Experimental mathematics and computational statistics

Bailey, David H.; Borwein, Jonathan M.

doi:10.1002/wics.1

Cited by 4 publications

(4 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The conjecture that certain fundamental constants -irrational numbers such as π, e, √ 2, and ln 2-are normal has attracted much scientific scrutiny (e.g., [Bailey and Borwein, 2009, Bailey and Crandall, 2001, Borwein et al, 2004). Aside from theoretical interest and practical application, the enduring fascination with this topic may be due in part to the paradoxical result that the digits sequences are perfectly predictable yet apparently appear random: "Plenty of arrangements in which design had a hand [...] would be quite indistinguishable in their results from those in which no design whatever could be traced.…”

Section: The Normality Of Irrational Numbersmentioning

confidence: 99%

“…Experimental mathematics focuses on data and computation in order to address and discover mathematical questions that have so far escaped formal proof [Bailey and Borwein, 2009]. In many cases, this means that mathematical conjectures are examined by studying their consequences for a large range of data; every time a consequence is confirmed this increases one's confidence in the veracity of the conjecture.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Bayesian Evidence Accumulation in Experimental Mathematics: A Case Study of Four Irrational Numbers

Gronau

Wagenmakers

2017

Experimental Mathematics

View full text Add to dashboard Cite

Many questions in experimental mathematics are fundamentally inductive in nature. Here we demonstrate how Bayesian inference -the logic of partial beliefs-can be used to quantify the evidence that finite data provide in favor of a general law. As a concrete example we focus on the general law which posits that certain fundamental constants (i.e., the irrational numbers π, e, √ 2, and ln 2) are normal; specifically, we consider the more restricted hypothesis that each digit in the constant's decimal expansion occurs equally often. Our analysis indicates that for each of the four constants, the evidence in favor of the general law is overwhelming. We argue that the Bayesian paradigm is particularly apt for applications in experimental mathematics, a field in which the plausibility of a general law is in need of constant revision in light of data sets whose size is increasing continually and indefinitely.

show abstract

Section: The Normality Of Irrational Numbersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Bayesian Evidence Accumulation in Experimental Mathematics: A Case Study of Four Irrational Numbers

Gronau

Wagenmakers

2017

Experimental Mathematics

View full text Add to dashboard Cite

show abstract

“…Let us elaborate. We first introduce the logistic map in its familiar form (Nicolis, 1995(Nicolis, , 1998Nicolis and Gaspard, 1994;MacKernan and Nicolis, 1994;Bailey and Borwein, 2009;Karamanos and Nicolis, 1999;Ebeling et al, 2001;Steuer et al, 2001;Karamanos, 2001;Karamanos and Kotsireas, 2002;Borwein and Karamanos, 2005):…”

Section: Logistic Mapmentioning

confidence: 99%

The many facets of Poincare recurrence theorem of the logistic map

Karamanos

Mistakidis

2012

Kybernetes

View full text Add to dashboard Cite

Purpose -The purpose of this paper is to illustrate the many aspects of Poincare recurrence time theorem for an archetype of a complex system, the logistic map. Design/methodology/approach -At the beginning of the twentieth century, Poincare's recurrence theorem had revolutionized modern mechanics and statistical physics. However, this theorem did not attract considerable attention, at least from a numerical and computational point of view. In a series of relatively recent papers, Balakrishnan, Nicolis and Nicolis have addressed the recurrence time problem in a firm basis, introducing notation, theory, and numerical studies. Motivated by this call, the paper proposes to illustrate the many aspects of Poincare recurrence time theorem for an archetype of a complex system, the logistic map. The authors propose here in different tests and computations, each one illuminating the many aspects of the problem of recurrence. The paper ends up with a short discussion and conclusions. Findings -In this paper, the authors obtain new results on computations, each one illuminating the many aspects of the problem of recurrence. One striking aspect of this detailed work, is that when the sizes of the cells in the phase space became considerable, then the recurrence times assume ordinary values. Originality/value -The paper extends previous results on chaotic maps to the logistic map, enhancing comprehension, making possible connections with number theory, combinatorics and cryptography. IntroductionIn its classical version, Poincare recurrence theorem refers to a one-parameter family F 0 of one-to-one measure-preserving transformations. It states that if C is a subset of the phase space G such that m(C) . 0 (m is a completely additive measure with m(G) ¼ 1), then for almost every point P [ C there exists a sufficiently large t such that F 0 P [ C.

show abstract

“…The calculation of the block entropies for the logistic map clearly reflects the subtle interplay of the transfer of information between the "syntactic" and "grammatical levels" (Nicolis, 1991). To advance in these topics, one has to resort to heavy computational statistics, also related to the existing information technology (Bailey and Borwein, 2009).…”

Section: Introductionmentioning

confidence: 99%

Statistical versus optimal partitioning for block entropies

2013

View full text Add to dashboard Cite

Purpose -Given a time-series, what is the best partitioning of the state space in order to obtain reasonable values for the block entropies? The purpose of this paper is to provide a simple answer (an algorithm), although approximative, in connection with symbolic dynamics and statistical properties of 1-d maps on the interval. Design/methodology/approach -The logistic map is examined as an archetype of a Complex System with different behaviors, namely: periodicity, order-to-chaos period-doubling transition, weak chaos, parametric intermittent chaos, developed chaos and fully developed chaos. For the logistic map the generating partition is known, and allows comparison with other prescriptions in the literature. The partitioning of the phase space with the easy generated bipartition induced by the mean value of a curve in the plane, gives results in good agreement (roughly up to a 20 per cent difference) with the results of the generating partition, if the trajectory of the system is in parametric intermittent chaos and in developed chaos (DC). In the case of fully developed chaos (FDC), the agreement is perfect. Findings -The authors confirm that a statistical partitioning is almost equivalent with the exact partitioning for the logistic map. Originality/value -The paper updates previous results and proposes a better understanding on the partitioning for symbolic dynamics.

show abstract

Experimental mathematics and computational statistics

Cited by 4 publications

References 21 publications

Bayesian Evidence Accumulation in Experimental Mathematics: A Case Study of Four Irrational Numbers

Bayesian Evidence Accumulation in Experimental Mathematics: A Case Study of Four Irrational Numbers

The many facets of Poincare recurrence theorem of the logistic map

Statistical versus optimal partitioning for block entropies

Contact Info

Product

Resources

About