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Journal articles on the topic 'Theorem of intermediate value (Theorem of Bolzano)'

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Published: 7 September 2021

Last updated: 1 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 fundamental results of continuity of functions of soft sets such as Bolzano’s theorem, intermediate value property, and fixed point theorem. Because the soft real numbers are not linearly ordered, several twists in the arguments are required for proving those results. In the context of differentiability of functions, some basic theorems like Rolle’s theorem and Lagrange’s mean value theorem are also extended in soft setting.

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2

Włodarczyk, Kazimierz. "Intermediate value theorems for holomorphic maps in complex Banach spaces." Mathematical Proceedings of the Cambridge Philosophical Society 109, no. 3 (May 1991): 539–40. http://dx.doi.org/10.1017/s0305004100069978.

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One of the most celebrated theorems of mathematical analysis is the intermediate value theorem of Bolzano which, in a simple case, states that a real-valued continuous map f of a closed interval [a, b], such that f(a) and f(b) have different signs, has a zero in (a, b). Recently, Shih in [5] observed that without loss of generality we may suppose that a 7 < 0 < b and f(a) < 0 < f(b) and, consequently, the condition f(a).f(b) < 0 becomes x.f(x) > 0 for x∈∂Ω where ∂Ω denotes the boundary of the interval Ω = (a, b); then the conclusion is that f has at least one zero in ω. It is a remarkable fact that Shih extends this form of Boizano's theorem to analytic maps in ℂ [5] and, subsequently, in ℂn [6]. He proved that if Ω is a bounded domain in ℂn containing the origin, is continuous in and analytic in Ω and Re for z∈∂Ω, then f has exactly one zero in Ω. In this paper we extend Shih's result to Banach spaces.

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3

ROSSER, J. BARKLEY. "ON THE FOUNDATIONS OF MATHEMATICAL ECONOMICS." New Mathematics and Natural Computation 08, no. 01 (March 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 weakening of certain assumptions to allow for more general results as shown by Rosser51 in his extension of Gödel's incompleteness theorem in his opening section. This paper considers these arguments in reverse order, moving from the matters of economics applications to the broader issue of constructivist mathematics, concluding by considering the views of Rosser on these matters, drawing both on his writings and on personal conversations with him.

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4

Bayoumi, Aboubakr. "Bolzano’s intermediate-value theorem for quasi-holomorphic maps." Central European Journal of Mathematics 3, no. 1 (March 2005): 76–82. http://dx.doi.org/10.2478/bf02475656.

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5

Kryszewski, Wojciech, and Jakub Siemianowski. "The Bolzano mean-value theorem and partial differential equations." Journal of Mathematical Analysis and Applications 457, no. 2 (January 2018): 1452–77. http://dx.doi.org/10.1016/j.jmaa.2017.01.040.

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6

Johnsonbaugh, Richard. "A Discrete Intermediate Value Theorem." College Mathematics Journal 29, no. 1 (January 1998): 42. http://dx.doi.org/10.2307/2687637.

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7

Johnsonbaugh, Richard, and Duane W. DeTemple. "A Discrete Intermediate Value Theorem." College Mathematics Journal 29, no. 1 (January 1998): 42. http://dx.doi.org/10.1080/07468342.1998.11973914.

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8

Huang, Xun-Cheng. "From Intermediate Value Theorem to Chaos." Mathematics Magazine 65, no. 2 (April 1, 1992): 91. http://dx.doi.org/10.2307/2690487.

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9

Huang, Xun-Cheng. "From Intermediate Value Theorem To Chaos." Mathematics Magazine 65, no. 2 (April 1992): 91–103. http://dx.doi.org/10.1080/0025570x.1992.11995989.

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10

Bridges, Douglas S. "A General Constructive Intermediate Value Theorem." Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 35, no. 5 (1989): 433–35. http://dx.doi.org/10.1002/malq.19890350509.

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11

Larson, Suzanne. "THE INTERMEDIATE VALUE THEOREM INf-RINGS." Communications in Algebra 30, no. 5 (May 31, 2002): 2469–504. http://dx.doi.org/10.1081/agb-120003479.

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12

Bayne, Richard, Terrence Edwards, and Myung H. Kwack. "A Common Generalization of the Intermediate Value Theorem and Rouché's Theorem." Missouri Journal of Mathematical Sciences 18, no. 1 (February 2006): 26–32. http://dx.doi.org/10.35834/2006/1801026.

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13

Wu, Z. "A Fixed Point Theorem, Intermediate Value Theorem, and Nested Interval Property." Analysis Mathematica 45, no. 2 (October 17, 2018): 443–47. http://dx.doi.org/10.1007/s10476-018-0612-3.

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14

HERZOG, GERD, and ROLAND LEMMERT. "BOUNDARY VALUE PROBLEMS VIA AN INTERMEDIATE VALUE THEOREM." Glasgow Mathematical Journal 50, no. 3 (September 2008): 531–37. http://dx.doi.org/10.1017/s0017089508004394.

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AbstractWe use an intermediate value theorem for quasi-monotone increasing functions to prove the existence of the smallest and the greatest solution of the Dirichlet problem u″ + f(t, u) = 0, u(0) = α, u(1) = β between lower and upper solutions, where f:[0,1] × E → E is quasi-monotone increasing in its second variable with respect to a regular cone.

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15

MASHINCHI, Mashallah. "An Intermediate Value Theorem in Neighbourhood Spaces." Tokyo Journal of Mathematics 09, no. 1 (June 1986): 181–86. http://dx.doi.org/10.3836/tjm/1270150984.

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16

WORDSWORTH, J. R. "An intermediate value theorem for asymptotic values." Mathematical Proceedings of the Cambridge Philosophical Society 138, no. 1 (January 2005): 129–34. http://dx.doi.org/10.1017/s0305004104007844.

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17

Chantasartrassmee, Avapa, and Narong Punnim. "An Intermediate Value Theorem for the Arboricities." International Journal of Mathematics and Mathematical Sciences 2011 (2011): 1–7. http://dx.doi.org/10.1155/2011/947151.

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LetGbe a graph. The vertex (edge) arboricity ofGdenoted bya(G)(a1(G))is the minimum number of subsets into which the vertex (edge) set ofGcan be partitioned so that each subset induces an acyclic subgraph. Letdbe a graphical sequence and letR(d)be the class of realizations ofd. We prove that ifπ∈{a,a1}, then there exist integersx(π)andy(π)such thatdhas a realizationGwithπ(G)=zif and only ifzis an integer satisfyingx(π)≤z≤y(π). Thus, for an arbitrary graphical sequencedandπ∈{a,a1}, the two invariantsx(π)=min(π,d):=min{π(G):G∈R(d)}and y(π)=max(π,d):=max{π(G):G∈R(d)}naturally arise and henceπ(d):={π(G):G∈R(d)}={z∈Z:x(π)≤z≤y(π)}.We writed=rn:=(r,r,…,r)for the degree sequence of anr-regular graph of ordern. We prove thata1(rn)={⌈(r+1)/2⌉}. We consider the corresponding extremal problem on vertex arboricity and obtainmin(a,rn)in all situations andmax(a,rn)for alln≥2r+2.

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18

Fierro, Raul, Carlos Martinez, and Claudio H. Morales. "The aftermath of the intermediate value theorem." Fixed Point Theory and Applications 2004, no. 3 (2004): 516570. http://dx.doi.org/10.1155/s1687182004310053.

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19

Matveev, M. N. "An intermediate value theorem for face polytopes." Lobachevskii Journal of Mathematics 37, no. 3 (May 2016): 307–15. http://dx.doi.org/10.1134/s1995080216030173.

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20

Herzog, Gerd. "An intermediate value theorem in ordered Banach spaces." Annales Polonici Mathematici 98, no. 1 (2010): 63–69. http://dx.doi.org/10.4064/ap98-1-4.

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21

Ardeshir, Mohammad, and Rasoul Ramezanian. "The double negation of the intermediate value theorem." Annals of Pure and Applied Logic 161, no. 6 (March 2010): 737–44. http://dx.doi.org/10.1016/j.apal.2009.06.005.

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22

Gaber, Iris, Arieh Lev, and Romina Zigdon. "Insights and Observations on Teaching the Intermediate Value Theorem." American Mathematical Monthly 126, no. 9 (October 21, 2019): 845–49. http://dx.doi.org/10.1080/00029890.2019.1647061.

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23

Duca, Dorel I., and Ovidiu T. Pop. "On the intermediate point in Cauchy's mean-value theorem." Mathematical Inequalities & Applications, no. 3 (2006): 375–89. http://dx.doi.org/10.7153/mia-09-37.

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24

Duca, Dorel I., and Ovidiu T. Pop. "Concerning the intermediate point in the mean value theorem." Mathematical Inequalities & Applications, no. 3 (2009): 499–512. http://dx.doi.org/10.7153/mia-12-38.

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25

Hendtlass, Matthew. "The intermediate value theorem in constructive mathematics without choice." Annals of Pure and Applied Logic 163, no. 8 (August 2012): 1050–56. http://dx.doi.org/10.1016/j.apal.2011.12.026.

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26

Zhu, Sanguo. "An intermediate-value theorem for the upper quantization dimension." Journal of Mathematical Analysis and Applications 348, no. 1 (December 2008): 389–94. http://dx.doi.org/10.1016/j.jmaa.2008.07.043.

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27

Castro Pérez, Jaime, Andrés González Nucamendi, and Gerardo Pioquinto Aguilar Sánchez. "Approximate integration through remarkable points using the Intermediate Value Theorem." Scientia et Technica 25, no. 1 (March 30, 2020): 142–49. http://dx.doi.org/10.22517/23447214.21641.

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Using the Intermediate Value Theorem we demonstrate the rules of Trapeze and Simpson's. Demonstrations with this approach and its generalization to new formulas are less laborious than those resulting from methods such as polynomial interpolation or Gaussian quadrature. In addition, we extend the theory of approximate integration by finding new approximate integration formulas. The methodology we used to obtain this generalization was to use the definition of the integral defined by Riemann sums. Each Riemann sum provides an approximation of the result of an integral. With the help of the Intermediate Value Theorem and a detailed analysis of the Middle Point, Trapezoidal and Simpson Rules we note that these rules of numerical integration are Riemann sums. The results we obtain with this analysis allowed us to generalize each of the rules mentioned above and obtain new rules of approximation of integrals. Since each of the rules we obtained uses a point in the interval we have called them according to the point of the interval we take. In conclusion we can say that the method developed here allows us to give new formulas of numerical integration and generalizes those that already exist.

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28

Perrucci, Daniel, and Marie-Françoise Roy. "Quantitative fundamental theorem of algebra." Quarterly Journal of Mathematics 70, no. 3 (May 15, 2019): 1009–37. http://dx.doi.org/10.1093/qmath/haz008.

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Abstract Using subresultants, we modify a real-algebraic proof due to Eisermann of the fundamental theorem of Algebra (FTA) to obtain the following quantitative information: in order to prove the FTA for polynomials of degree d, the intermediate value theorem (IVT) is required to hold only for real polynomials of degree at most d2. We also explain that the classical proof due to Laplace requires IVT for real polynomials of exponential degree. These quantitative results highlight the difference in nature of these two proofs.

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29

Tan, Yu Cheng, Chang Shou Deng, and Yan Liu. "Mean Evolutionary Algorithm Based on Intermediate Value Theorem of Continuous Function." Advanced Materials Research 989-994 (July 2014): 1686–91. http://dx.doi.org/10.4028/www.scientific.net/amr.989-994.1686.

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A new Evolutionary algorithm was presented based on intermediate value theorem of continuous function. The global search ability and the local search ability of this algorithm are well balanced, and operators used in the algorithm are simple, Further, small size of population scale is used. Initial numerical experiments show that the mean evolutionary algorithm is better than differential evolution algorithm in solving high dimension function optimization.

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30

Shamseddine, Khodr, and Martin Berz. "Intermediate Value Theorem for Analytic Functions on a Levi-Civita Field." Bulletin of the Belgian Mathematical Society - Simon Stevin 14, no. 5 (December 2007): 1001–15. http://dx.doi.org/10.36045/bbms/1197908910.

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31

Bau, Sheng, Benjamin van Niekerk, and David White. "An Intermediate Value Theorem for the Decycling Numbers of Toeplitz Graphs." Graphs and Combinatorics 31, no. 6 (December 4, 2014): 2037–42. http://dx.doi.org/10.1007/s00373-014-1492-3.

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32

Lindenstrauss, Elon, Yuval Peres, and Wilhelm Schlag. "Bernoulli convolutions and an intermediate value theorem for entropies ofK-partitions." Journal d'Analyse Mathématique 87, no. 1 (December 2002): 337–67. http://dx.doi.org/10.1007/bf02868480.

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33

Kabbouch, Oussama, and Mustapha Najmeddine. "The Reverse of the Intermediate Value Theorem in Some Topological Spaces." International Journal of Mathematics and Mathematical Sciences 2021 (April 10, 2021): 1–4. http://dx.doi.org/10.1155/2021/6320969.

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Any continuous function with values in a Hausdorff topological space has a closed graph and satisfies the property of intermediate value. However, the reverse implications are false, in general. In this article, we treat additional conditions on the function, and its graph for the reverse to be true.

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34

Gomes, Abel, and José Morgado. "A Generalized Regula Falsi Method for Finding Zeros and Extrema of Real Functions." Mathematical Problems in Engineering 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/394654.

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Many zero-finding numerical methods are based on the Intermediate Value Theorem, which states that a zero of a real function is bracketed in a given interval if and have opposite signs; that is, . But, some zeros cannot be bracketed this way because they do not satisfy the precondition . For example, local minima and maxima that annihilate may not be bracketed by the Intermediate Value Theorem. In this case, we can always use a numerical method for bracketing extrema, checking then whether it is a zero of or not. Instead, this paper introduces a single numerical method, calledgeneralized regula falsi(GRF) method to determine both zeros and extrema of a function. Consequently, it differs from the standardregula falsi methodin that it is capable of finding any function zero in a given interval even when the Intermediate Value Theorem is not satisfied.

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35

Liu, Dongyuan, and Zigen Ouyang. "Solvability of Third-Order Three-Point Boundary Value Problems." Abstract and Applied Analysis 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/793639.

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We are interested in the existence theorems for a third-order three-point boundary value problem. In the nonresonant case, using the Krasnosel’skii fixed point theorem, we obtain some sufficient conditions for the existence of the positive solutions. In addition, we focus on the resonant case, the boundary value problem being transformed into an integral equation with an undetermined parameter, and the existence conditions being obtained by the Intermediate Value Theorem.

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36

Patinkin, Seth. "Stirring our way to Sharkovsky's theorem." Bulletin of the Australian Mathematical Society 56, no. 3 (December 1997): 453–58. http://dx.doi.org/10.1017/s0004972700031245.

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The periodic-point or cycle structure of a continuous map of a topological space has been a subject of great interest since A.N. Sharkovsky completely explained the hierarchy of periodic orders of a continuous map f: R → R, where R is the real line. In this paper the topological idea of “stirring” is invoked in an effort to obtain a transparent proof of a generalisation of Sharkovsky's Theorem to continuous functions f: I → I where I is any interval. The stirring approach avoids all graph-theoretical and symbolic abstraction of the problem in favour of a more concrete intermediate-value-theorem-oriented analysis of cycles inside an interval.

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37

Corgnier, Luigi, Carla Massaza, and Paolo Valabrega. "Hensel's lemma and the intermediate value theorem over a non-Archimedean field." Journal of Commutative Algebra 9, no. 2 (June 2017): 185–211. http://dx.doi.org/10.1216/jca-2017-9-2-185.

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38

HENRIKSEN, MELVIN, SUZANNE LARSON, and JORGE MARTINEZ. "The Intermediate Value Theorem for Polynomials over Lattice-ordered Rings of Functions." Annals of the New York Academy of Sciences 788, no. 1 General Topol (May 1996): 108–23. http://dx.doi.org/10.1111/j.1749-6632.1996.tb36802.x.

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39

Julian, William H., Ray Milnes, and Fred Richman. "The intermediate value theorem: preimages of compact sets under uniformly continuous functions." Rocky Mountain Journal of Mathematics 18, no. 1 (March 1988): 25–36. http://dx.doi.org/10.1216/rmj-1988-18-1-25.

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40

Berger, Josef, Hajime Ishihara, Takayuki Kihara, and Takako Nemoto. "The binary expansion and the intermediate value theorem in constructive reverse mathematics." Archive for Mathematical Logic 58, no. 1-2 (May 10, 2018): 203–17. http://dx.doi.org/10.1007/s00153-018-0627-2.

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41

Vrahatis, Michael N. "Intermediate value theorem for simplices for simplicial approximation of fixed points and zeros." Topology and its Applications 275 (April 2020): 107036. http://dx.doi.org/10.1016/j.topol.2019.107036.

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42

Cameron, Thomas. "Spectral Bounds for Matrix Polynomials with Unitary Coefficients." Electronic Journal of Linear Algebra 30 (February 8, 2015): 585–91. http://dx.doi.org/10.13001/1081-3810.2911.

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It is well known that the eigenvalues of any unitary matrix lie on the unit circle. The purpose of this paper is to prove that the eigenvalues of any matrix polynomial, with unitary coefficients, lie inside the annulus A_{1/2,2) := {z â C | 1/2 < |z| < 2}. The foundations of this result rely on an operator version of Roucheâs theorem and the intermediate value theorem.

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43

Massaza, Carla, Lea Terracini, and Paolo Valabrega. "On an intermediate value theorem for polynomials and power series over a valued field." Communications in Algebra 45, no. 10 (December 23, 2016): 4528–41. http://dx.doi.org/10.1080/00927872.2016.1270955.

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44

Barany, Michael J. "Stuck in the Middle: Cauchy’s Intermediate Value Theorem and the History of Analytic Rigor." Notices of the American Mathematical Society 60, no. 10 (November 1, 2013): 1. http://dx.doi.org/10.1090/noti1049.

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45

David, Erika J., Kyeong Hah Roh, and Morgan E. Sellers. "Teaching the Representations of Concepts in Calculus: The Case of the Intermediate Value Theorem." PRIMUS 30, no. 2 (April 25, 2019): 191–210. http://dx.doi.org/10.1080/10511970.2018.1540023.

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46

Shamseddine, Khodr, and Todd Sierens. "On Locally Uniformly Differentiable Functions on a Complete Non-Archimedean Ordered Field Extension of the Real Numbers." ISRN Mathematical Analysis 2012 (April 17, 2012): 1–20. http://dx.doi.org/10.5402/2012/387053.

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We study the properties of locally uniformly differentiable functions on N, a non-Archimedean field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In particular, we show that locally uniformly differentiable functions are C1, they include all polynomial functions, and they are closed under addition, multiplication, and composition. Then we formulate and prove a version of the inverse function theorem as well as a local intermediate value theorem for these functions.

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47

Pop, Emilia-Loredana, Dorel Duca, and Augusta Raţiu. "Properties of the intermediate point from a mean value theorem of the integral calculus - II." General Mathematics 27, no. 1 (June 1, 2019): 29–36. http://dx.doi.org/10.2478/gm-2019-0003.

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Abstract In this paper we consider two continuous functions f, g : [a, b] → ℝ and we study for these ones, under which circumstances the intermediate point function is four order di erentiable at the point x = a and we calculate its derivative.

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48

Pop, Emilia-Loredana, Dorel Duca, and Augusta Raţiu. "Calculus for the intermediate point associated with a mean value theorem of the integral calculus." General Mathematics 28, no. 1 (June 1, 2020): 59–66. http://dx.doi.org/10.2478/gm-2020-0005.

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AbstractIf f, g: [a, b] → 𝕉 are two continuous functions, then there exists a point c ∈ (a, b) such that\int_a^c {f\left(x \right)} dx + \left({c - a} \right)g\left(c \right) = \int_c^b {g\left(x \right)} dx + \left({b - c} \right)f\left(c \right).In this paper, we study the approaching of the point c towards a, when b approaches a.

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49

Oman, Greg. "The Converse of the Intermediate Value Theorem: From Conway to Cantor to Cosets and Beyond." Missouri Journal of Mathematical Sciences 26, no. 2 (November 2014): 134–50. http://dx.doi.org/10.35834/mjms/1418931955.

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50

Palmgren, Erik. "Developments in Constructive Nonstandard Analysis." Bulletin of Symbolic Logic 4, no. 3 (September 1998): 233–72. http://dx.doi.org/10.2307/421031.

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AbstractWe develop a constructive version of nonstandard analysis, extending Bishop's constructive analysis with infinitesimal methods. A full transfer principle and a strong idealisation principle are obtained by using a sheaf-theoretic construction due to I. Moerdijk. The construction is, in a precise sense, a reduced power with variable filter structure. We avoid the nonconstructive standard part map by the use of nonstandard hulls. This leads to an infinitesimal analysis which includes nonconstructive theorems such as the Heine–Borel theorem, the Cauchy–Peano existence theorem for ordinary differential equations and the exact intermediate-value theorem, while it at the same time provides constructive results for concrete statements. A nonstandard measure theory which is considerably simpler than that of Bishop and Cheng is developed within this context.

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