Quicksort with unreliable comparisons: a probabilistic analysis

We provide a probabilistic analysis of the output of Quicksort when comparisons can err.Comment: 29 pages, 3 figure

Planar maps and continued fractions

We present an unexpected connection between two map enumeration problems. The first one consists in counting planar maps with a boundary of prescribed length. The second one consists in counting planar maps with two points at a prescribed distance. We show that, in the general class of maps with controlled face degrees, the solution for both problems is actually encoded into the same quantity, respectively via its power series expansion and its continued fraction expansion. We then use known techniques for tackling the first problem in order to solve the second. This novel viewpoint provides a constructive approach for computing the so-called distance-dependent two-point function of general planar maps. We prove and extend some previously predicted exact formulas, which we identify in terms of particular Schur functions.Comment: 47 pages, 17 figures, final version (very minor changes since v2

Antireflection of an absorbing substrate by an absorbing thin film at normal incidence

An absorbing substrate of complex refractive index n2 - jk2 at wavelength λ can be coated by an absorbing thin film of complex refractive index n1 - jk1 and thickness d to achieve zero reflection at normal incidence. For given n2,k2 multiple solutions (n1,k1,d/λ) are found that correspond to infinitely many distinct antireflection layers. This is demonstrated for a Si substrate at two wavelengths (6328 and 4420 Å). The response of these absorbing antireflection layers to changes of the angle of incidence from 0 to 45° and to changes of thickness of ±10% is also determined and compared to the limting case of a nonabsorbing antireflection layer

The topological structure of scaling limits of large planar maps

We discuss scaling limits of large bipartite planar maps. If p is a fixed integer strictly greater than 1, we consider a random planar map M(n) which is uniformly distributed over the set of all 2p-angulations with n faces. Then, at least along a suitable subsequence, the metric space M(n) equipped with the graph distance rescaled by the factor n to the power -1/4 converges in distribution as n tends to infinity towards a limiting random compact metric space, in the sense of the Gromov-Hausdorff distance. We prove that the topology of the limiting space is uniquely determined independently of p, and that this space can be obtained as the quotient of the Continuum Random Tree for an equivalence relation which is defined from Brownian labels attached to the vertices. We also verify that the Hausdorff dimension of the limit is almost surely equal to 4.Comment: 45 pages Second version with minor modification

Antireflection of an absorbing substrate by an absorbing thin film at normal incidence

An absorbing substrate of complex refractive index n2 - jk2 at wavelength λ can be coated by an absorbing thin film of complex refractive index n1 - jk1 and thickness d to achieve zero reflection at normal incidence. For given n2,k2 multiple solutions (n1,k1,d/λ) are found that correspond to infinitely many distinct antireflection layers. This is demonstrated for a Si substrate at two wavelengths (6328 and 4420 Å). The response of these absorbing antireflection layers to changes of the angle of incidence from 0 to 45° and to changes of thickness of ±10% is also determined and compared to the limting case of a nonabsorbing antireflection layer

Random trees between two walls: Exact partition function

We derive the exact partition function for a discrete model of random trees embedded in a one-dimensional space. These trees have vertices labeled by integers representing their position in the target space, with the SOS constraint that adjacent vertices have labels differing by +1 or -1. A non-trivial partition function is obtained whenever the target space is bounded by walls. We concentrate on the two cases where the target space is (i) the half-line bounded by a wall at the origin or (ii) a segment bounded by two walls at a finite distance. The general solution has a soliton-like structure involving elliptic functions. We derive the corresponding continuum scaling limit which takes the remarkable form of the Weierstrass p-function with constrained periods. These results are used to analyze the probability for an evolving population spreading in one dimension to attain the boundary of a given domain with the geometry of the target (i) or (ii). They also translate, via suitable bijections, into generating functions for bounded planar graphs.Comment: 25 pages, 7 figures, tex, harvmac, epsf; accepted version; main modifications in Sect. 5-6 and conclusio

Statistics of reduced words in locally free and braid groups: Abstract studies and application to ballistic growth model

We study numerically and analytically the average length of reduced (primitive) words in so-called locally free and braid groups. We consider the situations when the letters in the initial words are drawn either without or with correlations. In the latter case we show that the average length of the reduced word can be increased or lowered depending on the type of correlation. The ideas developed are used for analytical computation of the average number of peaks of the surface appearing in some specific ballistic growth modelComment: 29 pages, LaTeX, 7 separated Postscript figures (available on request), submitted to J. Phys. (A): Math. Ge

Genetics of anophthalmia and microphthalmia. Part 1, Non-syndromic anophthalmia/microphthalmia

Eye formation is the result of coordinated induction and differentiation processes during embryogenesis. Disruption of any one of these events has the potential to cause ocular growth and structural defects, such as anophthalmia and microphthalmia (A/M). A/M can be isolated or occur with systemic anomalies, when they may form part of a recognizable syndrome. Their etiology includes genetic and environmental factors; several hundred genes involved in ocular development have been identified in humans or animal models. In humans, around 30 genes have been repeatedly implicated in A/M families, although many other genes have been described in single cases or families, and some genetic syndromes include eye anomalies occasionally as part of a wider phenotype. As a result of this broad genetic heterogeneity, with one or two notable exceptions, each gene explains only a small percentage of cases. Given the overlapping phenotypes, these genes can be most efficiently tested on panels or by whole exome/genome sequencing for the purposes of molecular diagnosis. However, despite whole exome/genome testing more than half of patients currently remain without a molecular diagnosis. The proportion of undiagnosed cases is even higher in those individuals with unilateral or milder phenotypes. Furthermore, even when a strong gene candidate is available for a patient, issues of incomplete penetrance and germinal mosaicism make diagnosis and genetic counselling challenging. In this review, we present the main genes implicated in nonsyndromic human A/M phenotypes and, for practical purposes, classify them according to the most frequent or predominant phenotype each is associated with. Our intention is that this will allow clinicians to rank and prioritize their molecular analyses and interpretations according to the phenotypes of their patients