Growth rate for the expected value of a generalized random Fibonacci sequence

A random Fibonacci sequence is defined by the relation g_n = | g_{n-1} +/- g_{n-2} |, where the +/- sign is chosen by tossing a balanced coin for each n. We generalize these sequences to the case when the coin is unbalanced (denoting by p the probability of a +), and the recurrence relation is of the form g_n = |\lambda g_{n-1} +/- g_{n-2} |. When \lambda >=2 and 0 < p <= 1, we prove that the expected value of g_n grows exponentially fast. When \lambda = \lambda_k = 2 cos(\pi/k) for some fixed integer k>2, we show that the expected value of g_n grows exponentially fast for p>(2-\lambda_k)/4 and give an algebraic expression for the growth rate. The involved methods extend (and correct) those introduced in a previous paper by the second author

Équidistribution presque partout modulo 1 de suites oscillantes perturbées III : cas liouvillien multidimensionnel

AbstractWe extend the results of uniform distribution modulo 1 given in [B. Rittaud, Équidistribution presque partout modulo 1 de suites oscillantes perturbées, Bull. Soc. Math. France 128 (2000) 451–471; B. Rittaud, Équidistribution presque partout modulo 1 de suites oscillantes perturbées, II: Cas Liouvillien unidimensionnel, Colloq. Math. 96 (1) (2003) 55–73], which deal with sequences of the form (t(hnF(nΘ)+εnhn′))n, where (hn)n, (hn′)n and (hn/hn′)n are polynomially increasing sequences, (εn)n a bounded sequence, F:Rd→R essentially a C3-function Zd-periodic, Θ an element of Rd and t a real number. We remove the Diophantine hypothesis on Θ needed in [the first of above mentioned articles], and add a technical hypothesis on hn. We apply this result to the convergence of diagonal averages for d×d matrices

The Field Q and the Equality 0.999. . . = 1 from Combinatorics of Circular Words and History of Practical Arithmetics

We reconsider the classical equality 0.999 . . . = 1 with the tool of circular words, that is, finite words whose last letter is assumed to be followed by the first one. Such circular words are naturally embedded with algebraic structures that enlight this problematic equality, allowing it to be considered in Q rather than in R. We comment early history of such structures, that involves English teachers and accountants of the first part of the 18th century, who appear to be the firsts to assert the equality 0.999 . . . = 1. Their level of understanding show links with Dubinsky et al.’s apos theory in mathematics education. Eventually, we rebuilt the field Q from circular words, and provide an original proof of the fact that an algebraic integer is either an integer or an irrational number

Transgressing trough semantic prosody with lexical smoothing over

In our speeches we by-pass easily linguistic taboos with the implicit. It explains why the significant prosody (studied in phonopragmatic research) holds a so important place among the resources of multimodality. Thanks to the discretion of vocal signs, the speaker can transmit ideas, opinions, etc., that, if s/he expressed them with words, might shock the addressee, upset him/her or make him/her strongly react (particularly with aggressive or politically incorrect words); this could establish a breach in the conventions and the societal codes, in the rules of communication, or even a monoside communicative break-up. Vocal signs thus allow the speaker to avoid safely or almost safely many lexical taboos, as for example: to be hurtful, malevolent or rough without facing the consequences, to reduce the impact of the aggression, to let see lack of concern toward the interlocutor’s feeling, to pretend we agree with him/her. With examples taken from a corpus of French speakers conversations (recorded or not), the target is to point out what vocal signs are made of and what are their meanings, as well as their effects (expected or not) upon the hearer.Pour éviter les tabous linguistiques le plus facile est de recourir à l’implicite de l’énonciation. Cela explique l’importance de la place de la prosodie signifiante (objet de la phonopragmatique) parmi les ressources de la multimodalité. Car les signes vocaux permettent de transmettre discrètement ce qui, dit avec des mots, constituerait une entorse aux conventions, codes sociétaux, règles de communication, ou même une rupture monolatérale d’interaction ; avec l’explicite on risque en effet de choquer, d’ennuyer, d’impressionner désagréablement ou de faire réagir vivement. La permissivité offerte par la couche vocale aide l’énonciateur à contourner sans danger – ou presque – des interdits de toutes sortes, comme : désir de diminuer l’impact de la violence infligée, plaisir d’être blessant, malveillant ou brutal sans en subir les conséquences, indifférence aux ressentis de l’interlocuteur, ou même volonté de faire croire au consensus. L’analyse (signifiant sonore et signifié pragmatique) d’exemples oraux (enregistrés ou non) de locuteurs de français langue première est suivie de celle de leurs effets prévus ou imprévus sur l’écouteur

Comment exploiter les 'corpus-surprise' ?

To what extent non-recorded oral corpora may constitute objects of analysis of pragmatic meaning?These corpora are heard by chance: on the radio, on television, in the street, a shop, a means of transport or generally in any conversational interaction in which the linguist participates, but had not previously planned to record for his research. The problem of the use of these corpora in linguistics is all the more crucial since the aim, in phonopragmatics, is to discover the functions and significations of their phonic part. I shall attempt to answer the following questions:–The accuracy of the transcription with respect to the original. To what extent can we ignore our own phonological code, our regional variants, mastered/partly known styles of speech?–The reliability of the oral reproduction carried out by the linguist – for example, during a talk at a conference. What is his capacity for deferred mimicry?–The relation between a significant discrepancy and the elocutionary habits of the speaker.–The relation between the comprehension of the external auditors and the effect produced on the 'real' person addressed.Considering that transparency is (sometimes? often?) an illusion, I shall also examine what precautions should be taken so that these corpora offer guarantees as to the veracity

Code & order in polygonal billiards

Two polygons $P,Q$ are code equivalent if there are billiard orbits $u,v$ which hit the same sequence of sides and such that the projections of the orbits are dense in the boundaries $\partial P, \partial Q$. Our main results show when code equivalent polygons have the same angles, resp. are similar, resp. affinely similar

Hybrid quantum-classical dynamics of pure-dephasing systems

We consider the interaction dynamics of a classical oscillator and a quantum two-level system for different pure-dephasing Hamiltonians of the type $\widehat{H}(q,p)=H_C(q,p)\boldsymbol{1}+H_I(q,p)\widehat\sigma_z$. This type of systems represents a severe challenge for popular hybrid quantum-classical descriptions. For example, in the case of the common Ehrenfest model, the classical density evolution is shown to decouple entirely from the pure-dephasing quantum dynamics. We focus on a recently proposed hybrid wave equation that is based on Koopman's wavefunction description of classical mechanics. This model retains quantum-classical correlations whenever a coupling potential is present. Here, several benchmark problems are considered and the results are compared with those arising from fully quantum dynamics. A good agreement is found for a series of study cases involving harmonic oscillators with linear and quadratic coupling, as well as time-varying coupling parameters. In all these cases the classical evolution coincides exactly with the oscillator dynamics resulting from the fully quantum description. In the special case of time-independent coupling involving a classical oscillator with varying frequency, the quantum Bloch rotation exhibits peculiar features that escape from the hybrid description. In addition, nonlinear corrections to the harmonic Hamiltonian lead to an overall growth of decoherence at long times, which is absent in the fully quantum treatment.Comment: 24 pages, 10 figures. Contribution to the Special Collection "Koopman methods in classical and quantum-classical mechanics" in the Journal of Physics