The Depoissonisation Quintet: Rice-Poisson-Mellin-Newton-Laplace

This paper is devoted to the Depoissonisation process which is central in various analyses of the AofA domain. We first recall in Section 1 the two possible paths that may be used in this process, namely the Depoissonisation path and the Rice path. The two paths are rarely described for themselves in the literature, and general methodological results are often difficult to isolate amongst particular results that are more directed towards various applications. The main results for the Depoissonisation path are scattered in at least five papers, with a chronological order which does not correspond to the logical order of the method. The Rice path is also almost always presented with a strong focus towards possible applications. It is often very easy to apply, but it needs a tameness condition, which appears a priori to be quite restrictive, and is not deeply studied in the literature. This explains why the Rice path is very often undervalued. Second, the two paths are not precisely compared, and the situation creates various "feelings": some people see the tools that are used in the two paths as quite different, and strongly prefer one of the two paths; some others think the two paths are almost the same, with just a change of vocabulary. It is thus useful to compare the two paths and the tools they use. This will be done in Sections 2 and 3. We also "follow" this comparison on a precise problem, related to the analysis of tries, introduced in Section 1.7. The paper also exhibits in Section 4 a new framework, of practical use, where the tameness condition of Rice path is proven to hold. This approach, perhaps of independent interest, deals with the shifting of sequences and then the inverse Laplace transform, which does not seem of classical use in this context. It performs very simple computations. This adds a new method to the Depoissonisation context and explains the title of the paper. We then conclude that the Rice path is both of easy and practical use: even though (much?) less general than the Depoissonisation path, it is easier to apply

Process convergence for the complexity of Radix Selection on Markov sources

A fundamental algorithm for selecting ranks from a finite subset of an ordered set is Radix Selection. This algorithm requires the data to be given as strings of symbols over an ordered alphabet, e.g., binary expansions of real numbers. Its complexity is measured by the number of symbols that have to be read. In this paper the model of independent data identically generated from a Markov chain is considered. The complexity is studied as a stochastic process indexed by the set of infinite strings over the given alphabet. The orders of mean and variance of the complexity and, after normalization, a limit theorem with a centered Gaussian process as limit are derived. This implies an analysis for two standard models for the ranks: uniformly chosen ranks, also called grand averages, and the worst case rank complexities which are of interest in computer science. For uniform data and the asymmetric Bernoulli model (i.e. memoryless sources), we also find weak convergence for the normalized process of complexities when indexed by the ranks while for more general Markov sources these processes are not tight under the standard normalizations.Comment: main results significantly improved, 4 figure

Nonlinear dynamics of trapped beams

Die vorliegende Arbeit befasst sich mit der nichtlinearen Propagation von Licht in schwach führenden Wellenleitern und in der Atmosphäre. Bei der Lichtausbreitung im Wellenleiter leiten wir ein hinreichendes Kriterium für die Stabilität schwach nichtlinearer Wellenleitermoden ab. Der zweite Teil der Arbeit beschäftigt sich mit der Ausbreitung hochintensiver ultrakurzer Laserpulse in der Atmosphäre, insbesondere mit multipler Filamentierung

Light fields in the ocean

The problem of light fields in the ocean is in basic ocean optics. Twenty-six separate studies discuss: (1) the field of solar radiation in the ocean; (2) stationary and nonstationary light fields created in the sea by artificial sources; and (3) the use of optical methods to study biological and hydrodynamic characteristics of the sea

Proceedings of the 7th Sound and Music Computing Conference

Proceedings of the SMC2010 - 7th Sound and Music Computing Conference, July 21st - July 24th 2010

Secular cooling of the solid Earth, emergence of the continents, and evolution of Earth's external envelopes

The secular cooling of the mantle and of the continental lithosphere trigger an increase in the area of emerged land. The corollary increase in weathering and erosion processes has major consequences for the evolution of Earth's external envelopes. We developed a physical model to evaluate the area of emerged land as a function of mantle temperature, continental area, and of the distribution of continental elevations. Our numerical results show that less than 15% of Earth's surface consisted of emerged land by the end of the Archaean. This is consistent with many geological and geochemical observations. To estimate the secular cooling of the continental lithosphere, we combined thermo-mechanical models with fi eld observations. Our results, constrained by geological data, suggest that the Moho temperature has decreased by ~ 200ºC over 2.7 Ga in the Pilbara Craton. To evaluate the eff ect of continental growth on the evolution of the area of emerged land, we developed a model based on published thermal evolution models. Our results suggest that the area of emerged land was less than 5% of Earth's surface in the Archaean, and that it does not depend on crustal growth. This allows to reconcile the evolution of oceanic 87Sr/86Sr with early crustal growth models. Continents are enriched in phosphorus, which is essential to the biosphere. The emergence of the continents would thus have triggered an increase in the production of oxygen by photosynthetic micro-organisms, possibly contributing to the oxidation of the atmosphere 2.4 Ga ago

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Gaussian Distribution of Trie Depth for Strongly Tame Sources

International audienc