, it just shows that the number of summands is infinite and does not allows us to count them. The usage of the grossone methodology allows us to see that (as it happens in the case where the number of addends is finite) rearrangements do not change the result for any sum with a fixed infinite number of summands. There are considered some traditional summation techniques such as Ramanujan summation producing results where to divergent series containing infinitely many positive integers negative results are assigned. It is shown that the careful counting of the number of addends in infinite series allows us to avoid this kind of results if grossone-based numerals are used.

... In particular, relations of the new approach to bijections are studied in [19] and metamathematical investigations on the new theory and its non-contradictory can be found in [18]. Then, the new methodology has been applied for studying Euclidean and hyperbolic geometry (see [20,21]), percolation (see [12,13,44]), fractals (see [25,27,35,44]), numerical differentiation and optimization (see [4,28,33,47]), infinite series and the Riemann zeta function (see [29,34,46]), the first Hilbert problem, Turing machines, and lexicographic ordering (see [31,[41][42][43]39]), cellular automata (see [5][6][7]), ordinary differential equations (see [37,38]), etc. The interested reader is invited to have a look also at surveys [26,32,36] and the book [24] written in a popular way. ...

... In this paper only two applications where x-based numerals are useful have been discussed: infinite sets and Turing machines. More examples showing how these numerals can be successfully used can be found in the following publications: Euclidean and hyperbolic geometry (see [20,21]), percolation (see [12,13,44]), fractals (see [25,27,35,44]), infinite series and the Riemann zeta function (see [29,34,46]), the first Hilbert problem and lexicographic ordering (see [31,[41][42][43]39]), cellular automata (see [5][6][7]). ...

Traditional computers are able to work numerically with finite numbers only. The Infinity Computer patented recently in USA and EU gets over this limitation. In fact, it is a computational device of a new kind able to work numerically not only with finite quantities but with infinities and infinitesimals, as well. The new supercomputing methodology is not related to non-standard analysis and does not use either Cantor’s infinite cardinals or ordinals. It is founded on Euclid’s Common Notion 5 saying ‘The whole is greater than the part’. This postulate is 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 numerals belonging to a positional numeral system with an infinite radix described by a specific ad hoc introduced axiom. Numerous examples of the usage of the introduced computational tools are given during the lecture. In particular, algorithms for solving optimization problems and ODEs are considered among the computational applications of the Infinity Computer. Numerical experiments executed on a software prototype of the Infinity Computer are discussed.

... In particular, relations of the new approach to bijections are studied in [19] and metamathematical investigations on the new theory and its non-contradictory can be found in [18]. Then, the new methodology has been applied for studying Euclidean and hyperbolic geometry (see [20,21]), percolation (see [12,13,44]), fractals (see [25,27,35,44]), numerical differentiation and optimization (see [4,28,33,47]), infinite series and the Riemann zeta function (see [29,34,46]), the first Hilbert problem, Turing machines, and lexicographic ordering (see [31,[41][42][43]39]), cellular automata (see [5][6][7]), ordinary differential equations (see [37,38]), etc. The interested reader is invited to have a look also at surveys [26,32,36] and the book [24] written in a popular way. ...

... In this paper only two applications where x-based numerals are useful have been discussed: infinite sets and Turing machines. More examples showing how these numerals can be successfully used can be found in the following publications: Euclidean and hyperbolic geometry (see [20,21]), percolation (see [12,13,44]), fractals (see [25,27,35,44]), infinite series and the Riemann zeta function (see [29,34,46]), the first Hilbert problem and lexicographic ordering (see [31,[41][42][43]39]), cellular automata (see [5][6][7]). ...

In this paper, a recent computational methodology is described. It has been introduced with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework. It is based on the principle ‘The part is less than the whole’ applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The methodology uses as a computational device the Infinity Computer (patented in USA and EU) working numerically with infinite and infinitesimal numbers that can be written in a positional system with an infinite radix. On a number of examples dealing mainly with infinite sets and Turing machines with different infinite tapes it is shown that it becomes possible to execute a fine analysis of these mathematical objects. The accuracy of the obtained results is continuously compared with results obtained by traditional tools used to work with mathematical objects involving infinity.

... First of all, we mention numerous applications in local, global, and multi-criteria optimization and classification (see, e.g., [4,9,10,13] and references given therein). Then, we can indicate game theory (see, e.g., [12,16]), probability theory (see, e.g., [7,31,32,33]), fractals (see, e.g., [3,6,37]), infinite series (see [38,41,45]), Turing machines, cellular automata, and ordering (see, e.g., [11,33,36,42]), numerical differentiation and numerical solution of ordinary differential equations (see, e.g., [1,14,15,22] and references given therein), etc. ...

It is well known that the set of algebraic numbers (let us call it $A$) is countable. In this paper, instead of the usage of the classical terminology of cardinals proposed by Cantor, a recently introduced methodology using \G1-based infinite numbers is applied to measure the set $A$ (where the number \G1 is called \emph{grossone}). Our interest to this methodology is explained by the fact that in certain cases where cardinals allow one to say only whether a set is countable or it has the cardinality of the continuum, the \G1-based methodology can provide a more accurate measurement of infinite sets. In this article, lower and upper estimates of the number of elements of $A$ are obtained. Both estimates are expressed in \G1-based numbers.

... First of all, we mention numerous applications in local, global, and multicriteria optimization and classification (see, e.g., [4,9,10,13] and references given therein). Then, we can indicate game theory (see, e.g., [12,16]), probability theory (see, e.g., [7,[31][32][33]), fractals (see, e.g., [3,6,37]), infinite series (see [38,41,45]), Turing machines, cellular automata, and ordering (see, e.g., [11,33,36,42]), numerical differentiation and numerical solution of ordinary differential equations (see, e.g., [1,14,15,22] and references given therein), etc. ...

It is well known that the set of algebraic numbers (let us call it A) is countable. In this paper, instead of the usage of the classical terminology of cardinals proposed by Cantor, a recently introduced methodology using 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textcircled {1}$$\end{document}-based infinite numbers is applied to measure the set A (where the number 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textcircled {1}$$\end{document} is called grossone). Our interest to this methodology is explained by the fact that in certain cases where cardinals allow one to say only whether a set is countable or it has the cardinality of the continuum, the 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textcircled {1}$$\end{document}-based methodology can provide a more accurate measurement of infinite sets. In this article, lower and upper estimates of the number of elements of A are obtained. Both estimates are expressed in 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textcircled {1}$$\end{document}-based numbers.

... The dedicated web page [26] developed at the University of East Anglia, UK contains, among other things, a comprehensive teaching manual and a nice animation related to the Hilbert's paradox of the Grand Hotel. Then, we can indicate game theory and probability (see, e.g., [8,12,18,19,46,49,50]); local, global, and multiple criteria optimization (see [9,[13][14][15]22,55,61,67]), hyperbolic geometry and percolation (see [31,42]), fractals (see [4,6,57]), infinite series (see [58,66]), Turing machines, cellular automata, and supertasks (see [11,48,50,60]), numerical differentiation and numerical solution of ordinary differential equations (see [2,16,17,29]), etc. ...

In this article, some classical paradoxes of infinity such as Galileo’s paradox, Hilbert’s paradox of the Grand Hotel, Thomson’s lamp paradox, and the rectangle paradox of Torricelli are considered. In addition, three paradoxes regarding divergent series and a new paradox dealing with multiplication of elements of an infinite set are also described. It is shown that the surprising counting system of an Amazonian tribe, Pirahã, working with only three numerals (one, two, many) can help us to change our perception of these paradoxes. A recently introduced methodology allowing one to work with finite, infinite, and infinitesimal numbers in a unique computational framework not only theoretically but also numerically is briefly described. This methodology is actively used nowadays in numerous applications in pure and applied mathematics and computer science as well as in teaching. It is shown in the article that this methodology also allows one to consider the paradoxes listed above in a new constructive light.

... Выполняя переупорядочение, мы заметим, что после добавления /2 выделенных скобками сумм 1 + 1 -1 у нас заканчиваются положительные единицы и остается только /2 отрицательных единиц, которые нам нужно добавить, чтобы использовать все имеющиеся слагаемые, то есть: Отрицательные единицы не были видны в традиционных системах записи и, таким образом, казалось, что перестановки слагаемых приводят к изменению результата. Более подробное обсуждение этого и других рядов можно найти в работах [18,43], где подробно обсуждаются различные результаты, полученные как традиционным, так и новым подходом (в частности, поклонники дзета-функции Римана найдут в этих работах несколько страниц, посвященных этой теме). ...

This article describes a recently proposed methodology that allows one to work with infinitely large and infinitely small quantities on a computer. The approach uses a number of ideas that bring it closer to modern physics, in particular, the relativity of mathematical knowledge and its dependence on the tools used by mathematicians in their studies are discussed. It is shown that the emergence of new computational tools influences the way we perceive traditional mathematical objects, and also helps to discover new interesting objects and problems. It is discussed that many difficulties and paradoxes regarding infinity do not depend on its nature, but are the result of the weakness of the traditional numeral systems used to work with infinitely large and infinitely small quantities. A numeral system is proposed that not only allows one to work with these quantities analytically in a simpler and more intuitive way, but also makes possible practical calculations on the Infinity Computer, patented in a number of countries. Examples of measuring infinite sets with the accuracy of one element are given and it is shown that the new methodology avoids the appearance of some well-known paradoxes associated with infinity. Examples of solving a number of computational problems are given and some results of teaching the described methodology in Italy and Great Britain are discussed.

... Besides, this computational methodology is not linked to nonstandard analysis (see [14]). Recent developments have shown that grossone methodology provides access to a myriad of new practical possibilities, such as Infinity Computer patented in several countries (see [5,6]), optimization (see [15][16][17][18][19][20][21][22]) and going through infinite series (see [11,[23][24][25][26]), fractals and cellular automata (see [8,23,[27][28][29][30][31][32]), hyperbolic geometry and percolation (see [33][34][35]), the first Hilbert problem and Turing machines (see [11,36]), infinite decision making processes, game theory, and probability (see [37][38][39][40]), numerical differentiation and ordinary differential equations (see [9,10,41,42]), etc. ...

The problem of diffusion through the fluctuating medium, where processes of substance disintegration and reproduction are possible, was posed at the end of the last century. It was established that the action of multiplicative external noise on a system can result in qualitative reorganization of its dynamical behavior. When such reorganization leads to the appearance of a new stationary dynamic mode, it is customary to speak about a noise-induced phase or kinetic transition. In this paper the noise-induced kinetic transition in two-component environment where the interacting components have contrasting lifetimes and diffusion coefficients is considered. It is shown that the presence of an additional long-lived component can lead to a dramatic decrease in the system generation threshold. We called this effect the depository reproduction. Analytical consideration of the diffusion process in a fluctuating medium causes enormous difficulties even for a single component substance. Meanwhile, in some cases of practical interest, the problem consideration can be conducted using stochastic geometry and percolation theory in particular. In the present work the noise-induced kinetic transition in two-component distributed systems is studied by the tools of directed percolation. To present the depository reproduction effect more vividly we use a new numeral grossone that allows to express different infinitesimal and infinite numerals. It was shown that the reverse conversion of the long-lived component to the short-lived one ensures the survival of the system at significantly lower concentrations of production centers.

... In particular, in the papers [36,44] , it has been shown that the Infinity Computer is much faster than symbolic computations allowing one to obtain exact 2 solutions for different real-life problems as symbolic computations do. It has already been successfully applied for solving different real-world engineering problems: in handling ill-conditioning (see, e.g., [32,56] ), in optimization (see, e.g., [10,11,[15][16][17]64] ), differentiation and numerical solution to ODEs [3,35,36,49,50,55] , game theory [28,47] , probability and infinite series [9,63] , Turing machines and cellular automata [13,53] , etc. These aspects make the software solution particularly suitable for simulating real physical systems composed of heterogeneous components, also operating in distributed environments [24,31] . ...

This paper is dedicated to numerical computation of higher order derivatives in Simulink. In this paper, a new module has been implemented to achieve this purpose within the Simulink-based Infinity Computer solution, recently introduced by the authors. This module offers several blocks to calculate higher order derivatives of a function given by the arithmetic operations and elementary functions. Traditionally, this can be done in Simulink using finite differences only, for which it is well-known that they can be characterized by instability and low accuracy. Moreover, the proposed module allows to calculate higher order Lie derivatives embedded in the numerical solution to Ordinary Differential Equations (ODEs). Traditionally, Simulink does not offer any practical solution for this case without using difficult external libraries and methodologies, which are domain-specific, not general-purpose and have their own limitations. The proposed differentiation module bridges this gap, is simple and does not require any additional knowledge or skills except basic knowledge of the Simulink programming language. Finally, the block for constructing the Taylor expansion of the differentiated function is also proposed, adding so another efficient numerical method for solving ODEs and for polynomial approximation of the functions. Numerical experiments on several classes of test problems confirm advantages of the proposed solution.

... Besides the computation of derivatives, the ①-based methodology has been successfully applied in several areas of Mathematics and Computer Science: e.g., in optimization (see [27][28][29][30][31][32][33]) and going through infinite series (see, e.g., [24,34]), in modelling and numerical simulation [3,15,35,36], fractals and cellular automata (see [37,38]), the first Hilbert problem and Turing machines (see [24,39]), infinite decision making processes, game theory, and probability (see [40][41][42][43]), etc. ...

In this paper, we deal with the computation of Lie derivatives, which are required, for example, in some numerical methods for the solution of differential equations. One common way for computing them is to use symbolic computation. Computer algebra software, however, might fail if the function is complicated, and cannot be even performed if an explicit formulation of the function is not available, but we have only an algorithm for its computation. An alternative way to address the problem is to use automatic differentiation. In this case, we only need the implementation of the algorithm that evaluates the function in terms of its analytic expression in a programming language, but we cannot use this if we have only a compiled version of the function. In this paper, we present a novel approach for calculating the Lie derivative of a function, even in the case where its analytical expression is not available, that is based on the Infinity Computer arithmetic. A comparison with symbolic and automatic differentiation shows the potentiality of the proposed technique.

... Recently, there has been a large amount of research activity on the logical theory and applications of grossone. To name a few, see Caldarola (2018) Iudin et al. (2015), Lolli (2015), Margenstern (2011), Rizza (2019, Rizza (2018), Montagna et al. (2015), Sergeyev and Garro (2010), and Zhigljavsky (2012). This next section will describe a new application of grossone to infinite games. 2 ...

In his seminal work, Robert McNaughton [see McNaughton (Ann Pure Appl Log 65:149–184, 1993) and Khoussainov and Nerode (Automata theory and its applications. Birkhauser, Basel, 2001)] developed a model of infinite games played on finite graphs. This paper presents a new model of infinite games played on finite graphs using the grossone paradigm. The new grossone paradigm provides certain advantages such as allowing for draws, which are common in board games, and a more accurate and decisive method for determining the winner.

... He calls the system consisting of all such expressions as related algebraically via the two initials schematized above the Primary Arithmetic. He then introduces variables that may take such expressions as values and calls the system of expanded expressions the Primary Algebra, demonstrating standard results of consistency Evidence of the efficacy of the grossone approach is highlighted by its successful application to many fields of applied mathematics, including optimization (see Cococcioni et al. 2018Cococcioni et al. , 2020De Cosmis and De Leone 2012;Sergeyev et al. 2018;Iavernaro et al. 2020;Iudin et al. 2012;Sergeyev 2007Sergeyev , 2013bZhigljavsky 2012), fractals and cellular automata (see Caldarola 2018;D'Alotto 2012D'Alotto , 2015D'Alotto , 2013 as well as infinite decision-making processes, game theory, and probability (see Fiaschi and Cococcioni 2018;Rizza 2018Rizza , 2019, whereas the formal logical foundation of grossone has been investigated in Lolli (2012); Margenstern (2011); Montagna et al. (2015). The approach presented in this paper bears similarities to the application of grossone to Turing machines as described in Sergeyev and Garro (2010). ...

In this paper, we investigate some aspects of Spencer–Brown’s Calculus of Indications. Drawing from earlier work by Kauffman and Varela, we present a new categorical framework that allows to characterize the construction of infinite arithmetic expressions as sequences taking values in grossone.

... The mentioned computational system was introduced by Y.D. Sergeyev in the early 2000's: we refer the reader to [35,38,41,45] for detailed introductory surveys on the subject showing how to work numerically with infinite and infinitesimals numbers in a very easy and handy way, or to the book [36] as well, written in a popular manner. Such a new computational methodology has found recently many applications in a number of theoretical and computational research areas as optimization theory and numerical differentiation (see [15,18,27,40,53]), fractals (see [11,12,37,39,44,46]), hyperbolic geometry (see [26]), cellular automata and complex systems (see [16,17] and in the context of [5][6][7] under investigation), numerical series and Z-transform (see [14,52]), Hilbert problems, Turing machines and supertasks (see [30,32,42,47]), numerical solution of ordinary differential equations (see [27,40,43,48]), or even for speculative-didactic purposes (see [3,13,23,31]). ...

The Peano and the Hilbert curves, denoted by P and H respectively, are historically the first and some of the best known space-filling curves. They have a fractal structure, many variants (as the well-known Moore curve M or a probably new “looped” version \({\overline{H}}\) of H), and a huge number of applications in the most diverse fields of mathematics and experimental sciences. In this paper, we employ a recently proposed computational system, allowing numerical calculations with infinite and infinitesimal numbers, to investigate the behavior of such curves and to highlight the differences with the classical treatment. In particular, we perform several types of computations and give many examples based not only on the curves H and P, but also on their d-dimensional versions \(H^d\) and \(P^d\), respectively. Following our approach, it is easy to apply this new computational methodology to many other geometrical contexts, with interesting advantages such as summarizing in a single (infinite) number, representing the final result of a sequence of computations, much information both on the geometrical meaning of such a sequence and on the base geometrical structure itself.

... From the foundational point of view, grossone has been introduced as an infinite unit of measure equal to the number of elements of the set N of natural numbers (notice that the x-based computational methodology is not related to non-standard analysis (see [11]) and its non-contradictory has been studied in depth in [12,13,14]). From the practical point of view, this methodology has given rise both to a new supercomputer patented in several countries (see [15]) and called Infinity Computer and to a variety of applications starting from optimization (see [1,16,17,18,19,20,21,22]) and going through infinite series (see [2,23,24,25,26]), fractals and cellular automata (see [23,27,28,29,30,31]), hyperbolic geometry and percolation (see [32,33,34]), the first Hilbert problem and Turing machines (see [2,35,36]), infinite decision making processes, game theory, and probability (see [37,38,39,40,41,42]), numerical differentiation and ordinary differential equations (see [43,44,45,46,47]), etc. ...

In the previous work (see [1]) the authors have shown how to solve a Lexicographic Multi-Objective Linear Programming (LMOLP) problem using the Grossone methodology described in [2]. That algorithm, called GrossSimplex, was a generalization of the well-known simplex algorithm, able to deal numerically with infinitesimal/infinite quantities.
The aim of this work is to provide an algorithm able to solve a similar problem, with the addition of the constraint that some of the decision variables have to be integer. We have called this problem LMOMILP (Lexicographic Multi-Objective Mixed-Integer Linear Programming).
This new problem is solved by introducing the GrossBB algorithm, which is a generalization of the Branch-and-Bound (BB) algorithm. The new method is able to deal with lower-bound and upper-bound estimates which involve infinite and infinitesimal numbers (namely, Grossone-based numbers). After providing theoretical conditions for its correctness, it is shown how the new method can be coupled with the GrossSimplex algorithm described in [1], to solve the original LMOMILP problem. To illustrate how the proposed algorithm finds the optimal solution, a series of LMOMILP benchmarks having a known solution is introduced. In particular, it is shown that the GrossBB combined with the GrossSimplex is able solve the proposed LMOMILP test problems with up to 200 objectives.

... x . The x-based methodology has been successfully applied in several areas of Mathematics and Computer Science: single and multiple criteria optimization (see [8,11,12]), handling ill-conditioning (see [13,37]), numerical differentiation and solution of ordinary differential equations (see [1,38,40,31]), cellular automata (see [9,10]), Euclidean and hyperbolic geometry (see [22]), percolation (see [19,42]), fractals (see [5,33,42]), infinite series and the Riemann zeta function (see [32,34,43]), the first Hilbert problem and supertasks (see [24,29,34]), Turing machines and probability (see [25,34,35,36]), etc. ...

Multi-derivative one-step methods based upon Euler-Maclaurin integration formulae are considered for the solution of canonical Hamiltonian dynamical systems. Despite the negative result that simplecticity may not be attained by any multi-derivative Runge-Kutta methods, we show that Euler-MacLaurin formulae are all topologically conjugate to a symplectic formula. This feature entitles them to play a role in the context of geometric integration and, to make their implementation competitive with the existing integrators, we explore the possibility of computing the underlying higher order derivatives with the aid of the Infinity Computer.

... The ①-based methodology has been successfully applied in several areas of Mathematics and Computer Science: single and multiple criteria optimization and ill-conditioning (see Cococcioni et al. 2018;De Cosmis and De Leone 2012;De Leone 2018;Gaudioso et al. 2018;Sergeyev et al. 2018), cellular automata (see D'Alotto 2012D'Alotto , 2013, Euclidean and hyperbolic geometry (see Margenstern 2015), percolation (see Iudin et al. 2015;Vita et al. 2012), fractals (see Caldarola 2018;Sergeyev 2007Sergeyev , 2009aSergeyev , 2011cSergeyev , 2016Vita et al. 2012), infinite series and the Riemann zeta function (see Sergeyev 2009bSergeyev , 2011bSergeyev , 2015bSergeyev , 2017Sergeyev , 2018Zhigljavsky 2012), the first Hilbert problem and supertasks (see Rizza 2016;Sergeyev 2010aSergeyev , 2017, Turing machines and probability (see Rizza 2017;Sergeyev 2017;Garro 2010, 2013), numerical differentiation and solution of ordinary differential equations (see Amodio et al. 2017;Sergeyev 2011aSergeyev et al. 2016), etc. However, in the paper (Gutman et al. 2017) published in Foundations of Science 2 and in the papers (Gutman and Kutateladze 2008;Kutateladze 2011) published in two journals printed by the Institute where the authors of (Gutman and Kutateladze 2008;Kutateladze 2011) work, there are numerous attacks on the ①-based methodology and its author. ...

This commentary considers non-standard analysis and a recently introduced computational methodology based on the notion of \G1 (this symbol is called \emph{grossone}). The latter approach was developed with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework and in all the situations requiring these notions. Non-standard analysis is a classical purely symbolic technique that works with ultrafilters, external and internal sets, standard and non-standard numbers, etc. In its turn, the \G1-based methodology does not use any of these notions and proposes a more physical treatment of mathematical objects separating the objects from tools used to study them. It both offers a possibility to create new numerical methods using infinities and infinitesimals in floating-point computations and allows one to study certain mathematical objects dealing with infinity more accurately than it is done traditionally. In these notes, we explain that even though both methodologies deal with infinities and infinitesimals, they are independent and represent two different philosophies of Mathematics that are not in a conflict. It is proved that texts \cite{Flunks, Gutman_Kutateladze_2008, Kutateladze_2011} asserting that the \G1-based methodology is a part of non-standard analysis unfortunately contain several logical fallacies. Their attempt to prove that the \G1-based methodology is a part of non-standard analysis is similar to trying to show that constructivism can be reduced to the traditional mathematics.

... Numerical computations with functions that can assume infinite and infinitesimal values are executed using the Infinity Computing paradigm allowing one to work numerically with a variety of infinities and infinitesimals on a patented in Europe and USA new supercomputer called the Infinity Computer (see, e.g., surveys [24,28]). This computational methodology has already been successfully applied in optimization and numerical differentiation [3,5,6,26] and in a number of other theoretical and applied research areas such as, e.g., cellular automata [4], hyperbolic geometry [16], percolation [13], fractals [2,27], infinite series [25,38], Turing machines [22], numerical solution of ordinary differential equations [1,33], etc. In particular, in the recent paper [9], numerical infinities and infinitesimals from [24,28] have been successfully used to handle ill-conditioning in a multidimensional optimization problem. ...

The necessity to find the global optimum of multiextremal functions arises in many applied problems where finding local solutions is insufficient. One of the desirable properties of global optimization methods is strong homogeneity meaning that a method produces the same sequences of points where the objective function is evaluated independently both of multiplication of the function by a scaling constant and of adding a shifting constant. In this paper, several aspects of global optimization using strongly homogeneous methods are considered. First, it is shown that even if a method possesses this property theoretically, numerically very small and large scaling constants can lead to ill-conditioning of the scaled problem. Second, a new class of global optimization problems where the objective function can have not only finite but also infinite or infinitesimal Lipschitz constants is introduced. Third, the strong homogeneity of several Lipschitz global optimization algorithms is studied in the framework of the Infinity Computing paradigm allowing one to work numerically with a variety of infinities and infinitesimals. Fourth, it is proved that a class of efficient univariate methods enjoys this property for finite, infinite and infinitesimal scaling and shifting constants. Finally, it is shown that in certain cases the usage of numerical infinities and infinitesimals can avoid ill-conditioning produced by scaling. Numerical experiments illustrating theoretical results are described.

... This approach proposes a numeral system that allows one to use the same numerals in all the occasions we need infinities and infinitesimals. There are applications in numerical solution of ordinary differential equations (see [136,137,138,139]), the first Hilbert problem, Turing machines, and lexicographic ordering (see [140,141,142]), hyperbolic geometry, fractals, and percolation (see [143,144,145,146,147,148]), single and multiple criteria optimization (see [149,150,151,152]), infinite series and the Riemann zeta function (see [153,154,155]), cellular automata (see [156]), etc. ...

Unconventional computing is about breaking boundaries in thinking, acting and computing. Typical topics of this non-typical field include, but are not limited to physics of computation, non-classical logics, new complexity measures, novel hardware, mechanical, chemical and quantum computing. Unconventional computing encourages a new style of thinking while practical applications are obtained from uncovering and exploiting principles and mechanisms of information processing in and functional properties of, physical , chemical and living systems; in particular, efficient algorithms are developed , (almost) optimal architectures are designed and working prototypes of future computing devices are manufactured. This article includes idiosyn-cratic accounts of 'unconventional computing' scientists reflecting on their personal experiences, what attracted them to the field, their inspirations and discoveries.

... We embed into the treatment of the optimization problem (1) a recently introduced computational methodology allowing one to work numerically with infinities and infinitesimals in a handy way (see the detailed survey [33] and the informative book [36]). This computational methodology has already been successfully applied in optimization and numerical differentiation [8,42], and in a number of other theoretical and computational research areas such as cellular automata [7], Euclidean and hyperbolic geometry [28], percolation [19], fractals [37], infinite series and the Riemann zeta function [41], the first Hilbert problem, Turing machines, and supertasks [31], numerical solution of ordinary differential equations [1,29,38], etc. ...

The objective of the paper is to evaluate the impact of the infinity computing paradigm on practical solution of nonsmooth unconstrained optimization problems, where the objective function is assumed to be convex and not necessarily differentiable. For such family of problems, the occurrence of discontinuities in the derivatives may result in failures of the algorithms suited for smooth problems.
We focus on a family of nonsmooth optimization methods based on a variable metric approach, and we use the infinity computing techniques for numerically dealing with some quantities which can assume values arbitrarily small or large, as a consequence of nonsmoothness. In particular we consider the case, treated in the literature, where the metric is defined via a diagonal matrix with positive entries.
We provide the computational results of our implementation on a set of benchmark test-problems from scientific literature.

... In the present paper it is proposed to analyze LMOLP problems using a recently introduced computational methodology allowing one to work numerically with infinities and infinitesimals in a handy way (see for a detailed introduction surveys [19,23,28,30,32] and the book [17] written in a popular way). This computational methodology has already been successfully applied in optimization and numerical differentiation (see [3,6,24,40]) and in a number of other theoretical and computational research areas such as cellular automata (see [4,5]), Euclidean and hyperbolic geometry (see [11,12]), percolation (see [8,9,38]), fractals (see [18,20,26,31,38]), infinite series and the Riemann zeta function (see [21,25,39]), the first Hilbert problem, Turing machines, and supertasks (see [15,22,33,34]), numerical solution of ordinary differential equations (see [1,13,27,35]), etc. ...

Numerous problems arising in engineering applications can have several objectives to be satisfied. An important class of problems of this kind is lexicographic multi-objective problems where the first objective is incomparably more important than the second one which, in its turn, is incomparably more important than the third one, etc. In this paper, Lexicographic Multi-Objective Linear Programming (LMOLP) problems are considered. To tackle them, traditional approaches either require solution of a series of linear programming problems or apply a scalarization of weighted multiple objectives into a single-objective function. The latter approach requires finding a set of weights that guarantees the equivalence of the original problem and the single-objective one and the search of correct weights can be very time consuming. In this work a new approach for solving LMOLP problems using a recently introduced computational methodology allowing one to work numerically with infinities and infinitesimals is proposed. It is shown that a smart application of infinitesimal weights allows one to construct a single-objective problem avoiding the necessity to determine finite weights. The equivalence between the original multi-objective problem and the new single-objective one is proved. A simplex-based algorithm working with finite and infinitesimal numbers is proposed, implemented, and discussed. Results of some numerical experiments are provided.

... On the other hand, the new methodology has been successfully applied for studying cellular automata (see [D'Alotto (2012[D'Alotto ( , 2013[D'Alotto ( , 2015]), Euclidean and hyperbolic geometry (see [Margenstern (2012[Margenstern ( , 2015]), percolation (see [Iudin et al. (2012[Iudin et al. ( , 2015; Vita et al. (2012)]), fractals (see [Sergeyev (2007[Sergeyev ( , 2009a[Sergeyev ( , 2011c[Sergeyev ( , 2015a; Vita et al. (2012)]), numerical differentiation and optimization (see [De Cosmis and De Leone (2012); Sergeyev (2009bSergeyev ( , 2011a;Žilinskas (2012)]), infinite series and the Riemann zeta function (see [Sergeyev (2009c[Sergeyev ( , 2011b; Zhigljavsky (2012)]), the first Hilbert problem, Turing machines, and lexicographic ordering (see [Sergeyev (2010b); Sergeyev and Garro (2010; Sergeyev (2015c)]), ordinary differential equations (see [Sergeyev ( , 2015b]), etc. The interested reader is invited to have a look also at surveys [Sergeyev (2008a[Sergeyev ( , 2010c] and the book [Sergeyev (2003[Sergeyev ( , 2d electronic ed. ...

... In particular, relations of the new approach to bijections are studied in [12] and metamathematical investigations on the new theory and its non-contradictory can be found in [10]. On the other hand, the new methodology has been successufully applied in many areas such as cellular automata [2][3][4], Euclidean and hyperbolic geometry [13,14], percolation [6,7,41], fractals [18,21,28,34,41], numerical differentiation and optimization [26,44], infinite series and the Riemann zeta function [23,27,43], the first Hilbert problem, Turing machines, and lexicographic ordering [24,[35][36][37]32], ordinary differential equations [1,29,31,38], etc. The interested reader is invited to have refer to the surveys [20,25] and to the book [17] for a introduction for general public. ...

A novel and interesting approach to infinite and infinitesimal numbers was recently proposed in a series of papers and a book by Y. Sergeyev. This novel numeral system is based on the use of a new infinite unit of measure (the number grossone, indicated by the numeral ①), the number of elements of the set, N, of natural numbers. Based on the use of ①, De Cosmis and De Leone [5] have proposed a new exact differentiable penalty function for constrained optimization problems. In this paper these results are specialized to the important case of quadratic problems with linear constraints. Moreover, the crucial role of Constraint Qualification conditions (well know in constraint minimization literature) is also discussed with reference to the new proposed penalty function. Keywords Nonlinear optimization · Grossone · Penalty Functions In a series of papers and in a book [17,19,22,39], Yaroslav Sergeyev proposed an interesting and fresh approach to infinite and infinitesimal numbers whose peculiar characteristic is the attention to numerical aspects and to applications. This novel numeral system is based on the use of a new infinite unit of measure (the numeral grossone, indicated by ①), the number of elements of the set, N, of natural numbers. Grossone is introduced through the following three properties: – Infinity. Any finite natural number n is less than grossone, i.e., n < ①. – Identity. The following relationships link ① to the identity elements 0 end 1 0 · ① = ① · 0 = 0, ① − ① = 0, ① ① = 1, ① 0 = 1, 1 ① = 1, 0 ① = 0 (1)

... In particular, connections of the new approach with bijections is studied in [28] and metamathematical investigations on the new theory and its non-contradictory can be found in [27]. The new methodology has been successfully applied for studying percolation and biological processes (see [21,22,62,48]), infinite series (see [46,64]), hyperbolic geometry (see [29,30]), fractals (see [21,22,40,45,48]), numerical differentiation and optimization (see [15,54]), the first Hilbert problem, Turing machines, and lexicographic ordering (see [52,58,59,60,55]), cellular automata (see [13,14]), ordinary differential equations (see [56,57]), etc. ...

In this chapter, a number of traditional models related to the percolation theory is taken into consideration: site percolation, gradient percolation, and forest-fire model. They are studied by means of a new computational methodology that gives a possibility to work with finite, infinite, and infinitesimal quantities numerically by using a new kind of a computer—the Infinity Computer
—introduced recently. It is established that in light of the new arithmetic using grossone-based numerals the phase transition point in site percolation and gradient percolation appears as a critical interval, rather than a critical point. Depending on the ‘microscope’ we use, this interval could be regarded as finite, infinite, or infinitesimal interval. By applying the new approach we show that in vicinity of the percolation threshold we have many different infinite clusters
instead of one infinite cluster that appears in traditional considerations. With respect to the cellular automaton forest-fire model, two traditional versions of the model are studied: a real forest-fire model where fire catches adjacent trees in the forest in the step by step manner and a simplified version with instantaneous combustion. By applying the new approach there is observed that in both situations we deal with the same model but with different time resolutions. We show that depending on ‘microscope’ we use, the same cellular automaton forest-fire model reveals either the instantaneous forest combustion or the step by step firing.

... The new methodology has been successfully used in several fields. We can mention numerical differentiation and optimization (see [8,38,52]), models for percolation and biological processes (see [17,18,40,50]), hyperbolic geometry (see [25,26]), fractals (see [17,18,30,33,40,47]), infinite series (see [19,34,39,51]), the first Hilbert problem, lexicographic ordering, and Turing machines (see [36,43,45,48,49]), cellular automata (see [9][10][11]), etc. ...

New algorithms for the numerical solution of Ordinary Differential Equations (ODEs) with initial condition are proposed. They are designed for work on a new kind of a supercomputer – the Infinity Computer, – that is able to deal numerically with finite, infinite and infinitesimal numbers. Due to this fact, the Infinity Computer allows one to calculate the exact derivatives of functions using infinitesimal values of the stepsize. As a consequence, the new methods described in this paper are able to work with the exact values of the derivatives, instead of their approximations.

... In particular, connections of the new approach with bijections are studied in [19] and metamathematical investigations on the theory and its non-contradictory identification can be found in [18]. The new methodology has been successfully used in such fields as numerical differentiation and optimization (see [9,39,56]), fractals (see [13,14,31,34,41,48]), models for percolation and biological processes (see [13,14,52,41]), hyperbolic geometry (see [20,21]), infinite series (see [16,35,40,55]), set theory, lexicographic ordering, and Turing machines (see [37,46,44,49,50,50]), cellular automata (see [10,11,12]), etc. ...

A well-known drawback of algorithms based on Taylor series formulae is that the explicit calculation of higher order derivatives formally is an over-elaborate task. To avoid the analytical computation of the successive derivatives, numeric and automatic differentiation are usually used. A recent alternative to these techniques is based on the calculation of higher derivatives by using the Infinity Computer — a new computational device allowing one to work numerically with infinities and infinitesimals. Two variants of a one-step multi-point method closely related to the classical Taylor formula of order three are considered. It is shown that the new formula is order three accurate, though requiring only the first two derivatives of (rather than three if compared with the corresponding Taylor formula of order three). To get numerical evidence of the theoretical results, a few test problems are solved by means of the new methods and the obtained results are compared with the performance of Taylor methods of order up to four.

In this paper, we present a summary of Homotopy Type Theory with Voevodsky’s Univalent Axiom and discuss the existence of a universe of types satisfying the latter. We take up the idea of the observability of the behavior of computations, as defined in the grossone framework, in which for every finite natural number n, . In this context, the number of complete computable sequences that can be enumerated by imaginary Turing machines is equal to . We replace the notion of ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document}-topos of ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document}-groupoids by a -topos of -groupoids as a model of Type Theory. We find that in this setting there exists a universe of types satisfying univalence, ensuring the completeness of the sequences generated by type-theoretic computations.

There exist many applications where it is necessary to approximate numerically derivatives of a function f(x) which is given by a computer procedure. A novel way to efficiently compute exact derivatives (the word “exact” means here with respect to the accuracy of the implementation of f(x)) is presented in this Chapter. It uses a new kind of a supercomputer—the Infinity Computer—able to work numerically with different finite, infinite, and infinitesimal numbers. Numerical examples illustrating these concepts and numerical tools are given. In particular, the field of Lipschitz global optimization having a special interest in exact numerical differentiation is considered in cases where there exists a code for computing f(x) but a code for its derivative \(f'(x)\) is not available. In addition, it is supposed that the first derivative \(f'(x)\) satisfies the Lipschitz condition. Algorithms using smooth piece-wise quadratic support functions and their convergence conditions are discussed. All the methods are implemented both in the traditional floating-point arithmetic and in the Infinity Computing framework.

In his seminal work, Robert McNaughton (see [14] and [10]) developed a model of infinite games played on finite graphs. Here is presented a new model of infinite games played on finite graphs using the Grossone paradigm. The new Grossone model provides certain advantages such as allowing for draws, which are common in board games, and a more accurate and decisive method for determining the winner.

We discuss the fruitful impact of the infinity computing paradigm on the practical solution of convex nonsmooth optimization problems. We consider a class of unconstrained nonsmooth optimization methods based on a variable metric approach, where the use of the infinity computing techniques allows one to numerically deal with quantities which can take arbitrarily small or large values, as a consequence of nonsmoothness. In particular, choosing a diagonal matrix with positive entries as a metric, we modify the so called Diagonal Bundle algorithm by means of matrix updates based on the infinity computing paradigm, and we provide the computational results obtained on a set of benchmark academic test-problems.

The present chapter studies the impact of scaling on global optimization algorithms. In particular, the notion of strong homogeneity is under study. A method is strongly homogeneous if it produces the same sequences of evaluation points independently both of multiplication of the objective function by a scaling constant and of adding a shifting constant. It is shown that even if a method possesses this property theoretically, numerically very small and large scaling constants can lead to ill-conditioning of the scaled problem. A new class of global optimization problems where the objective function can have not only finite but also infinite or infinitesimal Lipschitz constants is described. The strong homogeneity of several Lipschitz global optimization algorithms is then addressed within the Infinity Computing framework. It is finally shown that the usage of numerical infinities and infinitesimals can in certain cases avoid ill-conditioning produced by scaling.

This Chapter surveys a recent computational methodology allowing one to work with infinities and infinitesimals numerically on a supercomputer called the Infinity Computer that has been patented in several countries. This methodology applies the principle The whole is greater than the part to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). It is shown that, from the theoretical point of view, the methodology allows one to consider infinite and infinitesimal quantities more accurately w.r.t. traditional approaches such as Cantor’s cardinals and non-standard analysis of Robinson. On the other hand, the methodology has a pronounced numerical character that gives an opportunity to construct algorithms of a completely new type and run them on the Infinity Computer (there already exist numerous applications not only in optimization but also in numerical differentiation, ODEs, game theory, etc.). In this Chapter, some key aspects of the methodology are described and several examples are given. Due to the breadth of the subject, the main attention is limited to methodological and practical aspects required to grasp applications in optimization contained in the other Chapters of the book. The material is presented in an accessible form that does not require any additional knowledge exceeding the first year university course of mathematical analysis.

Numerical computing represents a critical aspect ofconventional computer architecture. Traditional computers adopt the IEEE 754-1985 binary floating-point standard to represent andwork with real numbers. Due to the architectural limitations of traditional computers, it is impossible to handle infinite and infinitesimal quanti- ties numerically. This chapter is devoted to the Infinity Computer, a supercomputer that permits to execute numerical computation with finite, infinite, and infinitesimal numbers. The accessible software simulator of the Infinity Computer is adopted in many industrial and research domains for addressing important real-world issues, where precision plays a crucial aspect. However, the Infinity Computer simulator is not suitable for handling problems in control theory and dynamics, where visual programming environments like Simulink are commonly used. In this context, the chapter presents the Simulink-based Solution for the Infinity Computer, a novel solu- tion that allows one to exploit the Infinity Computer arithmetic within the Simulink environment.

We tackle the problem of separating two finite sets of samples by means of a spherical surface, focusing on the case where the center of the sphere is fixed. Such approach reduces to the minimization of a convex and nonsmooth function of just one variable (the radius), revealing very effective in terms of computational time. In particular, we analyze the case where the center of the sphere is selected far from both the two sets, embedding the grossone idea and obtaining a kind of linear separation. Some numerical results are presented on classical binary data sets drawn from the literature.

Numerical computing is a key part of the traditional computer architecture. Almost all traditional computers implement the IEEE 754-1985 binary floating point standard to represent and work with numbers. The architectural limitations of traditional computers make impossible to work with infinite and infinitesimal quantities numerically. This paper is dedicated to the Infinity Computer, a new kind of a supercomputer that allows one to perform numerical computations with finite, infinite, and infinitesimal numbers. The already available software simulator of the Infinity Computer is used in different research domains for solving important real-world problems, where precision represents a key aspect. However, the software simulator is not suitable for solving problems in control theory and dynamics, where visual programming tools like Simulink are used frequently. In this context, the paper presents an innovative solution that allows one to use the Infinity Computer arithmetic within the Simulink environment. It is shown that the proposed solution is user-friendly, general purpose, and domain independent.

We present here an approach to the analysis of the truth values of Peirce’s α-graphs without the restriction of finite number of elements (cuts and characters) on the Sheet of Assertion. We show that the ensuing structure in which such graphs are objects constitutes a topos. While the computation of the truth value of a graph in the topos can be an infinite process, we show that using the concept of grossone (①) the subobject classifier of the topos allows to determine a truth value for each graph.

In his seminal work, Robert McNaughton (see [1] and [7]) developed a model of infinite games played on finite graphs. This paper presents a new model of infinite games played on finite graphs using the Grossone paradigm. The new Grossone model provides certain advantages such as allowing for draws, which are common in board games, and a more accurate and decisive method for determining the winner when a game is played to infinite duration.

Multi-derivative one-step methods based upon Euler–Maclaurin integration formulae are considered for the solution of canonical Hamiltonian dynamical systems. Despite the negative result that simplecticity may not be attained by any multi-derivative Runge–Kutta methods, we show that the Euler–MacLaurin method of order p is conjugate-symplectic up to order p+2. This feature entitles them to play a role in the context of geometric integration and, to make their implementation competitive with the existing integrators, we explore the possibility of computing the underlying higher order derivatives with the aid of the Infinity Computer.

This paper proposes an application of the Infinite Unit Axiom and grossone, introduced by Yaroslav Sergeyev (see [19, 20, 21, 22, 23]), to the development and classification of two-dimensional cellular automata. This application establishes, by the application of grossone, a new and more precise nonarchimedean metric on the space of definition for two-dimensional cellular automata, whereby the accuracy of computations is increased. Using this new metric, open disks are defined and the number of points in each disk computed. The forward dynamics of a cellular automaton map are also studied by defined sets. It is also shown that using the Infinite Unit Axiom, the number of configurations that follow a given configuration, under the forward iterations of the cellular automaton map, can now be computed and hence a classification scheme developed based on this computation.

The Sierpinski curve is one of the most known space-filling curves and one with the highest number of applications. We present a recently proposed computational methodology based on the infinite quantity called grossone to investigate the behavior of two different constructions of the Sierpinski curve. We emphasize that, adopting this point of view, we have infinitely many Sierpinski curves depending, contrarily to traditional analysis, on each specific starting configuration. Of particular interest are some power series expansions in the new infinitesimal quantities emerging from the study of the considered curves.

This paper proposes an application of the Infinite Unit Axiom and grossone, introduced by Yaroslav Sergeyev (see [7] - [12]), to the development and classification of one and two-dimensional cellular automata. By the application of grossone, new and more precise nonarchimedean metrics on the space of definition for one and two-dimensional cellular automata are established. These new metrics allow us to do computations with infinitesimals. Hence configurations in the domain space of cellular automata can be infinitesimally close (but not equal). That is, they can agree at infinitely many places. Using the new metrics, open disks are defined and the number of points in each disk computed. The forward dynamics of a cellular automaton map are also studied by defined sets. It is also shown that using the Infinite Unit Axiom, the number of configurations that follow a given configuration, under the forward iterations of cellular automaton maps, can now be computed and hence a classification scheme developed based on this computation.

It is well known that traditional computers work numerically with finite numbers only and situations where a use of infinite or infinitesimal quantities is required are studied mainly theoretically by human beings. In this paper, a recently introduced computational methodology that has been proposed with the intention to change this differentiation is discussed. It is based on the principle ‘The part is less than the whole’ applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The methodology uses as a computational device the Infinity Computer (patented in USA, EU, and Russian Federation) working numerically with infinite and infinitesimal numbers that can be written using a numeral positional system with an infinite base. On a number of examples it is shown that it becomes possible both to execute computations of a new type and to simplify computations where infinity and/or infinitesimals are required.

A new computational methodology for computations with infinite and infinitesimal quantities is described. It is based on the principle ‘The part is less than the whole’ introduced by Ancient Greeks and observed in the physical world. It is applied to all sets and processes (finite and infinite) and all numbers (finite, infinite, and infinitesimal). It is shown that it becomes possible to work with all of them in a unique framework (different from non-standard analysis) allowing one to easily manage mathematical situations that traditionally create difficulties (divergences of various kind, indeterminate forms, etc.) and to construct mathematical models of physical phenomena of a new type.

This paper proposes an application of the Infinite Unit Axiom and grossone, introduced by Yaroslav Sergeyev (see [17]-[21]), to the development and classification of two-dimensional cellular automata. This application establishes, by the application of grossone, a new and more precise nonarchimedean metric on the space of definition for two-dimensional cellular automata, whereby the accuracy of computations is increased. Using this new metric, open disks are defined and the number of points in each disk computed. The forward dynamics of a cellular automaton map are also studied by defined sets. It is also shown that using the Infinite Unit Axiom of Sergeyev, the number of configurations that follow a given configuration, under the forward iterations of the cellular automaton map, can now be computed and hence a classification scheme developed based on this computation.

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 new methodology has allowed us to introduce the Infinity Computer working with such numbers (its simulator has already been realized). Examples dealing with divergent series, infinite sets, and limits are given.

A recently developed computational methodology for executing numerical calculations with infinities and infinitesimals is described. 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 uses strongly physical ideas emphasizing interrelations that hold between a mathematical object under observation and the tools used for this observation. It is shown how a new numeral system allowing one to express different infinite and infinitesimal quantities in a unique framework can be used for theoretical and computational purposes. Numerous examples dealing with infinite sets, divergent series, limits, and probability theory are given.

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 new methodology has
allowed us to introduce the Infinity Computer working with such numbers (its
simulator has already been realized). Examples dealing with divergent series,
infinite sets, and limits are given.

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 \textit{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
execute computations of a new type and open new horizons for creating new
mathematical models where a computational usage of infinite and/or
infinitesimal numbers can be useful. A number of numerical examples showing the
potential of the new approach and dealing with divergent series, limits,
probability theory, linear algebra, and calculation of volumes of objects
consisting of parts of different dimensions are given.

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 of a computer–the Infinity Computer–able to work numerically with all of them. An introduction to the theory of physical and mathematical continuity and differentiation (including subdifferentials) for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains is developed in the paper. This theory allows one to work with derivatives that can assume not only finite but infinite and infinitesimal values, as well. It is emphasized that the newly introduced notion of the physical continuity allows one to see the same mathematical object as a continuous or a discrete one, in dependence on the wish of the researcher, i.e., as it happens in the physical world where the same object can be viewed as a continuous or a discrete in dependence on the instrument of the observation used by the researcher. Connections between pure mathematical concepts and their computational realizations are continuously emphasized through the text. Numerous examples are given.

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Arithmetic of Infinity

- D Ya
- Sergeyev

Ya. D. Sergeyev, Arithmetic of Infinity, Edizioni Orizzonti Meridionali, CS, 2003.