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Journal articles on the topic 'Bolzano Weierstrass theorem'

Author: Grafiati

Published: 4 June 2021

Last updated: 5 February 2022

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1

Thakur, Ramkrishna, and S. K. Samanta. "A Study on Some Fundamental Properties of Continuity and Differentiability of Functions of Soft Real Numbers." Advances in Fuzzy Systems 2018 (2018): 1–8. http://dx.doi.org/10.1155/2018/6429572.

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We introduce a new type of functions from a soft set to a soft set and study their properties under soft real number setting. Firstly, we investigate some properties of soft real sets. Considering the partial order relation of soft real numbers, we introduce concept of soft intervals. Boundedness of soft real sets is defined, and the celebrated theorems like nested intervals theorem and Bolzano-Weierstrass theorem are extended in this setting. Next, we introduce the concepts of limit, continuity, and differentiability of functions of soft sets. It has been possible for us to study some fundame

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2

Filipów, Rafał, Nikodem Mrożek, Ireneusz Recław, and Piotr Szuca. "Ideal convergence of bounded sequences." Journal of Symbolic Logic 72, no. 2 (2007): 501–12. http://dx.doi.org/10.2178/jsl/1185803621.

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AbstractWe generalize the Bolzano-Weierstrass theorem (that every bounded sequence of reals admits a convergent subsequence) on ideal convergence. We show examples of ideals with and without the Bolzano-Weierstrass property, and give characterizations of BW property in terms of submeasures and extendability to a maximal P-ideal. We show applications to Rudin-Keisler and Rudin-Blass orderings of ideals and quotient Boolean algebras. In particular we show that an ideal does not have BW property if and only if its quotient Boolean algebra has a countably splitting family.

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3

Eidolon, Katrina, and Greg Oman. "A Short Proof of the Bolzano–Weierstrass Theorem." College Mathematics Journal 48, no. 4 (2017): 288–89. http://dx.doi.org/10.4169/college.math.j.48.4.288.

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4

Conidis, Chris J. "Comparing theorems of hyperarithmetic analysis with the arithmetic Bolzano-Weierstrass theorem." Transactions of the American Mathematical Society 364, no. 9 (2012): 4465–94. http://dx.doi.org/10.1090/s0002-9947-2012-05416-x.

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5

Robertson, Neill. "The metrisability of precompact sets." Bulletin of the Australian Mathematical Society 43, no. 1 (1991): 131–35. http://dx.doi.org/10.1017/s0004972700028847.

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A uniform space is trans-separable if every uniform cover has a countable subcover. We show that a uniform space is trans-separable if it contains a suitable family of precompact sets. Applying this result to locally convex spaces, we are able to deduce that the precompact subsets of a wide class of spaces are metrisable. The proof of our main Theorem is based on a cardinality argument, and is reminiscent of the classical Bolzano-Weierstrass Theorem.

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6

Brattka, Vasco, Guido Gherardi, and Alberto Marcone. "The Bolzano–Weierstrass Theorem is the jump of Weak Kőnig’s Lemma." Annals of Pure and Applied Logic 163, no. 6 (2012): 623–55. http://dx.doi.org/10.1016/j.apal.2011.10.006.

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7

ROSSER, J. BARKLEY. "ON THE FOUNDATIONS OF MATHEMATICAL ECONOMICS." New Mathematics and Natural Computation 08, no. 01 (2012): 53–72. http://dx.doi.org/10.1142/s1793005712400029.

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Kumaraswamy Vela Velupillai74 presents a constructivist perspective on the foundations of mathematical economics, praising the views of Feynman in developing path integrals and Dirac in developing the delta function. He sees their approach as consistent with the Bishop constructive mathematics and considers its view on the Bolzano-Weierstrass, Hahn-Banach, and intermediate value theorems, and then the implications of these arguments for such "crown jewels" of mathematical economics as the existence of general equilibrium and the second welfare theorem. He also relates these ideas to the weaken

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8

Moore, Gregory H. "Historians and Philosophers of Logic: Are They Compatible? The Bolzano-Weierstrass Theorem as a Case Study." History and Philosophy of Logic 20, no. 3-4 (1999): 169–80. http://dx.doi.org/10.1080/01445349950044125.

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9

Schmerl, James H. "A reflection principle and its applications to nonstandard models." Journal of Symbolic Logic 60, no. 4 (1995): 1137–52. http://dx.doi.org/10.2307/2275878.

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Some methods of constructing nonstandard models work only for particular theories, such as ZFC, or CA + AC (which is second order number theory with the choice scheme). The examples of this which motivated the results of this paper occur in the main theorems of [5], which state that if T is any consistent extension of either ZFC0 (which is ZFC but with only countable replacement) or CA + AC and if κ and λ are suitably chosen cardinals, then T has a model which is κ-saturated and has the λ-Bolzano-Weierstrass property. (Compare with Theorem 3.5.) Another example is a result from [12] which stat

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10

Brattka, Vasco, Andrea Cettolo, Guido Gherardi, Alberto Marcone, and Matthias Schröder. "Addendum to: “The Bolzano–Weierstrass theorem is the jump of weak Kőnig's lemma” [Ann. Pure Appl. Logic 163 (6) (2012) 623–655]." Annals of Pure and Applied Logic 168, no. 8 (2017): 1605–8. http://dx.doi.org/10.1016/j.apal.2017.04.004.

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11

Khan, L. A., and A. B. Thaheem. "On the equivalence of the Heine-Borel and the Bolzano-Weierstrass theorems." International Journal of Mathematical Education in Science and Technology 31, no. 4 (2000): 620–22. http://dx.doi.org/10.1080/002073900412714.

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12

Mehmood, Arif, Saleem Abdullah, Mohammed M. Al-Shomrani, Muhammad Imran Khan, and Orawit Thinnukool. "Some Results in Neutrosophic Soft Topology Concerning Neutrosophic Soft ∗ b Open Sets." Journal of Function Spaces 2021 (May 24, 2021): 1–15. http://dx.doi.org/10.1155/2021/5544319.

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In this article, new generalised neutrosophic soft open known as neutrosophic soft ∗ b open set is introduced in neutrosophic soft topological spaces. Neutrosophic soft ∗ b open set is generated with the help of neutrosophic soft semiopen and neutrosophic soft preopen sets. Then, with the application of this new definition, some soft neutrosophical separation axioms, countability theorems, and countable space can be Hausdorff space under the subjection of neutrosophic soft sequence which is convergent, the cardinality of neutrosophic soft countable space, engagement of neutrosophic soft counta

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13

Ragazzo, C. Grotta. "The motion of a vortex on a closed surface of constant negative curvature." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2206 (2017): 20170447. http://dx.doi.org/10.1098/rspa.2017.0447.

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The purpose of this work is to present an algorithm to determine the motion of a single hydrodynamic vortex on a closed surface of constant curvature and of genus greater than one. The algorithm is based on a relation between the Laplace–Beltrami Green function and the heat kernel. The algorithm is used to compute the motion of a vortex on the Bolza surface. This is the first determination of the orbits of a vortex on a closed surface of genus greater than one. The numerical results show that all the 46 vortex equilibria can be explicitly computed using the symmetries of the Bolza surface. Som

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14

Kreuzer, Alexander P., and Ulrich Kohlenbach. "Term extraction and Ramsey's theorem for pairs." Journal of Symbolic Logic 77, no. 3 (2012): 853–95. http://dx.doi.org/10.2178/jsl/1344862165.

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AbstractIn this paper we study with proof-theoretic methods the function(al)s provably recursive relative to Ramsey's theorem for pairs and the cohesive principle (COH).Our main result on COH is that the type 2 functional provably recursive fromare primitive recursive. This also provides a uniform method to extract bounds from proofs that use these principles. As a consequence we obtain a new proof of the fact thatis-conservative over PRA.Recent work of the first author showed thatis equivalent to a weak variant of the Bolzano-Weierstraß principle. This makes it possible to use our results to

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15

Sanders, Sam. "Countable sets versus sets that are countable in reverse mathematics." Computability, September 6, 2021, 1–31. http://dx.doi.org/10.3233/com-210313.

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The program Reverse Mathematics (RM for short) seeks to identify the axioms necessary to prove theorems of ordinary mathematics, usually working in the language of second-order arithmetic L 2 . A major theme in RM is therefore the study of structures that are countable or can be approximated by countable sets. Now, countable sets must be represented by sequences here, because the higher-order definition of ‘countable set’ involving injections/bijections to N cannot be directly expressed in L 2 . Working in Kohlenbach’s higher-order RM, we investigate various central theorems, e.g. those due to

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