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Journal articles on the topic 'Weierstrass theorem -The Catalan'

Author: Grafiati

Published: 4 June 2025

Last updated: 23 June 2025

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1

Mohammed, Ghanim. "A SHORT ELEMENTARY PROOF OF THE BEAL CONJECTURE WITH DEDUCTION OF THE FERMAT LAST THEOREM." GLOBAL JOURNAL OF ADVANCED ENGINEERING TECHNOLOGIES AND SCIENCES 8, no. 1 (2021): 1–16. https://doi.org/10.5281/zenodo.4568087.

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The present short paper, which is an amelioration of my previous article “confirmation of the Beal-Brun-Tijdeman-Zagier conjecture” published by the GJETS in 20/11/2019 [15], confirms the Beal’s conjecture, remained open since 1914 and saying that:       The proof uses elementary tools of mathematics, such as the L’Hôpital rule, the Bolzano-Weierstrass theorem, the intermediate value theorem and the growth properties of certain elementary functions. The proof uses also the Catalan-Mihailescu theorem [18] [19] and some methods developed in my paper on the Fermat last theorem [14] published by the GJAETS in 10/12/2018. The particular case of the Fermat last theorem is deduced

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2

Laville, Guy, and Ivan Ramadanoff. "Stone-Weierstrass theorem." Banach Center Publications 37, no. 1 (1996): 189–94. http://dx.doi.org/10.4064/-37-1-189-194.

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3

Nesterenko, Yu V. "Lindemann–Weierstrass Theorem." Moscow University Mathematics Bulletin 76, no. 6 (2021): 239–43. http://dx.doi.org/10.3103/s0027132221060073.

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4

Wu, Hueytzen J. "New Stone-Weierstrass Theorem." Advances in Pure Mathematics 06, no. 13 (2016): 943–47. http://dx.doi.org/10.4236/apm.2016.613071.

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5

Portilla, Ana, Yamilet Quintana, José M. Rodrı́guez, and Eva Tourı́s. "Weierstrass’ theorem with weights." Journal of Approximation Theory 127, no. 1 (2004): 83–107. http://dx.doi.org/10.1016/j.jat.2004.01.003.

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6

Rao, N. V. "The Stone-Weierstrass Theorem Revisited." American Mathematical Monthly 112, no. 8 (2005): 726. http://dx.doi.org/10.2307/30037574.

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7

Rao, N. V. "The Stone-Weierstrass Theorem Revisited." American Mathematical Monthly 112, no. 8 (2005): 726–29. http://dx.doi.org/10.1080/00029890.2005.11920244.

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8

Niemiec, Piotr. "Strengthened Stone-Weierstrass type theorem." Opuscula Mathematica 31, no. 4 (2011): 645. http://dx.doi.org/10.7494/opmath.2011.31.4.645.

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9

Martínez-Legaz, Juan Enrique. "On Weierstrass extreme value theorem." Optimization Letters 8, no. 1 (2012): 391–93. http://dx.doi.org/10.1007/s11590-012-0587-0.

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10

Prolla, J. B. "On the Weierstrass-Stone Theorem." Journal of Approximation Theory 78, no. 3 (1994): 299–313. http://dx.doi.org/10.1006/jath.1994.1080.

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11

Ghanouchi, Jamel. "About Catalan-Mihailescu Theorem." Asian Journal of Algebra 2, no. 1 (2008): 11–16. http://dx.doi.org/10.3923/aja.2009.11.16.

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12

Ghanouchi, Jamel. "About Catalan-Mihailescu Theorem*." Asian Journal of Algebra 3, no. 2 (2010): 53–58. http://dx.doi.org/10.3923/aja.2010.53.58.

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13

Výborný, Rudolf. "The Weierstrass theorem on polynomial approximation." Mathematica Bohemica 130, no. 2 (2005): 161–66. http://dx.doi.org/10.21136/mb.2005.134132.

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14

Amini-Harandi, A., M. Fakhar, and H. R. Hajisharifi. "Some Generalizations of the Weierstrass Theorem." SIAM Journal on Optimization 26, no. 4 (2016): 2847–62. http://dx.doi.org/10.1137/15m1054997.

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15

Gajek, L., and D. Zagrodny. "Weierstrass theorem for monotonically semicontinuous functions." Optimization 29, no. 3 (1994): 199–203. http://dx.doi.org/10.1080/02331939408843949.

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16

Miller, Daniel J. "A preparation theorem for Weierstrass systems." Transactions of the American Mathematical Society 358, no. 10 (2006): 4395–439. http://dx.doi.org/10.1090/s0002-9947-06-04190-0.

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17

Ferens-Sorotskiy, E. V. "Weierstrass preparation theorem for noncommutative rings." Journal of Mathematical Sciences 161, no. 4 (2009): 597–601. http://dx.doi.org/10.1007/s10958-009-9587-8.

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18

Portilla, Ana, Yamilet Quintana, José M. Rodríguez, and Eva Tourís. "Weighted Weierstrass' theorem with first derivatives." Journal of Mathematical Analysis and Applications 334, no. 2 (2007): 1167–98. http://dx.doi.org/10.1016/j.jmaa.2006.12.066.

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19

Drwalewska, A. "A Generalization of the Weierstrass Theorem." Zeitschrift für Analysis und ihre Anwendungen 15, no. 3 (1996): 759–63. http://dx.doi.org/10.4171/zaa/727.

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20

Hemdaoui, M. "Extension of the Weierstrass factorization theorem." Rendiconti del Circolo Matematico di Palermo 49, no. 3 (2000): 435–44. http://dx.doi.org/10.1007/bf02904256.

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21

Rodrı́guez, José M. "Weierstrass' Theorem in Weighted Sobolev Spaces." Journal of Approximation Theory 108, no. 2 (2001): 119–60. http://dx.doi.org/10.1006/jath.2000.3490.

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22

Lin, Chin-Hung. "Some Combinatorial Interpretations and Applications of Fuss-Catalan Numbers." ISRN Discrete Mathematics 2011 (November 14, 2011): 1–8. http://dx.doi.org/10.5402/2011/534628.

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Fuss-Catalan number is a family of generalized Catalan numbers. We begin by two definitions of Fuss-Catalan numbers and some basic properties. And we give some combinatorial interpretations different from original Catalan numbers. Finally we generalize the Jonah's theorem as its applications.

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23

Isabu, Hillary Amonyela, and Michael Onyango Ojiema. "On Some Aspects of Compactness in Metric Spaces." African Scientific Annual Review 1, Mathematics 1 (2024): 18–27. http://dx.doi.org/10.51867/asarev.maths.1.1.2.

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In this paper, we investigate the generalizations of the concepts from Heine-Borel Theorem and the Bolzano-Weierstrass Theorem to metric spaces. We show that the metric space X is compact if every open covering has a finite subcovering. This abstracts the Heine-Borel property. Indeed, the Heine-Borel Theorem states that closed bounded subsets of the real line R are compact. In this study, we rephrase compactness in terms of closed bounded subsets of the real line R, that is, the Bolzano-Weierstrass theorem. Let X be any closed bounded subset of the real line. Then any sequence (xn) of the points of X has a subsequence converging to a point of X. We have used these interesting theorems to characterize compactness in metric spaces.

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24

Reggio, Luca. "Beth definability and the Stone-Weierstrass Theorem." Annals of Pure and Applied Logic 172, no. 8 (2021): 102990. http://dx.doi.org/10.1016/j.apal.2021.102990.

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25

Elkhadiri, Abdelhafed, and Hassan Sfouli. "Weierstrass division theorem in quasianalytic local rings." Studia Mathematica 185, no. 1 (2008): 83–86. http://dx.doi.org/10.4064/sm185-1-5.

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26

Thuong, Tran van. "Some Applications of the Stone-Weierstrass Theorem." Missouri Journal of Mathematical Sciences 3, no. 2 (1991): 79–83. http://dx.doi.org/10.35834/1991/0302079.

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27

Konderak, Jerzy J. "A Weierstrass representation theorem for Lorentz surfaces." Complex Variables, Theory and Application: An International Journal 50, no. 5 (2005): 319–32. http://dx.doi.org/10.1080/02781070500032895.

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28

de Lima, Amanda, and Daniel Smania. "Central limit theorem for generalized Weierstrass functions." Stochastics and Dynamics 19, no. 01 (2019): 1950002. http://dx.doi.org/10.1142/s0219493719500023.

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Let [Formula: see text] be a [Formula: see text] expanding map of the circle and let [Formula: see text] be a [Formula: see text] function. Consider the twisted cohomological equation [Formula: see text] which has a unique bounded solution [Formula: see text]. We show that [Formula: see text] is either [Formula: see text] or continuous but nowhere differentiable. If [Formula: see text] is nowhere differentiable then the Newton quotients of [Formula: see text], after an appropriated normalization, converges in distribution (with respect to the unique absolutely continuous invariant probability of [Formula: see text]) to the normal distribution. In particular, [Formula: see text] is not a Lipschitz continuous function on any subset with positive Lebesgue measure.

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29

Gzyl, Henryk, and Jose Luis Palacios. "The Weierstrass Approximation Theorem and Large Deviations." American Mathematical Monthly 104, no. 7 (1997): 650. http://dx.doi.org/10.2307/2975059.

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30

Stoica, George. "Components Used in the Stone–Weierstrass Theorem." American Mathematical Monthly 127, no. 7 (2020): 658. http://dx.doi.org/10.1080/00029890.2020.1763123.

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31

Benko, David, and András Kroó. "A Weierstrass-type theorem for homogeneous polynomials." Transactions of the American Mathematical Society 361, no. 03 (2008): 1645–65. http://dx.doi.org/10.1090/s0002-9947-08-04625-4.

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32

Klingler, L., and M. Marshall. "E-sequences and the Stone–Weierstrass Theorem." Journal of Number Theory 133, no. 5 (2013): 1525–36. http://dx.doi.org/10.1016/j.jnt.2012.10.003.

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33

Gzyl, Henryk, and José Luis Palacios. "The Weierstrass Approximation Theorem and Large Deviations." American Mathematical Monthly 104, no. 7 (1997): 650–53. http://dx.doi.org/10.1080/00029890.1997.11990694.

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34

Chabert, Jean-Luc, and Giulio Peruginelli. "Adelic versions of the Weierstrass approximation theorem." Journal of Pure and Applied Algebra 222, no. 3 (2018): 568–84. http://dx.doi.org/10.1016/j.jpaa.2017.04.020.

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35

García-Máynez, Adalberto, Margarita Gary, and Adolfo Pimienta. "An application of the Stone-Weierstrass theorem." Proyecciones (Antofagasta) 42, no. 5 (2023): 1211–20. http://dx.doi.org/10.22199/issn.0717-6279-5576.

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Let (X, τ) be a topological space, we will denote by |X|,|X|K, |X|τ and |X|δ, the cardinalities of X; the family of compacts in X; the family of closed in X, and the family of Gδ-closed in X, respectively. The purpose of this work is to establish relationships between these four numbers and conditions under which two of them coincide or one of them is ≤ c, where c denotes, as usual, the cardinality of the set of real numbers R. We will use the Stone-Weierstrass theorem to prove that: Let (X, τ) be a compact Hausdorff topological space, then |X|δ ≤ |X|ℵ0

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36

Marcheva, Plamena I., and Stoil I. Ivanov. "Convergence Analysis of a Modified Weierstrass Method for the Simultaneous Determination of Polynomial Zeros." Symmetry 12, no. 9 (2020): 1408. http://dx.doi.org/10.3390/sym12091408.

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In 2016, Nedzhibov constructed a modification of the Weierstrass method for simultaneous computation of polynomial zeros. In this work, we obtain local and semilocal convergence theorems that improve and complement the previous results about this method. The semilocal result is of significant practical importance because of its computationally verifiable initial condition and error estimate. Numerical experiments to show the applicability of our semilocal theorem are also presented. We finish this study with a theoretical and numerical comparison between the modified Weierstrass method and the classical Weierstrass method.

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37

Marcheva, Plamena I., Ivan K. Ivanov, and Stoil I. Ivanov. "On the Q-Convergence and Dynamics of a Modified Weierstrass Method for the Simultaneous Extraction of Polynomial Zeros." Algorithms 18, no. 4 (2025): 205. https://doi.org/10.3390/a18040205.

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In the present paper, we prove a new local convergence theorem with initial conditions and error estimates that ensure the Q-quadratic convergence of a modification of the famous Weierstrass method. Afterward, we prove a semilocal convergence theorem that is of great practical importance owing to its computable initial condition. The obtained theorems improve and complement all existing such kind of convergence results about this method. At the end of the paper, we provide three numerical examples to show the applicability of our semilocal theorem to some physics problems. Within the examples, we propose a new algorithm for the experimental study of the dynamics of the simultaneous methods and compare the convergence and dynamical behaviors of the modified and the classical Weierstrass methods.

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38

Frennemo, Lennart. "General $n$-dimensional Tauberian problems with application to the Laplace- and Stieltjes transforms." MATHEMATICA SCANDINAVICA 91, no. 2 (2002): 269. http://dx.doi.org/10.7146/math.scand.a-14390.

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A general theorem on the closure of translates in certain weighted spaces in $R^n$ is proved and as a consequence a general $n$-dimensional Tauberian theorem. This is applied to the $n$-dimensional Laplace transform and to the one-dimensional Stieltjes- and Weierstrass transforms.

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39

II, J. H. Lindsey. "A Simple Proof of the Weierstrass Approximation Theorem." American Mathematical Monthly 98, no. 5 (1991): 429. http://dx.doi.org/10.2307/2323862.

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40

Beukers, F., J. P. Bezivin, and P. Robba. "An Alternative Proof of the Lindemann-Weierstrass Theorem." American Mathematical Monthly 97, no. 3 (1990): 193. http://dx.doi.org/10.2307/2324683.

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41

Boel, Soren, Toke Meier Carlsen, and Niels Richard Hansen. "A Useful Strengthening of the Stone-Weierstrass Theorem." American Mathematical Monthly 108, no. 7 (2001): 642. http://dx.doi.org/10.2307/2695271.

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42

Sevost'yanov, E., and A. Markysh. "On Sokhotski–Casorati–Weierstrass theorem on metric spaces." Complex Variables and Elliptic Equations 64, no. 12 (2018): 1973–93. http://dx.doi.org/10.1080/17476933.2018.1557155.

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43

McAfee, R. Preston, and Philip J. Reny. "A Stone-Weierstrass theorem without closure under suprema." Proceedings of the American Mathematical Society 114, no. 1 (1992): 61. http://dx.doi.org/10.1090/s0002-9939-1992-1091186-2.

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44

SHEPPARD, B. "A STONE–WEIERSTRASS THEOREM FOR ${\rm JB}^*$ -TRIPLES." Journal of the London Mathematical Society 65, no. 02 (2002): 381–96. http://dx.doi.org/10.1112/s0024610701003039.

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45

Sheppard, B. "A Stone-weierstrass Theorem for Postliminal JB-Algebras." Quarterly Journal of Mathematics 52, no. 4 (2001): 507–18. http://dx.doi.org/10.1093/qjmath/52.4.507.

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46

Quintana, Yamilet. "On Hilbert extensions of Weierstrass' theorem with weights." Journal of Function Spaces and Applications 8, no. 2 (2010): 201–13. http://dx.doi.org/10.1155/2010/645369.

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In this paper we study the set of ℊ-valued functions which can be approximated by ℊ-valued continuous functions in the normLℊ∞(I,w), whereI⊂ℝis a compact interval, ℊ is a separable real Hilbert space and w is a certain ℊ-valued weakly measurable weight. Thus, we obtain a new extension of the celebrated Weierstrass approximation theorem.

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47

Banaschewski, Bernhard, and Christopher J. Mulvey. "A constructive proof of the Stone-Weierstrass theorem." Journal of Pure and Applied Algebra 116, no. 1-3 (1997): 25–40. http://dx.doi.org/10.1016/s0022-4049(96)00160-0.

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48

Asgarova, Aida Kh. "On a generalization of the Stone–Weierstrass theorem." Annales mathématiques du Québec 42, no. 1 (2017): 1–6. http://dx.doi.org/10.1007/s40316-017-0081-2.

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49

Lubinsky, D. S. "WEIERSTRASS' THEOREM IN THE TWENTIETH CENTURY: A SELECTION." Quaestiones Mathematicae 18, no. 1-3 (1995): 91–130. http://dx.doi.org/10.1080/16073606.1995.9631789.

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50

Beukers, F., J. P. Bézivin, and P. Robba. "An Alternative Proof of the Lindemann-Weierstrass Theorem." American Mathematical Monthly 97, no. 3 (1990): 193–97. http://dx.doi.org/10.1080/00029890.1990.11995573.

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