Comments on selected fundamental aspects of microarray analysis

Microarrays are becoming a ubiquitous tool of research in life sciences. However, the working principles of microarray-based methodologies are often misunderstood or apparently ignored by the researchers who actually perform and interpret experiments. This in turn seems to lead to a common over-expectation regarding the explanatory and/or knowledge-generating power of microarray analyses. In this note we intend to explain basic principles of five (5) major groups of analytical techniques used in studies of microarray data and their interpretation: the principal component analysis (PCA), the independent component analysis (ICA), the t-test, the analysis of variance (ANOVA), and self organizing maps (SOM). We discuss answers to selected practical questions related to the analysis of microarray data. We also take a closer look at the experimental setup and the rules, which have to be observed in order to exploit microarrays efficiently. Finally, we discuss in detail the scope and limitations of microarray-based methods. We emphasize the fact that no amount of statistical analysis can compensate for (or replace) a well thought through experimental setup. We conclude that microarrays are indeed useful tools in life sciences but by no means should they be expected to generate complete answers to complex biological questions. We argue that even well posed questions, formulated within a microarray-specific terminology, cannot be completely answered with the use of microarray analyses alone.     

Une Conjecture de Lebesgue

Let A > 0 be an integer. The equation x⁵–y⁵ = Az⁵ wasfirst studied by Dirichlet and Lebesgue. Lebesgue conjecturedin 1843 that if A has no prime divisors of the form 10k+1, theequation has no solutions except the visible ones. Partial resultswere obtained by Lebesgue and by Terjanian in 1987. The purposeof the paper is to prove Lebesgue's conjecture. The main toolused is the method known as the elliptic Chabauty method.     

Random Euclidean embeddings in spaces of bounded volume ratio

Let $\text{(}\text{R}{}^{\mathrm{N}}\text{,6·6)}$ be the space $\text{R}{}^{\mathrm{N}}$ equipped with a norm 6·6 whose unit ball has a bounded volume ratio with respect to the Euclidean unit ball. Let Γ be any random N×n matrix with N>n, whose entries are independent random variables satisfying some moment assumptions. We show that with high probability Γ is a good isomorphism from the n-dimensional Euclidean space $\text{(}\text{R}{}^{\mathrm{n}}\text{,|·|)}$ onto its image in $\text{(}\text{R}{}^{\mathrm{N}}\text{,6·6)}$: there exist α,β>0 such that for all $\text{x∈}\text{R}{}^{\mathrm{n}}$, $\text{α}\text{N}\text{|x|?6Γx6?β}\text{N}\text{|x|}$. This solves a conjecture of Schechtman on random embeddings of ?₂ⁿ into ?₁^N. To cite this article: A. Litvak et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Paradoxe de Klein pour l'équation de Klein–Gordon chargée : superradiance et opérateur de diffusion

We develop the scattering theory for the charged Klein–Gordon equation on ${\mathbb{R}}_t\times {\mathbb{R}}_x$, when the electrostatic potential $A(x)$ has different asymptotics $a^{\pm }$ as $x\to \pm \infty$. In this case, the conserved energy is not positive definite (Klein Paradox). We construct the spectral representation for the harmonic equation, and we establish the existence of a Scattering Operator the symbol of which has a norm strictly larger than 1, for the frequencies in $(a^{?}\text{,}{a}^{+})$. These results can be applied to the DeSitter–Reissner–Nordstrøm metric, to justify the notion of superradiance of the charged black-holes. To cite this article: A. Bachelot, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

At-sea measurements of sound penetration into sediments using a buried vertical synthetic array

Acoustic bottom penetration experiments were carried out in a medium-grain sandy bottom at a site in St. Andrews Bay, Florida. These investigations used a new buried, vertical, one-dimensional synthetic array system where a small hydrophone was water-jetted into the sediment to a depth of approximately 2 m. Once buried, this hydrophone was mounted to a vertical robotics stage that translated the hydrophone upward in 1-cm increments. A broadband (3 to 80 kHz) spherical source, positioned 50 cm above the sediment-water interface, was used to insonify the sediment. Measurements were made with insonification angles above and below the critical angle by changing the horizontal distance of the source relative to the insertion point. This new measurement system is detailed, and results are presented that include temporal, frequency, and wavenumber analysis for natural and roughened interfaces. The measured compressional sound speed and attenuation are shown to be self-consistent using the Kramers-Kronig relation. Furthermore, only a single fast compressional wave was observed. There was no observation of a second slower compressional wave as predicted by some applications of the Biot model to unconsolidated water-saturated porous media.     

Three-dimensional wavelet-based multifractal method: the need for revisiting the multifractal description of turbulence dissipation data

We generalize the wavelet transform modulus maxima (WTMM) method to multifractal analysis of 3D random fields. This method is calibrated on synthetic 3D monofractal fractional Brownian fields and on 3D multifractal singular cascade measures as well as their random function counterpart obtained by fractional integration. Then we apply the 3D WTMM method to the dissipation field issued from 3D isotropic turbulence simulations. We comment on the need to revisit previous box-counting analyses which have failed to estimate correctly the corresponding multifractal spectra because of their intrinsic inability to master nonconservative singular cascade measures.     

Parabolic Capacity and Soft Measures for Nonlinear Equations

We first introduce, using a functional approach, the notion of capacity related to the parabolic p-Laplace operator. Then we prove a decomposition theorem for measures (in space and time) that do not charge the sets of null capacity. We apply this result to prove existence and uniqueness of renormalized solutions for nonlinear parabolic initial boundary-value problems with such measures as right-hand side.     

Quantum Pseudo-Telepathy

Quantum information processing is at the crossroads of physics, mathematics and computer science. It is concerned with what we can and cannot do with quantum information that goes beyond the abilities of classical information processing devices. Communication complexity is an area of classical computer science that aims at quantifying the amount of communication necessary to solve distributed computational problems. Quantum communication complexity uses quantum mechanics to reduce the amount of communication that would be classically required. Pseudo-telepathy is a surprising application of quantum information processing to communication complexity. Thanks to entanglement, perhaps the most nonclassical manifestation of quantum mechanics, two or more quantum players can accomplish a distributed task with no need for communication whatsoever, which would be an impossible feat for classical players. After a detailed overview of the principle and purpose of pseudo-telepathy, we present a survey of recent and not-so-recent work on the subject. In particular, we describe and analyse all the pseudo-telepathy games currently known to the authors. Supported in Part by Canada’s Natural Sciences and Engineering Research Council (NSERC), the Canada Research Chair programme and the Canadian Institute for Advanced Research (CIAR). Supported in part by a scholarship from Canada’s NSERC. Supported in part by Canada’s NSERC Québec’s Fonds de recherche sur la nature et les technologies (FQRNT), the CIAR and the Mathematics of Information Technology and Complex Systems Network (MITACS).