Singularly continuous spectrum of a self-similar Laplacian on the half-line

International audience Sampling from a random discrete distribution induced by a 'stick-breaking' process is considered. Under a moment condition, it is shown that the asymptotics of the sequence of occupancy numbers, and of the small-parts counts (singletons, doubletons, etc) can be read off from a limiting model involving a unit Poisson point process and a self-similar renewal process on the half-line.

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Monotonicity in the averaging process

Journal of Physics A Mathematical and Theoretical ◽

10.1088/1751-8121/ac354f ◽

2021 ◽

Author(s):

Eli Ben-Naim ◽

Paul L Krapivsky

Keyword(s):

Continuous Spectrum ◽

Elementary Step ◽

Statistical Properties ◽

Two Agents ◽

Interacting Agents ◽

Self Similar

Abstract We investigate an averaging process that describes how interacting agents approach consensus through binary interactions. In each elementary step, two agents are selected at random and they reach compromise by adopting their opinion average. We show that the fraction of agents with a monotonically decreasing opinion decays as $e^{-\alpha t}$, and that the exponent $\alpha=\tfrac{1}{2}-\tfrac{1+\ln \ln 2}{4\ln 2}$ is selected as the extremum from a continuous spectrum of possible values. The opinion distribution of monotonic agents is asymmetric, and it becomes self-similar at large times. Furthermore, the tails of the opinion distribution are algebraic, and they are characterized by two distinct and nontrivial exponents. We also explore statistical properties of agents with an opinion strictly above average.

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C. The Continuous Spectrum of Pulsating Variable Stars

Symposium - International Astronomical Union ◽

10.1017/s007418090001901x ◽

1967 ◽

Vol 28 ◽

pp. 177-206

Author(s):

J. B. Oke ◽

C. A. Whitney

Keyword(s):

Continuous Spectrum ◽

Variable Stars ◽

Velocity Fields ◽

The Other ◽

Line Spectrum ◽

Direct Information ◽

Thermodynamic Structure ◽

Diagnostic Interpretation ◽

Pulsating Variable Stars ◽

The Continuum

Pecker:The topic to be considered today is the continuous spectrum of certain stars, whose variability we attribute to a pulsation of some part of their structure. Obviously, this continuous spectrum provides a test of the pulsation theory to the extent that the continuum is completely and accurately observed and that we can analyse it to infer the structure of the star producing it. The continuum is one of the two possible spectral observations; the other is the line spectrum. It is obvious that from studies of the continuum alone, we obtain no direct information on the velocity fields in the star. We obtain information only on the thermodynamic structure of the photospheric layers of these stars–the photospheric layers being defined as those from which the observed continuum directly arises. So the problems arising in a study of the continuum are of two general kinds: completeness of observation, and adequacy of diagnostic interpretation. I will make a few comments on these, then turn the meeting over to Oke and Whitney.

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