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Published: 4 June 2021
Last updated: 29 July 2024

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1

Деев, Г. Е., and С. В. Ермаков. "Bi-Infinite Calculating Automaton." Успехи кибернетики / Russian Journal of Cybernetics, no. 3(11) (September 30, 2022): 52–62. http://dx.doi.org/10.51790/2712-9942-2022-3-3-6.

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на основе свойства экстравертности построен и рассмотрен абстрактный автомат, осуществляющий умножение на 3(4) в четверичной системе счисления; помимо этого, он вычисляет бесконечное число родственных операций. Умножитель на 3(4) взят для примера из-за его простоты. Устройство бесконечно, отчего оно является, в первую очередь, объектом теоретического исследования.Тем не менее оно имеет и практическую ценность, поскольку с его помощью обнаруживаются возможности реальных вычислительных процессов. В частности, решается вопрос о максимально быстрых вычислениях. Устройство по своей конструкции необычно, т.к. представляет собой Т-образный крест двух бесконечностей: бесконечности по состояниям («горизонтальная» бесконечность) и бесконечности по входному алфавиту («вертикальная» бесконечность), откуда и проистекает название: би-бесконечный. Аналогичные би-бесконечные устройства порождаются многими другими важнейшими вычислительными устройствами. Поэтому переход к би-бесконечности позволяет осуществить углубленное проникновение в суть вычислительных процессов. Конечные срезы всех би-бесконечных устройств реализуемы в В-технологии. using the concept of extroversion, we designed and studied an abstract automaton that performs multiplication by 3(4) in the quadratic number system; besides, it computes an infinite number of related operations. The multiplier by 3(4) is used as an example for simplicity. The device is infinite, so the research is mostly theoretical. Nevertheless, it also has some practical value because it reveals the capabilities of real-life computational processes. In particular, it helps find the fastest possible calculations. The device design is unusual. It is a T-shaped cross of two infinities: the infinity of the states (“horizontal”) and the infinity of the input alphabet (“vertical”). That is why the name: bi-infinity automation. Similar bi-infinite devices are generated by many other critical computing devices. Therefore, the transition to bi-infinity helps better understand the essence of computational processes. B-technology can implement some finite slices of each bi-infinite device.
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2

Holub, Štěpán. "Words with unbounded periodicity complexity." International Journal of Algebra and Computation 24, no. 06 (September 2014): 827–36. http://dx.doi.org/10.1142/s0218196714500362.

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If an infinite non-periodic word is uniformly recurrent or is of bounded repetition, then the limit of its periodicity complexity is infinity. Moreover, there are uniformly recurrent words with the periodicity complexity arbitrarily high at infinitely many positions.
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3

Love, William P. "Infinity: The Twilight Zone of Mathematics." Mathematics Teacher 82, no. 4 (April 1989): 284–92. http://dx.doi.org/10.5951/mt.82.4.0284.

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The concept of infinity has fascinated the human race for thousands of years. Who among us has never been awed by the mysterious and often paradoxical nature of the infinite? The ancient Greeks were fascinated by infinity, and they struggled with its nature. They left for us many unanswered questions including Zeno's famous paradoxes. The concept of infinity is with us today, and many ideas in modern mathematics are dependent on the infinitely large or the infinitely small. But most people's ideas about infinity are very vague and unclear, existing in that fuzzy realm of the twilight zone
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4

Friedlander, Alex. "Stories about Never-Ending Sums." Mathematics Teaching in the Middle School 15, no. 5 (December 2009): 274–80. http://dx.doi.org/10.5951/mtms.15.5.0274.

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Infinity and infinitely small numbers pique the curiosity of middle school students. Examples such as the story of Achilles and the Tortoise promote questions about domain, representations, and infinite sums–all of which may not get answered until students reach high school.
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5

Wood, Daniel A. "On the Implications of the Idea of Infinity for Postmodern Fundamental Theology." Pacifica: Australasian Theological Studies 25, no. 1 (February 2012): 67–81. http://dx.doi.org/10.1177/1030570x1202500106.

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This essay provides a dimensional analysis of the various manners in which mathematics, phenomenology, and theology claim to make present or mediate infinity. Edmund Husserl's 1935 lecture, Philosophy and the Crisis of European Humanity, because engaged with each discipline to various degrees, will function as our primary, preparatory text. Husserl's discussion of the ideal objects of mathematics and the Greek attitude will call for further analysis of the relation between mathematics and infinity. Similarly, intentional infinities, insofar as related to transcendental phenomenology, will be compared to Jean-Luc Marion's distinct phenomenology of the icon. Next, the ways in which the infinite God is conceptualised by Husserl and Marion will be juxtaposed in order to demonstrate their disparate, theological thinking. Finally, the notion of multiple infinities will be analogically extended from set theory to the discursive wholes of mathematics, phenomenology, and theology in order to suggest a novel understanding of the role of the infinite within postmodern fundamental theology.
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6

Katz, Mikhail, David Sherry, and Monica Ugaglia. "When Does a Hyperbola Meet Its Asymptote? Bounded Infinities, Fictions, and Contradictions in Leibniz." Revista Latinoamericana de Filosofía 49, no. 2 (November 9, 2023): 241–58. http://dx.doi.org/10.36446/rlf2023359.

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In his 1676 text De Quadratura Arithmetica, Leibniz distinguished infinita terminata from infinita interminata. The text also deals with the notion, originating with Desargues, of the point of intersection at infinite distance for parallel lines. We examine contrasting interpretations of these notions in the context of Leibniz’s analysis of asymptotes for logarithmic curves and hyperbolas. We point out difficulties that arise due to conflating these notions of infinity. As noted by Rodríguez Hurtado et al., a significant difference exists between the Cartesian model of magnitudes and Leibniz’s search for a qualitative model for studying perspective, including ideal points at infinity. We show how respecting the distinction between these notions enables a consistent interpretation thereof.
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7

Burgin, Mark. "Introduction to Hyperspaces." International Journal of Pure Mathematics 7 (February 8, 2021): 36–42. http://dx.doi.org/10.46300/91019.2020.7.5.

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The development of mathematics brought mathematicians to infinite structures. This process started with transcendent real numbers and infinite sequences going through infinite series to transfinite numbers to nonstandard numbers to hypernumbers. From mathematics, infinity came to physics where physicists have been trying to get rid of infinity inventing a variety of techniques for doing this. In contrast to this, mathematicians as well as some physicists suggested ways to work with infinity introducing new mathematical structures such distributions and extrafunctions. The goal of this paper is to extend mathematical tools for treating infinity by considering hyperspaces and developing their theory.
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8

Strydom, Piet. "Infinity, infinite processes and limit concepts." Philosophy & Social Criticism 43, no. 8 (August 29, 2017): 793–811. http://dx.doi.org/10.1177/0191453717692845.

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9

Sabatier, Jocelyn. "Fractional Order Models Are Doubly Infinite Dimensional Models and thus of Infinite Memory: Consequences on Initialization and Some Solutions." Symmetry 13, no. 6 (June 21, 2021): 1099. http://dx.doi.org/10.3390/sym13061099.

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Using a small number of mathematical transformations, this article examines the nature of fractional models described by fractional differential equations or pseudo state space descriptions. Computation of the impulse response of a fractional model using the Cauchy method shows that they exhibit infinitely small and high time constants. This impulse response can be rewritten as a diffusive representation whose Fourier transform permits a representation of a fractional model by a diffusion equation in an infinite space domain. Fractional models can thus be viewed as doubly infinite dimensional models: infinite as distributed with a distribution in an infinite domain. This infinite domain or the infinitely large time constants of the impulse response reveal a property intrinsic to fractional models: their infinite memory. Solutions to generate fractional behaviors without infinite memory are finally proposed.
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10

Magnot, Jean-Pierre. "The Mean Value for Infinite Volume Measures, Infinite Products, and Heuristic Infinite Dimensional Lebesgue Measures." Journal of Mathematics 2017 (2017): 1–14. http://dx.doi.org/10.1155/2017/9853672.

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One of the goals of this article is to describe a setting adapted to the description of means (normalized integrals or invariant means) on an infinite product of measured spaces with infinite measure and of the concentration property on metric measured spaces, inspired from classical examples of means. In some cases, we get a linear extension of the limit at infinity. Then, the mean value on an infinite product is defined, first for cylindrical functions and secondly taking the uniform limit. Finally, the mean value for the heuristic Lebesgue measure on a separable infinite dimensional topological vector space (e.g., on a Hilbert space) is defined. This last object, which is not the classical infinite dimensional Lebesgue measure but its “normalized” version, is shown to be invariant under translation, scaling, and restriction.
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11

Leone, Alexandre. "O Infinito Atual em Or Hashem de Hasdai Crescas (1340 -1411)." Circumscribere: International Journal for the History of Science 25 (July 9, 2020): 1–39. http://dx.doi.org/10.23925/1980-7651.2020v25;p01-39.

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This article focuses on the concept of the "infinite in act" of the medieval Jewish philosopher Has-dai Crescas (1340–1411), formulated in the book Or Hashem (1410) to Maimonides' first three propositions, as set out in the second part of the Guide of the Perplexed. Maimonides' theses aim to deny the possibility of the current infinite as an immaterial or material magnitude, as an infinite set of finite beings and as an infinite series of cause and effect. After a brief exposition of the trajectory of the concepts of infinity in the different Jewish wisdom traditions received in the Middle Ages, we indicate how the argument for the current idea of infinity in Crescas dialogues with them. From this dialogue, the concept of the infinite emerges as a singularity updated parallel to the real as an infinite vacuum, a place of coexistence of infinite universes, and as an actual divine infinite like Kavod, Glory, which fills the infinite universe and as an immanent cause of the infinite series of cause and effect that constitutes the eternal existence of contingent beings. In the critique of the third Maimonidian proposition, the first cause is described as an ontological and immanent cause of the infinite series of causes and effects. In this discussion, Crescas points to an idea of God very different from that developed by Maimonides. Here we have the medieval Jewish debate between defenders of divine transcendence and defenders of immanence. This theme is important for the understanding of the reception of Hasdai Crescas' work by Picco Della Mirandolla, Bruno and Espinosa.
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12

CARRARA, MASSIMILIANO, and ENRICO MARTINO. "ON THE INFINITE IN MEREOLOGY WITH PLURAL QUANTIFICATION." Review of Symbolic Logic 4, no. 1 (September 17, 2010): 54–62. http://dx.doi.org/10.1017/s1755020310000158.

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In “Mathematics is megethology,” Lewis reconstructs set theory using mereology and plural quantification (MPQ). In his recontruction he assumes from the beginning that there is an infinite plurality of atoms, whose size is equivalent to that of the set theoretical universe. Since this assumption is far beyond the basic axioms of mereology, it might seem that MPQ do not play any role in order to guarantee the existence of a large infinity of objects. However, we intend to demonstrate that mereology and plural quantification are, in some ways, particularly relevant to a certain conception of the infinite. More precisely, though the principles of mereology and plural quantification do not guarantee the existence of an infinite number of objects, nevertheless, once the existence of any infinite object is admitted, they are able to assure the existence of an uncountable infinity of objects. So, if—as Lewis maintains—MPQ were parts of logic, the implausible consequence would follow that, given a countable infinity of individuals, logic would be able to guarantee an uncountable infinity of objects.
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13

Patel, Gaurav Singh, and Saurabh Kumar Gautam. "Ramanujan: The New Sum of All Natural Numbers." International Journal for Research in Applied Science and Engineering Technology 10, no. 2 (February 28, 2022): 1272–74. http://dx.doi.org/10.22214/ijraset.2022.40511.

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Abstract: As we know that Sir Ramanujan gave the solution of sum of all natural numbers up to infinity and said that the sum of all natural numbers till infinity is -1/12. I studied on this topic and found that if we try to solve the infinite series in a slightly different way, then we get the answer of its sum different from -1/12, so this is what I have written in this paper that such Ramanujan Sir, what was the mistake in solving the infinite series, which by solving it in a slightly different way from the same concept, we get different answers. Keywords: Ramanujan the new sum of all natural numbers, Infinite series solution, Gaurav singh patel, Infinite series sum, Natural number sum.
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14

Posy, Carl. "Intuition and Infinity: A Kantian Theme with Echoes in the Foundations of Mathematics." Royal Institute of Philosophy Supplement 63 (October 2008): 165–93. http://dx.doi.org/10.1017/s135824610800009x.

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Kant says patently conflicting things about infinity and our grasp of it. Infinite space is a good case in point. In his solution to the First Antinomy, he denies that we can grasp the spatial universe as infinite, and therefore that this universe can be infinite; while in the Aesthetic he says just the opposite: ‘Space is represented as a given infinite magnitude’(A25/B39). And he rests these upon consistently opposite grounds. In the Antinomy we are told that we can have no intuitive grasp of an infinite space, and in the Aesthetic he says that our grasp of infinite space is precisely intuitive.
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15

Ginzburg, Tatiana. "Paradoxes Of Infinity And Foundations Of Transpersonal Psychology." Integral Transpersonal Journal 6, no. 6 (February 2015): 60–71. http://dx.doi.org/10.32031/itibte_itj_6-g2.

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Transpersonal psychology’s uniqueness comes from the point of infinity of the psyche, as the subject of the field. Jung being one of the predecessors of transpersonal psychology confirms the infinity of the psyche very clearly. But this has created another problem. What are the borders of the subject if it is infinite? And if psyche is infinite, how can we grasp it as whole? Can it be fully cognized? Or to give the opposite point of view, is it unknowable? In the search for the borders of the subject of transpersonal psychology, we are attempting to reflect on the paradoxes of infinity. As it turns out, the concept of “actual infinity” (opened by Georg Cantor in the late 19th century) allow us to create a new perspective in solving the infinity problems in psychology. The question arises that if the psyche is infinite, can the psyche be cognized? The idea of psyche being actually infinite allows us to resolve the issue of cognizability of the psyche in principle. This issue is whether a possibility exists for a person to complete the process of self-exploration. Such a solution may lay a new foundation for Transpersonal Psychology on a non-classical scientific basis. KEYWORDS Infinity, transpersonal psychology, self-exploration, enlightenment, perfection, unknowability, knowability,cognizability
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16

Hong, Zhang. "On Hegel’s View of Dialectical Infinity." Journal of Research in Philosophy and History 6, no. 1 (January 23, 2023): p8. http://dx.doi.org/10.22158/jrph.v6n1p8.

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It is well known that the problem of finity and infinity is the basic problem of mathematics, and it is also the basic problem of Philosophy. From the perspective of philosophy and mathematics, this paper comprehensively reviews and analyzes Hegel’s view of dialectical infinity, introduces Engels’discussion on infinity, deeply analyzes the characteristics of the thought of actual infinity, and points out: Hegel’s thought of real infinity is completely different from the thought of actual infinity, the Being of infinity (objective infinity) is not equal to the completed infinity (subjective infinity), the mathematical limit is a real infinity, and real infinity is the inner law of infinite things and truth; the view of actual infinity views the objective material world from the viewpoint of static rather than motion, denying the contradiction between finity and infinity, so it is actually a downright idealist. In this paper, the author puts forward the Infinite Exchange Paradox, which strongly questions the idea of actual infinity in Hilbert Hotel Problem, and points out the internal irreconcilable contradiction in the idea of actual infinity. At the same time, we made a detailed comparison of Hegel’s view of infinity and the view of mathematical infinity, and on this basis, the author gives a complete definition of the view of dialectical infinity: abandoning the wrong aspects of the potential infinity and actual infinity, and actively absorbing correct aspects of both, that is, not only to recognize the existence and knowability of infinite objectivity, but also to admit the imcompletion of infinite process. The reexcavation of Hegel’s view of dialectical infinity and the criticism of the actual infinity thought aim to find possible philosophical solutions for Russell’s Paradox and the problem of Continuum Hypothesis.
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17

Blanc, Jill Le. "Infinity in Theology and Mathematics." Religious Studies 29, no. 1 (March 1993): 51–62. http://dx.doi.org/10.1017/s0034412500022046.

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Can we apply the same concepts to both the finite and the infinite? Is there something distinctive about the infinite that prevents attribution to it of concepts that we can attribute to the finite? If so, then this could be a reason for our difficulties in talking about God – God is infinite, and our concepts, applying, as they do, to the finite objects of our experience, cannot be ‘extended’ to the infinite. God's infinity is sometimes used as an explanation of theological difficulties like the problem of evil or the paradoxes of omnipotence: we do not really know what we mean when we attribute infinite goodness or power to God.
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18

Heng, Zeyu. "All, One, Infinite: The Interpenetration and Opposition Between the Conceptual and Subjective Empirical Aspects of Mathematics." Highlights in Science, Engineering and Technology 88 (March 29, 2024): 228–35. http://dx.doi.org/10.54097/bga5n985.

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The philosophical exploration of the concepts of unity and infinity has always had a significant influence on mathematicians. Hegel posits that the achievement and realization of the infinite state of completeness within the finite realm is contingent upon the use of pure intellect and conceptual frameworks, rather than relying on perceptual intuition. Similar to Hilbert's exploration of the concept of infinity in his dissertation On the Infinite, it may be seen that the actualization of infinity is unattainable in reality. Instead, the finite can only comprehend and achieve infinity via abstract and purely conceptual reasoning, rather than through any kind of perceptual experience of the physical world. Hence, the attribute of ultimate perfection characterizes the concept of "true infinity," but any attempt to attain such perfection within limited boundaries is certain to be unsuccessful. This study delves into the exploration of the historical beginnings of the concept of infinity in the field of mathematics. The concept of absolute perfection is unattainable in the real world due to its reliance on the presence of infinite entities.
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19

Le Coz, Stefan, Dong Li, and Tai-Peng Tsai. "Fast-moving finite and infinite trains of solitons for nonlinear Schrödinger equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 145, no. 6 (November 23, 2015): 1251–82. http://dx.doi.org/10.1017/s030821051500030x.

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We study infinite soliton trains solutions of nonlinear Schrödinger equations, i.e. solutions behaving as the sum of infinitely many solitary waves at large time. Assuming the composing solitons have sufficiently large relative speeds, we prove the existence and uniqueness of such a soliton train. We also give a new construction of multi-solitons (i.e. finite trains) and prove uniqueness in an exponentially small neighbourhood, and we consider the case of solutions composed of several solitons and kinks (i.e. solutions with a non-zero background at infinity).
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20

Yang, Rui Liang, and Cai Xia Zhu. "Condition Number of Acoustic Infinite Element." Advanced Materials Research 181-182 (January 2011): 926–31. http://dx.doi.org/10.4028/www.scientific.net/amr.181-182.926.

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Various shape function and weight function of infinite element are researched and summarized into eight methods, and then various infinite element methods can be summarized as general equation, the condition number of which can reflect merits of infinite method. Condition number of various methods versus frequency and the node number are calculated in this paper. Finally, most optimal infinite element method is summed up. The infinite element method [1-12] is among the most successful techniques used to solve boundary-value problems on unbounded domains and whose solutions satisfy some condition at infinity. Two ideas make the infinite element method attractive: the idea of partition and the idea of approximation. The partition idea covers unbounded domains by attaching infinite strips to finite element partitions of bounded domains. More mature versions of infinite element method involved the approximation idea. These ideas make it possible that the finite element/infinite element method yields significantly greater computational efficiency than other methods such as the boundary element method. There have been a large number of infinite element methods, in which some methods have obvious advantages and some methods have fewer advantages. However, there is less research literature about merits of various infinite element methods appear at home and abroad. Thus, condition number of matrix equation is applied to verify merits of various infinite methods in this paper.
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21

Kapanadze, G. "Boundary Value Problems of Bending of A Plate for an Infinite Doubly-Connected Domain Bounded by Broken Lines." Georgian Mathematical Journal 7, no. 3 (September 2000): 513–21. http://dx.doi.org/10.1515/gmj.2000.513.

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Abstract A problem of bending of a plate is considered for an infinite doubly-connected domain bounded by two convex broken lines when the plate boundary is hinge-supported and normally bending moments are applied to the points at infinity. A similar reasoning is used to study a problem of bending of a plate for an infinite domain bounded by a convex polygon and a rectilinear cut or for an infinite domain with two rectilinear cuts.
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22

Song, Bo, and Victor O. K. Li. "A Revisit of Infinite Population Models for Evolutionary Algorithms on Continuous Optimization Problems." Evolutionary Computation 28, no. 1 (March 2020): 55–85. http://dx.doi.org/10.1162/evco_a_00249.

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Infinite population models are important tools for studying population dynamics of evolutionary algorithms. They describe how the distributions of populations change between consecutive generations. In general, infinite population models are derived from Markov chains by exploiting symmetries between individuals in the population and analyzing the limit as the population size goes to infinity. In this article, we study the theoretical foundations of infinite population models of evolutionary algorithms on continuous optimization problems. First, we show that the convergence proofs in a widely cited study were in fact problematic and incomplete. We further show that the modeling assumption of exchangeability of individuals cannot yield the transition equation. Then, in order to analyze infinite population models, we build an analytical framework based on convergence in distribution of random elements which take values in the metric space of infinite sequences. The framework is concise and mathematically rigorous. It also provides an infrastructure for studying the convergence of the stacking of operators and of iterating the algorithm which previous studies failed to address. Finally, we use the framework to prove the convergence of infinite population models for the mutation operator and the [Formula: see text]-ary recombination operator. We show that these operators can provide accurate predictions for real population dynamics as the population size goes to infinity, provided that the initial population is identically and independently distributed.
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23

Duncombe, Matthew. "Infinite Regress Arguments as per impossibile Arguments in Aristotle: De Caelo 300 a 30– b 1, Posterior Analytics 72 b 5–10, Physics V.2 225 b 33–226 a 10." Rhizomata 10, no. 2 (January 1, 2023): 262–82. http://dx.doi.org/10.1515/rhiz-2022-0015.

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Abstract Infinite regress arguments are a powerful tool in Aristotle, but this style of argument has received relatively little attention. Improving our understanding of infinite regress arguments has become pressing since recent scholars have pointed out that it is not clear whether Aristotle’s infinite regress arguments are, in general, effective or indeed what the logical structure of these arguments is. One obvious approach would be to hold that Aristotle takes infinite regress arguments to be per impossibile arguments, which derive an infinite sequence. Due to his finitism, Aristotle then rejects such a sequence as impossible. This paper argues that this obvious approach does not work, even for its most amenable cases. The paper argues instead that infinite regress arguments involve domain-specific infinities, and so there is not a general finitism which underpins infinite regress arguments in Aristotle, but rather domain-specific reasons that there cannot be an infinite number of entities in each domain in which Aristotle invokes an infinite regress argument.
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Zarepour, Mohammad Saleh. "God, Personhood, and Infinity: Against a Hickian Argument." European Journal for Philosophy of Religion 12, no. 1 (March 25, 2020): 61. http://dx.doi.org/10.24204/ejpr.v12i1.2987.

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Criticizing Richard Swinburne’s conception of God, John Hick argues that God cannot be personal because infinity and personhood are mutually incompatible. An essential characteristic of a person, Hick claims, is having a boundary which distinguishes that person from other persons. But having a boundary is incompatible with being infinite. Infinite beings are unbounded. Hence God cannot be thought of as an infinite person. In this paper, I argue that the Hickian argument is flawed because boundedness is an equivocal notion: in one sense it is not essential to personhood, and in another sense—which is essential to personhood—it is compatible with being infinite.
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BALIBREA, F., and J. SMÍTAL. "A CHARACTERIZATION OF THE SET Ω (f)\ ω (f) FOR CONTINUOUS MAPS OF THE INTERVAL WITH ZERO TOPOLOGICAL ENTROPY." International Journal of Bifurcation and Chaos 05, no. 05 (October 1995): 1433–35. http://dx.doi.org/10.1142/s0218127495001113.

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We give a characterization of the set of nonwandering points of a continuous map f of the interval with zero topological entropy, attracted to a single (infinite) minimal set Q. We show that such a map f can have a unique infinite minimal set Q and an infinite set B ⊂ Ω (f)\ ω (f) (of nonwandering points that are not ω-limit points) attracted to Q and such that B has infinite intersections with infinitely many disjoint orbits of f.
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Button, J. O. "Growth in Infinite Groups of Infinite Subsets." Algebra Colloquium 22, no. 02 (April 15, 2015): 333–48. http://dx.doi.org/10.1142/s1005386715000292.

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Given an infinite group G, we consider the finitely additive invariant measure defined on finite unions of cosets of finite index subgroups. We show that this shares many properties with the size of subsets of a finite group, for instance we can obtain equivalent results on the Ruzsa distance and product free sets. In particular, if G has infinitely many finite index subgroups, then it has subsets S of measure arbitrarily close to 1/2 with square S2 having measure less than 1. We also examine properties of the Ruzsa distance on the set of finite index subgroups of an infinite group, whereupon it becomes a genuine metric.
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Strauss, Danie. "The Philosophy of the Cosmonomic Idea and the Philosophical Foundations of Mathematics." Philosophia Reformata 86, no. 1 (January 7, 2021): 29–47. http://dx.doi.org/10.1163/23528230-bja10014.

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Abstract Since the discovery of the paradoxes of Zeno, the problem of infinity was dominated by the meaning of endlessness—a view also adhered to by Herman Dooyeweerd. Since Aristotle, philosophers and mathematicians distinguished between the potential infinite and the actual infinite. The main aim of this article is to highlight the strengths and limitations of Dooyeweerd’s philosophy for an understanding of the foundations of mathematics, including Dooyeweerd’s quasi-substantial view of the natural numbers and his view of the other types of numbers as functions of natural numbers. Dooyeweerd’s rejection of the actual infinite is turned upside down by the exploring of an alternative perspective on the interrelations between number and space in support of the idea of infinite totalities, or infinite wholes. No other trend has succeeded in justifying the mathematical use of the actual infinite on the basis of an analysis of the intermodal coherence between number and space.
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Arnal-Palacián, Mónica. "Infinite limit of a function at infinity and its phenomenology." Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia 14 (December 31, 2022): 25–41. http://dx.doi.org/10.24917/20809751.14.3.

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In this paper we aim to characterise and define the phenomena of the infinite limit of a function at infinity. Based on the intuitive and formal approaches, we obtain as results five phenomena organised by a definition of this limit: intuitive unlimited growth of a function, for plus and minus infinity, and intuitive unlimited decrease of a function, for plus and minus infinity (intuitive approach), and the round-trip phenomenon of infinite limit functions (formal approach). All this is intended to help overcome the difficulties that pre-university students have with the concept of limit, contributing from phenomenology, Advanced and Elementary Mathematical Thinking, and APOS theory. Keywords: limit, infinity, functions, phenomenology, Advanced Mathematical Thinking, APOS
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29

FRAMPTON, PAUL H. "CYCLIC UNIVERSE AND INFINITE PAST." Modern Physics Letters A 22, no. 34 (November 10, 2007): 2587–92. http://dx.doi.org/10.1142/s0217732307025698.

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We address two questions about the past for infinitely cyclic cosmology. The first is whether it can contain an infinite length null geodesic into the past in view of the Borde–Guth–Vilenkin (BGV) "no-go" theorem, The second is whether, given that a small fraction of spawned universes fail to cycle, there is an adequate probability for a successful universe after an infinite time. We give positive answers to both questions and then show that in infinite cyclicity the total number of universes has been infinite for an arbitrarily long time.
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30

Davidson, Rodney F. "Waves below first cutoff in a duct." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 29, no. 4 (April 1988): 448–60. http://dx.doi.org/10.1017/s0334270000005944.

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AbstractThe two-dimensional Helmholtz equation is studied for an infinite region with two semi-infinite plates extending to infinity in opposite directions and a finite duct in the overlapping region. The solution technique leads to coupled Wiener-Hopf equations, and subsequently to an infinite set of simultaneous linear equations. As an example, an asymptotic expansion is calculated and graphed for the case when the duct length divided by duct width is large enough to ensure damping of all but the zero mode wave in the duct.
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31

Singh, Anand P., and Garima Tomar. "On Levels of fast escaping sets and Spider's Web of transcendental entire functions." New Zealand Journal of Mathematics 49 (December 31, 2019): 1–9. http://dx.doi.org/10.53733/25.

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Let f be a transcendental entire function and let I(f) be the points which escape to infinity under iteration. Bergweiler and Hinkkanen introduced the fast escaping sets A(f) and subsequently, Rippon and Stallard introduced `Levels' of fast escaping sets . These sets under some restriction have the properties of "infinite spider's web" structure. Here we give some topological properties of the infinite spider's web and show some of the transcendental entire functions whose levels of the fast escaping sets have infinite spider's web structure.
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32

Guesmia, Aissa. "Non-exponential and polynomial stability results of a Bresse system with one infinite memory in the vertical displacement." Nonautonomous Dynamical Systems 4, no. 1 (October 26, 2017): 78–97. http://dx.doi.org/10.1515/msds-2017-0008.

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Abstract The asymptotic stability of one-dimensional linear Bresse systems under infinite memories was obtained by Guesmia and Kafini [10] (three infinite memories), Guesmia and Kirane [11] (two infinite memories), Guesmia [9] (one infinite memory acting on the longitudinal displacement) and De Lima Santos et al. [6] (one infinite memory acting on the shear angle displacement). When the kernel functions have an exponential decay at infinity, the obtained stability estimates in these papers lead to the exponential stability of the system if the speeds ofwave propagations are the same, and to the polynomial one with decay rate otherwise. The subject of this paper is to study the case where only one infinite memory is considered and it is acting on the vertical displacement. As far as we know, this case has never studied before in the literature. We show that this case is deeply different from the previous ones cited above by proving that the exponential stability does not hold even if the speeds of wave propagations are the same and the kernel function has an exponential decay at infinity. Moreover, we prove that the system is still stable at least polynomially where the decay rate depends on the smoothness of the initial data. For classical solutions, this decay rate is arbitrarily close to . The proof is based on a combination of the energy method and the frequency domain approach to overcome the new mathematical difficulties generated by our system.
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33

Zarepour, Mohammad Saleh. "Avicenna on Mathematical Infinity." Archiv für Geschichte der Philosophie 102, no. 3 (September 25, 2020): 379–425. http://dx.doi.org/10.1515/agph-2017-0032.

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AbstractAvicenna believed in mathematical finitism. He argued that magnitudes and sets of ordered numbers and numbered things cannot be actually infinite. In this paper, I discuss his arguments against the actuality of mathematical infinity. A careful analysis of the subtleties of his main argument, i. e., The Mapping Argument, shows that, by employing the notion of correspondence as a tool for comparing the sizes of mathematical infinities, he arrived at a very deep and insightful understanding of the notion of mathematical infinity, one that is much more modern than we might expect. I argue, moreover, that Avicenna’s mathematical finitism is interwoven with his literalist ontology of mathematics, according to which mathematical objects are properties of existing physical objects.
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34

Zemanian, A. "Infinite electrical networks with finite sources at infinity." IEEE Transactions on Circuits and Systems 34, no. 12 (December 1987): 1518–34. http://dx.doi.org/10.1109/tcs.1987.1086090.

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35

CALLAHAN, J. "Mastery of the Infinite: To Infinity and Beyond." Science 237, no. 4815 (August 7, 1987): 666–67. http://dx.doi.org/10.1126/science.237.4815.666.

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36

Zhu, Wujia, Yi Lin, Ningsheng Gong, and Guoping Du. "Problem of infinity between predicates and infinite sets." Kybernetes 37, no. 3/4 (April 11, 2008): 526–33. http://dx.doi.org/10.1108/03684920810863516.

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37

Laakkonen, Petteri, and Seppo Pohjolainen. "Directed structure at infinity for infinite-dimensional systems." International Journal of Control 84, no. 4 (April 2011): 702–15. http://dx.doi.org/10.1080/00207179.2011.572999.

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38

Keyl, M., D. Schlingemann, and R. F. Werner. "Infinitely entangled states." Quantum Information and Computation 3, no. 4 (July 2003): 281–306. http://dx.doi.org/10.26421/qic3.4-1.

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For states in infinite dimensional Hilbert spaces entanglement quantities like the entanglement of distillation can become infinite. This leads naturally to the question, whether one system in such an infinitely entangled state can serve as a resource for tasks like the teleportation of arbitrarily many qubits. We show that appropriate states cannot be obtained by density operators in an infinite dimensional Hilbert space. However, using techniques for the description of infinitely many degrees of freedom from field theory and statistical mechanics, such states can nevertheless be constructed rigorously. We explore two related possibilities, namely an extended notion of algebras of observables, and the use of singular states on the algebra of bounded operators. As applications we construct the essentially unique infinite analogue of maximally entangled states, and the singular state used heuristically in the fundamental paper of Einstein, Rosen and Podolsky.
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39

Scott, David D. "The boundless riches of God." Theology in Scotland 27, no. 2 (November 26, 2020): 25–41. http://dx.doi.org/10.15664/tis.v27i2.2138.

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This paper explores the concept of infinity in mathematics and its relation to theological considerations. It begins by seeking to answer the question of whether mathematical enquiry into the character of infinity may cast some light on the infinite character of God. Drawing on the work of Euclid, Cantor, and Gödel in particular, it considers concepts of potential and actual infinity and how mathematical discoveries have implications for (i) the relation of the finite and infinite (which has theological implications for the incarnation); (ii) the relation of theory and reality; (iii) the future scope of discovery and invention; and (iv) further reflection on the givenness of revelation.
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40

Mamolo, Ami. "INTUITIONS OF "INFINITE NUMBERS": INFINITE MAGNITUDE VS. INFINITE REPRESENTATION." Mathematics Enthusiast 6, no. 3 (July 1, 2009): 305–30. http://dx.doi.org/10.54870/1551-3440.1156.

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41

Özöğür-Akyüz, S., and G. W. Weber. "Infinite kernel learning via infinite and semi-infinite programming." Optimization Methods and Software 25, no. 6 (December 2010): 937–70. http://dx.doi.org/10.1080/10556780903483349.

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42

Ferreira, Sidnei. "Infinite legacy." Residência Pediátrica 9, no. 1 (2019): 5. http://dx.doi.org/10.25060/residpediatr-2019.v9n1-01.

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43

Ciobotaru, Corina. "Infinitely Generated Hecke Algebras with Infinite Presentation." Algebras and Representation Theory 23, no. 6 (December 13, 2019): 2275–93. http://dx.doi.org/10.1007/s10468-019-09939-8.

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44

Diaz-Espinoza, Irving Aaron, José Antonio Juárez-López, and Estela Juárez-Ruiz. "Exploring a Mathematics Teacher’s Conceptions of Infinity: The Case of Louise." Indonesian Journal of Mathematics Education 6, no. 1 (April 30, 2023): 1–6. http://dx.doi.org/10.31002/ijome.v6i1.560.

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Several papers studied infinity from the difficulties that students and teachers show in developing the concept. For this study, it was considered the analysis of the equality 0.999 … = 1. Mainly, this research aims to show that a mathematics teacher presents erroneous conceptions just like a student; that is, both students and teachers have the same difficulties in the concept of infinity. To this aim, a semi-structured interview was conducted with an in-service mathematics teacher in Tlaxcala, Mexico. The purpose of this research is to exhibit a high school math teacher’s misconceptions about the concept of infinity. In general, misconceptions found here can be divided into four groups: without a clear picture of the concept of infinity, an infinite periodic decimal number cannot be a representation of a finite number, a decreasing infinite sum cannot lead to a finite number and an infinite process is limited in real life is finite and has ended. The results obtained were compared with those already available in references about the difficulties with students and teachers, finding that the results shown here are like those reported in the literature. This highlights the need to overcome the teacher’s conceptions of infinity in future research.
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45

Frid, Anna E. "Infinite permutations vs. infinite words." Electronic Proceedings in Theoretical Computer Science 63 (August 17, 2011): 13–19. http://dx.doi.org/10.4204/eptcs.63.2.

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46

Johnson, Craig, and Brian Mullen. "Infinite Diversity in Infinite Combinations." Psychological Inquiry 3, no. 2 (April 1992): 166–68. http://dx.doi.org/10.1207/s15327965pli0302_17.

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47

Lindenmayer, David B. "Infinite perspectives on ‘Infinite Nature’." Trends in Ecology & Evolution 22, no. 2 (February 2007): 61. http://dx.doi.org/10.1016/j.tree.2006.11.010.

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48

Loeb, Peter A., and David A. Ross. "Infinite products of infinite measures." Illinois Journal of Mathematics 49, no. 1 (January 2005): 153–58. http://dx.doi.org/10.1215/ijm/1258138311.

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49

Vallentyne, Peter. "Infinite Utility and Temporal Neutrality." Utilitas 6, no. 2 (November 1994): 193–99. http://dx.doi.org/10.1017/s0953820800001576.

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Suppose that time is infinitely long towards the future, and that each feasible action produces a finite amount of utility at each time. Then, under appropriate conditions, each action produces an infinite amount of utility. Does this mean that utilitarianism lacks the resources to discriminate among such actions? Since each action produces the same infinite amount of utility, it seems that utilitarianism must judge all actions permissible, judge all actions impermissible, or remain completely silent. If the future is infinite, that is, the prospects for utilitarianism look bleak.
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50

Boza, L., A. Diánez, and A. Márquez. "On infinite outerplanar graphs." Mathematica Bohemica 119, no. 4 (1994): 381–84. http://dx.doi.org/10.21136/mb.1994.126112.

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