scholarly journals ON AN APPLICATION OF A VARIANT OF THE CLOSED GRAPH THEOREM AND THE SECANT METHOD

1993 ◽  
Vol 24 (3) ◽  
pp. 251-267
Author(s):  
IOANNIS K. ARGYROS
Keyword(s):  
Lipschitz Condition ◽  
Error Bounds ◽  
Mathematical Induction ◽  
Secant Method ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Hölder Continuous ◽  
Nondiscrete Mathematical Induction

The method of nondiscrete mathematical induction is used to find error bounds for the Secant method. We assume only that the operator has Holder continuous derivatives. In the case the Frechet­ derivative of the operator satisfies a Lipschitz condition our results reduce to the ones obtained by F. Potra (Num. Math. 1982).

scholarly journals Convergence by Nondiscrete Mathematical Induction of a Two Step Secant's Method

2007 ◽  
Vol 37 (2) ◽  
pp. 359-369 ◽  
Author(s):  
S. Amat ◽  
C. Bermúdez ◽  
S. Busquier ◽  
J. Gretay
Keyword(s):  
Mathematical Induction ◽  
Nondiscrete Mathematical Induction

scholarly journals AN ERROR ANALYSIS OF STIRLING'S METHOD IN BANACH SPACES

1993 ◽  
Vol 24 (2) ◽  
pp. 115-133
Author(s):  
IOANNIS K. ARGYROS
Keyword(s):  
Error Analysis ◽  
Banach Spaces ◽  
Error Estimates ◽  
Simple Proof ◽  
Mathematical Induction ◽  
Nondiscrete Mathematical Induction ◽  
Better Than

The method of nondiscrete mathematical induction is applied to Stirling's method. The method yields a very simple proof of the convergence and error estimates which are generally better than those given in the literature.

scholarly journals On Newton's method and nondiscrete mathematical induction

1988 ◽  
Vol 38 (1) ◽  
pp. 131-140 ◽  
Author(s):  
Ioannis K. Argyros
Keyword(s):  
Lipschitz Condition ◽  
Error Bounds ◽  
Newton’S Method ◽  
Newton's Method ◽  
Mathematical Induction ◽  
Fréchet Derivative ◽  
Hölder Continuous ◽  
Frechet Derivative ◽  
Nondiscrete Mathematical Induction ◽  
Sharp Error

The method of nondiscrete mathematical induction is used to find sharp error bounds for Newton's method. We assume only that the operator has Hölder continuous derivatives. In the case when the Fréchet-derivative of the operator satisfies a Lipschitz condition, our results reduce to the ones obtained by Ptak and Potra in 1972.

Sourcebook in the Mathematics of Medieval Europe and North Africa

2016 ◽  
Keyword(s):  
North Africa ◽  
Cultural Influences ◽  
Mathematical Induction ◽  
Cross Cultural ◽  
General Introduction ◽  
Extensive Survey ◽  
Medieval Europe ◽  
Ceva’S Theorem ◽  
Arabic Speaking ◽  
First Time

Medieval Europe was a meeting place for the Christian, Jewish, and Islamic civilizations, and the fertile intellectual exchange of these cultures can be seen in the mathematical developments of the time. This book presents original Latin, Hebrew, and Arabic sources of medieval mathematics, and shows their cross-cultural influences. Most of the Hebrew and Arabic sources appear here in translation for the first time. Readers will discover key mathematical revelations, foundational texts, and sophisticated writings by Latin, Hebrew, and Arabic-speaking mathematicians, including Abner of Burgos's elegant arguments proving results on the conchoid—a curve previously unknown in medieval Europe; Levi ben Gershon's use of mathematical induction in combinatorial proofs; Al-Muʾtaman Ibn Hūd's extensive survey of mathematics, which included proofs of Heron's Theorem and Ceva's Theorem; and Muhyī al-Dīn al-Maghribī's interesting proof of Euclid's parallel postulate. The book includes a general introduction, section introductions, footnotes, and references.

Regions-based Two-dimensional Continua: The Euclidean Case

2018 ◽  
Author(s):  
Geoffrey Hellman ◽  
Stewart Shapiro
Keyword(s):  
Mathematical Induction ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Entire Space ◽  
Euclidean Case ◽  
Free Basis ◽  
One Dimensional ◽  
Archimedean Property ◽  
The One

This chapter develops a Euclidean, two-dimensional, regions-based theory. As with the semi-Aristotelian account in Chapter 2, the goal here is to recover the now orthodox Dedekind–Cantor continuum on a point-free basis. The chapter derives the Archimedean property for a class of readily postulated orientations of certain special regions, what are called “generalized quadrilaterals” (intended as parallelograms), by which the entire space is covered. Then the chapter generalizes this to arbitrary orientations, and then establishes an isomorphism between the space and the usual point-based one. As in the one-dimensional case, this is done on the basis of axioms which contain no explicit “extremal clause”, and we have no axiom of induction other than ordinary numerical (mathematical) induction.

FOSTERING AN APPRECIATION OF MATHEMATICAL INDUCTION

PRIMUS ◽  
1995 ◽  
Vol 5 (3) ◽  
pp. 218-228
Author(s):  
Lynn Kiaer
Keyword(s):  
Mathematical Induction
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