Specular sets

We introduce the notion of specular sets which are subsets of groups called here specular and which form a natural generalization of free groups. These sets are an abstract generalization of the natural codings of linear involutions. We prove several results concerning the subgroups generated by return words and by maximal bifix codes in these sets.Comment: arXiv admin note: substantial text overlap with arXiv:1405.352

Local Complexity of Delone Sets and Crystallinity

This paper characterizes when a Delone set X is an ideal crystal in terms of restrictions on the number of its local patches of a given size or on the hetereogeneity of their distribution. Let N(T) count the number of translation-inequivalent patches of radius T in X and let M(T) be the minimum radius such that every closed ball of radius M(T) contains the center of a patch of every one of these kinds. We show that for each of these functions there is a `gap in the spectrum' of possible growth rates between being bounded and having linear growth, and that having linear growth is equivalent to X being an ideal crystal. Explicitly, for N(T), if R is the covering radius of X then either N(T) is bounded or N(T) >= T/2R for all T>0. The constant 1/2R in this bound is best possible in all dimensions. For M(T), either M(T) is bounded or M(T) >= T/3 for all T>0. Examples show that the constant 1/3 in this bound cannot be replaced by any number exceeding 1/2. We also show that every aperiodic Delone set X has M(T) >= c(n)T for all T>0, for a certain constant c(n) which depends on the dimension n of X and is greater than 1/3 when n > 1.Comment: 26 pages. Uses latexsym and amsfonts package

Palindromic complexity of trees

We consider finite trees with edges labeled by letters on a finite alphabet $\varSigma$. Each pair of nodes defines a unique labeled path whose trace is a word of the free monoid $\varSigma^*$. The set of all such words defines the language of the tree. In this paper, we investigate the palindromic complexity of trees and provide hints for an upper bound on the number of distinct palindromes in the language of a tree.Comment: Submitted to the conference DLT201

Ascite fébrile chez la femme, ne pas méconnaitre une tumeur de Krukenberg

Les tumeurs de Krukenberg (TK) se définissent comme des métastases ovariennes d'un cancer, le plus souvent digestif. Elles représentent 5 à 15% des tumeurs malignes ovariennes. Notre objectif était de décrire les caractéristiques épidémiologiques, diagnostiques, thérapeutiques et évolutives.Nous rapportons deux observations de tumeur de Krukenberg découvertes à l'occasion de l'exploration d'une ascite fébrile.Il s'agit de deux patientes multipares âgées respectivement de 32 ans et 50 ans. Les signes d'appel étaient essentiellement digestifs. La découverte de ces métastases ovariennes était survenue à distance des foyers primitifs. L'atteinte des ovaires était bilatérale dans le premier cas et unilatérale droite dans le second cas. Le diagnostic est apporté par la tomodensitométrie abdominopelvienne dans les deux cas. La fibroscopie oesogastroduodénale avait permis de retrouver le foyer primitif respectivement sous forme d'un processus bourgeonnant et d'un ulcère en position antrale avec des stigmates d'hémorragies. L'examen anatomopathologique des biopsies réalisées mettait en évidence un adénocarcinome tubuleux moyennement différencié de l'estomac avec composante mucineuse dans la première observation et un  dénocarcinome de type intestinal moyennement différencié dans la seconde. Le traitement chirurgical confirme le diagnostic histologique. Dans notre série, le traitement n'a pu être que symptomatique en raison de l'existence constante d'une carcinose péritonéale et de l'altération profonde de l'état général. Les deux patientes ont été confiées à l'institut de cancérologie pour une chimiothérapie palliative. La première est décédée 1 mois après. La tumeur de Krukenberg est une maladie rare. Le diagnostic est facilité par la radiologie et confirmé par l'histologie. Son pronostic demeure encore très sombre. Le seul espoir réside dans les mesures préventives

Abcès de la prostate de découverte fortuite : A propos de deux cas

Introduction : l’abcès prostatique est une pathologie de plus en plus rare. Sa symptomatologie n’est pas spécifique. L’échographie endorectale et le scanner pelvien occupent une place de choix dans le diagnostic. Le traitement repose sur l’antibiothérapie adaptée et sur le drainage percutané par voie transpérinéale, transrectale ou endoscopique. Observation : nous rapportons deux cas d’abcès de prostate, l’un des patients est âgé de 47 ans et l’autre 61 ans. Dans les deux cas, le diagnostic a été de découverte fortuite en per opératoire. Le scanner abdominopelvien a posé le diagnostic de kyste de prostate chez l’un et une hypertrophie bénigne de la prostate chez l’autre par une échographie réno-vésico-prostatique. Le traitement a consisté en un drainage par chirurgie ouverte avec une antibiothérapie adaptée. L’évolution a été favorable dans les deux cas. Conclusion : l’abcès prostatique est une pathologie rare et sa symptomatologie clinique n’est pas spécifique. Dans notre cas, le diagnostic n’a pu être confirmé qu’en peropératoire et le traitement par chirurgie ouverte avec un bon résultat

Patterns in rational base number systems

Number systems with a rational number $a/b > 1$ as base have gained interest in recent years. In particular, relations to Mahler's 3/2-problem as well as the Josephus problem have been established. In the present paper we show that the patterns of digits in the representations of positive integers in such a number system are uniformly distributed. We study the sum-of-digits function of number systems with rational base $a/b$ and use representations w.r.t. this base to construct normal numbers in base $a$ in the spirit of Champernowne. The main challenge in our proofs comes from the fact that the language of the representations of integers in these number systems is not context-free. The intricacy of this language makes it impossible to prove our results along classical lines. In particular, we use self-affine tiles that are defined in certain subrings of the ad\'ele ring $\mathbb{A}_\mathbb{Q}$ and Fourier analysis in $\mathbb{A}_\mathbb{Q}$. With help of these tools we are able to reformulate our results as estimation problems for character sums

On the Language of Standard Discrete Planes and Surfaces

International audienceA standard discrete plane is a subset of Z^3 verifying the double Diophantine inequality mu =< ax+by+cz < mu + omega, with (a,b,c) != (0,0,0). In the present paper we introduce a generalization of this notion, namely the (1,1,1)-discrete surfaces. We first study a combinatorial representation of discrete surfaces as two-dimensional sequences over a three-letter alphabet and show how to use this combinatorial point of view for the recognition problem for these discrete surfaces. We then apply this combinatorial representation to the standard discrete planes and give a first attempt of to generalize the study of the dual space of parameters for the latter [VC00]

Order in glassy systems

A directly measurable correlation length may be defined for systems having a two-step relaxation, based on the geometric properties of density profile that remains after averaging out the fast motion. We argue that the length diverges if and when the slow timescale diverges, whatever the microscopic mechanism at the origin of the slowing down. Measuring the length amounts to determining explicitly the complexity from the observed particle configurations. One may compute in the same way the Renyi complexities K_q, their relative behavior for different q characterizes the mechanism underlying the transition. In particular, the 'Random First Order' scenario predicts that in the glass phase K_q=0 for q>x, and K_q>0 for q<x, with x the Parisi parameter. The hypothesis of a nonequilibrium effective temperature may also be directly tested directly from configurations.Comment: Typos corrected, clarifications adde

Quantum Return Probability for Substitution Potentials

We propose an effective exponent ruling the algebraic decay of the average quantum return probability for discrete Schrodinger operators. We compute it for some non-periodic substitution potentials with different degrees of randomness, and do not find a complete qualitative agreement with the spectral type of the substitution sequences themselves, i.e., more random the sequence smaller such exponent.Comment: Latex, 13 pages, 6 figures; to be published in Journal of Physics

Generic Continuous Spectrum for Ergodic Schr"odinger Operators

We consider discrete Schr"odinger operators on the line with potentials generated by a minimal homeomorphism on a compact metric space and a continuous sampling function. We introduce the concepts of topological and metric repetition property. Assuming that the underlying dynamical system satisfies one of these repetition properties, we show using Gordon's Lemma that for a generic continuous sampling function, the associated Schr"odinger operators have no eigenvalues in a topological or metric sense, respectively. We present a number of applications, particularly to shifts and skew-shifts on the torus.Comment: 14 page