A metric space connected with generalized means

Cargo, G. T.; Shisha, Oved

doi:10.1016/0021-9045(69)90041-0

Cited by 10 publications

(12 citation statements)

References 2 publications

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“…However, the absolute difference of power means have been shown to represent a distance, or metric, on the set of all continuous and monotonic functions in the domain [z c , z h ] [5].…”

Section: Metric Space Of Power Mean Differencesmentioning

confidence: 99%

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Thermodynamics of an ideal generalized gas: II

Lavenda

2005

Naturwissenschaften

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The property that power means are monotonically increasing functions of their order is shown to be the basis of the second laws not only for processes involving heat conduction, but also for processes involving deformations. This generalizes earlier work involving only pure heat conduction and underlines the incomparability of the internal energy and adiabatic potentials when expressed as powers of the adiabatic variable. In an L-potential equilibration, the final state will be one of maximum entropy, whereas in an entropy equilibration, the final state will be one of minimum L. Unlike classical equilibrium thermodynamic phase space, which lacks an intrinsic metric structure insofar as distances and other geometrical concepts do not have an intrinsic thermodynamic significance in such spaces, a metric space can be constructed for the power means: the distance between means of different order is related to the Carnot efficiency. In the ideal classical gas limit, the average change in the entropy is shown to be proportional to the difference between the Shannon and Rényi entropies for nonextensive systems that are multifractal in nature. The L potential, like the internal energy, is a Schur convex function of the empirical temperature, which satisfies Jensen's inequality, and serves as a measure of the tendency to uniformity in processes involving pure thermal conduction.

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“…However, the absolute difference of power means have been shown to represent a distance, or metric, on the set of all continuous and monotonic functions in the domain [z c , z h ] [5].…”

Section: Metric Space Of Power Mean Differencesmentioning

confidence: 99%

“…The properties of the metric space of equivalent classes on the set of all continuous and monotonic functions have been elucidated in Cargo and Shisha [5]. In particular, the metric space is separable since there is a countable subset everywhere dense in it, like that of the real line.…”

Section: Metric Space Of Power Mean Differencesmentioning

confidence: 99%

Thermodynamics of an ideal generalized gas: II

Lavenda

2005

Naturwissenschaften

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“…4]). In particular, in the 1960s Cargo and Shisha [3] introduced a metric among them. Namely, if f and g are both continuous, strictly monotone and have the same domain, then one can define a distance ρ A [f ] , A [ The original definition was established in a different way, nevertheless this wording is equivalent.…”

Section: Introductionmentioning

confidence: 99%

“…The main goal of the present paper is to establish some lower bounds of the distance ρ. In [3], there was presented a number of majorizations of ρ. In particular few of them concern lower boundaries.…”

Section: Introductionmentioning

confidence: 99%

“…By [3,Theorem 6.1] the maximal value of the distance between quasiarithmetic means is obtained for a vector of length two with some weights (it obviously does not imply that it cannot be obtained for any other vector of entries). More precisely in the definition above we can assume, without loss of generality, that the vector a has length two.…”

Section: Introductionmentioning

confidence: 99%

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Lower estimation of the difference between quasi-arithmetic means

Pasteczka¹

2017

Aequat. Math.

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Abstract. In the 1960s Cargo and Shisha introduced a metric in a family of quasi-arithmetic means defined on a common interval as the maximal possible difference between these means taken over all admissible vectors with corresponding weights. During the years 2013-2016 we proved that, having two quasi-arithmetic means, we can majorize the distance between them in terms of the Arrow-Pratt index. In this paper we are going to prove that this operator can also be used to establish certain lower bounds of this distance.Mathematics Subject Classification. 26E60, 26D15, 26D07.

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Quasi-arithmetic-type invariant means on probability space

Derȩgowska

Pasteczka

2021

Aequat. Math.

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A metric space connected with generalized means

Cited by 10 publications

References 2 publications

Thermodynamics of an ideal generalized gas: II

Thermodynamics of an ideal generalized gas: II

Lower estimation of the difference between quasi-arithmetic means

Quasi-arithmetic-type invariant means on probability space

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