1,350 research outputs found
Unified multi-tupled fixed point theorems involving mixed monotone property in ordered metric spaces
In the present article, we introduce a unified notion of multi-tupled fixed
points and utilize the same to prove some existence and uniqueness unified
multi-tupled fixed point theorems for Boyd-Wong type nonlinear contractions
satisfying generalized mixed monotone property in ordered metric spaces. Our
results unify several classical and well-known n-tupled (including coupled,
tripled and quadrupled ones) fixed point results existing in the literature.Comment: arXiv admin note: substantial text overlap with arXiv: 1601.0251
From Euclidean Geometry to Knots and Nets
This document is the Accepted Manuscript of an article accepted for publication in Synthese. Under embargo until 19 September 2018. The final publication is available at Springer via https://doi.org/10.1007/s11229-017-1558-x.This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if (a) it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, (b) the information thus displayed is not metrical and (c) it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions.Peer reviewe
Report on some recent advances in Diophantine approximation
A basic question of Diophantine approximation, which is the first issue we
discuss, is to investigate the rational approximations to a single real number.
Next, we consider the algebraic or polynomial approximations to a single
complex number, as well as the simultaneous approximation of powers of a real
number by rational numbers with the same denominator. Finally we study
generalisations of these questions to higher dimensions. Several recent
advances have been made by B. Adamczewski, Y. Bugeaud, S. Fischler, M. Laurent,
T. Rivoal, D. Roy and W.M. Schmidt, among others. We review some of these
works.Comment: to be published by Springer Verlag, Special volume in honor of Serge
Lang, ed. Dorian Goldfeld, Jay Jorgensen, Dinakar Ramakrishnan, Ken Ribet and
John Tat
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