The Symmetric Derivative

Larson, L. E.

doi:10.2307/1999226

Cited by 4 publications

(1 citation statement)

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“…The resulting data set contains 9,273,720 data points (1,854,744 for each fish) including all the ten trials. Velocity was numerically derived from position using the symmetric difference quotient two-point estimation [39]. Heading was then computed as the four-quadrant inverse tangent of velocity and used to compute transfer entropy.…”

Section: Data Extraction and Pre-processingmentioning

confidence: 99%

Informative and misinformative interactions in a school of fish

et al. 2018

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It is generally accepted that, when moving in groups, animals process information to coordinate their motion. Recent studies have begun to apply rigorous methods based on Information Theory to quantify such distributed computation. Following this perspective, we use transfer entropy to quantify dynamic information flows locally in space and time across a school of fish during directional changes around a circular tank, i.e. U-turns. This analysis reveals peaks in information flows during collective U-turns and identifies two different flows: an informative flow (positive transfer entropy) based on fish that have already turned about fish that are turning, and a misinformative flow (negative transfer entropy) based on fish that have not turned yet about fish that are turning. We also reveal that the information flows are related to relative position and alignment between fish, and identify spatial patterns of information and misinformation cascades. This study offers several methodological contributions and we expect further application of these methodologies to reveal intricacies of self-organisation in other animal groups and active matter in general. * emanuele.crosato@sydney.edu.au arXiv:1705.01213v1 [q-bio.QM] 3 May 2017 Nagy et al. [55] used a variety of correlation functions to measure directional dependencies between the velocities of pairs of pigeons flying in flocks of up to ten individuals, reconstructing the leadership network of the flock. As has been shown later, this network does not correspond to the hierarchy between birds [56]. Information transfer has been extensively studied in flocks of starlings, by observing the propagation of direction changes across the flocks [20,19,2]. More recently, Rosenthal et al. [69] attempted to determine a communication structure of a school of fish during its collective evasion manoeuvres manifested through cascades of behavioural change. A functional mapping between sensory inputs and motor responses was inferred by tracking fish position and body posture, and calculating visual fields.Rather than consider semantic or pragmatic information, many contemporary studies employ rigorous information theoretic measures that quantify information as uncertainty reduction, following Shannon [24], in order to deal with the stochastic, continuous and noisy nature of intrinsic information processing in natural systems [28]. Distributed information processing is typically dissected into three primitive functions: the transmission, storage and modification of information [38]. Information dynamics is a recent framework characterising and measuring each of the primitives information-theoretically [49,41]. In viewing the state update dynamics of a random process as an information processing event, this framework performs an information regression in accounting for where the information to predict that state update can be found by an observer, first identifying predictive information from the past of the process as information storage, then predictive information from other sour...

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Section: Data Extraction and Pre-processingmentioning

confidence: 99%

Informative and misinformative interactions in a school of fish

et al. 2018

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Qualitative differentiation

Evans¹,

Larson²

1983

Trans. Amer. Math. Soc.

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Abstract. Qualitative dérivâtes and derivatives, as well as qualitative symmetric dérivâtes and derivatives, are studied in the paper. Analogues of several results known for ordinary dérivâtes and derivatives are obtained in the qualitative setting.1. Introduction. The notions of qualitative limits, qualitative continuity, and qualitative derivatives were introduced by S. Marcus [13][14][15]. The purpose of the present paper is to examine qualitative differentiation and qualitative symmetric differentiation and, in particular, to present analogues of results known to hold for ordinary differentiation, symmetric differentiation, approximate differentiation, and approximate symmetric differentiation. Loosely speaking, qualitative differentiation and qualitative symmetric differentiation may be thought of as category analogues of approximate differentiation and approximate symmetric differentiation, where the set neglected near a point in the computation of difference quotients is of first category at the point in the former setting instead of density zero at the point as in the latter.We state our definitions in §2. In §3 we examine qualitative derivatives and dérivâtes. There we show that a qualitatively differentiable function on the real line is actually differentiable everywhere and obtain what may be viewed as qualitative analogues of the Denjoy-Young-Saks theorem [18]. In §4 we consider qualitative symmetric derivatives and dérivâtes. We show that with mild continuity restrictions on the primitive, a qualitative symmetric derivative must belong to Baire class one and actually be the symmetric derivative of a closely related function except at countably many points. A monotonicity theorem and related results are given.

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The generalized Zahorski class structure of symmetric derivatives

Larson¹

1984

Trans. Amer. Math. Soc.

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ABSTRACT. A generalized Zahorski class structure is demonstrated for symmetric derivatives. A monotonicity theorem is proved and a condition sufficient to ensure that a symmetric derivative has the Darboux property is presented.l. Preliminaries. In 1950 Z. Zahorski [9) began a classification of the derivatives of continuous functions based upon the structure of their associated sets. He defined a descending sequence of subclasses of the Darboux-Baire one functions which form a stratification of the class of derivatives he considered. Kundu [6), in 1976, generalized the Zahorski classes in order to extend the theorems of Zahorski to include symmetric derivatives which have the Darboux property and continuous primitives. Our purpose in this work is to replace Kundu's requirement that the primitives be continuous by the more general requirement of measurability and, when possible, to remove the requirement that the symmetric derivatives have the Darboux property.It will be assumed here that, unless it is specifically mentioned otherwise, all functions are finite valued and have their domains contained in R, the real numbers. exists, whether finite or infinite, then I is symmetrically differentiable at x and its symmetric derivative is denoted I S( x).The complement of an arbitrary set A C R is denoted by A C and its closure is A. For simplicity we write (AY as Xc. If A is measurable, then its measure is written as IAI·

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The Symmetric Derivative

Cited by 4 publications

References 9 publications

Informative and misinformative interactions in a school of fish

Informative and misinformative interactions in a school of fish

Qualitative differentiation

The generalized Zahorski class structure of symmetric derivatives

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