Higher order Cheeger inequalities for Steklov eigenvalues

We prove a lower bound for the $k$-th Steklov eigenvalues in terms of an isoperimetric constant called the $k$-th Cheeger-Steklov constant in three different situations: finite spaces, measurable spaces, and Riemannian manifolds. These lower bounds can be considered as higher order Cheeger type inequalities for the Steklov eigenvalues. In particular it extends the Cheeger type inequality for the first nonzero Steklov eigenvalue previously studied by Escobar in 1997 and by Jammes in 2015 to higher order Steklov eigenvalues. The technique we develop to get this lower bound is based on considering a family of accelerated Markov operators in the finite and mesurable situations and of mass concentration deformations of the Laplace-Beltrami operator in the manifold setting which converges uniformly to the Steklov operator. As an intermediary step in the proof of the higher order Cheeger type inequality, we define the Dirichlet-Steklov connectivity spectrum and show that the Dirichlet connectivity spectra of this family of operators converges to (or bounded by) the Dirichlet-Steklov spectrum uniformly. Moreover, we obtain bounds for the Steklov eigenvalues in terms of its Dirichlet-Steklov connectivity spectrum which is interesting in its own right and is more robust than the higher order Cheeger type inequalities. The Dirichlet-Steklov spectrum is closely related to the Cheeger-Steklov constants

Método eletroforético rápido para detecção da adulteração do leite caprino com leite bovino.

Resumo: Avaliaram-se os metodos de eletroforese em gel de poliacrilamida (PAGE) em presenca de ureia (ureia-PAGE) e dodecil sulfato de sodio (SDS-PAGE) para identificar a adulteracao do leite de cabra pela adicao do leite de vaca. Um metodo foi otimizado para preparacao do caseinato de sodio em poucos minutos para analise eletroforetica. Ureia-PAGE foi o metodo mais apropriado para identificacao desse tipo de fraude, em decorrencia da presenca da caseina a1 com migracao mais rapida no leite bovino. A presenya da as1-caseina bovina foi detectada a partir da adicao de 2,5% de leite de vaca utilizando ureia-PAGE. O limite de deteccao, a repetibilidade, 0 tempo para execucao indicaram que esse metodo pode ser aplicado como rotina no controle de qualidade do leite de cabra recebido pelas industrias de processamento. [Fast electrophoretic detection method of adulteration of caprine milk by bovine milk]. Abstract: Polyacrylamide gel electrophoresis (PAGE) in presence of urea (urea-PAGE) or sodium dodecyl sulfate (SDS-PAGE) was evaluated to detect the presence of cow milk added to goat milk. A method was optimized to prepare sodium caseinate from milk in few minutes. After that, the sodium caseinate was analyzed by PAGE. The urea-PAGE was the most appropriated method to identify adulteration as caprine and bovine as1-caseins displayed different migration rates. When cow milk was added to goat milk at different proportions, the presence of bovine as1-casein was detected in the mixture by urea-PAGE for a minimal proportion of 2.5% of cow milk added to goat milk. The good sensitivity, the repeatability and the short time for execution indicate that the described method will be able to be routinely applied for the quality control of goat milk in dairy industry

On slowdown and speedup of transient random walks in random environment

We consider one-dimensional random walks in random environment which are transient to the right. Our main interest is in the study of the sub-ballistic regime, where at time $n$ the particle is typically at a distance of order $O(n^\kappa)$ from the origin, $\kappa\in(0,1)$. We investigate the probabilities of moderate deviations from this behaviour. Specifically, we are interested in quenched and annealed probabilities of slowdown (at time $n$, the particle is at a distance of order $O(n^{\nu_0})$ from the origin, $\nu_0\in (0,\kappa)$), and speedup (at time $n$, the particle is at a distance of order $n^{\nu_1}$ from the origin, $\nu_1\in (\kappa,1)$), for the current location of the particle and for the hitting times. Also, we study probabilities of backtracking: at time $n$, the particle is located around $(-n^\nu)$, thus making an unusual excursion to the left. For the slowdown, our results are valid in the ballistic case as well.Comment: 43 pages, 4 figures; to appear in Probability Theory and Related Field

Intertwining relations for one-dimensional diffusions and application to functional inequalities

International audienceFollowing the recent work [13] fulfilled in the discrete case, we pro- vide in this paper new intertwining relations for semigroups of one-dimensional diffusions. Various applications of these results are investigated, among them the famous variational formula of the spectral gap derived by Chen and Wang [15] together with a new criterion ensuring that the logarithmic Sobolev inequality holds. We complete this work by revisiting some classical examples, for which new estimates on the optimal constants are derived

Evolution of major milk proteins in Mus musculus and Mus spretus mouse species: a genoproteomic analysis

<p>Abstract</p> <p>Background</p> <p>Due to their high level of genotypic and phenotypic variability, <it>Mus spretus </it>strains were introduced in laboratories to investigate the genetic determinism of complex phenotypes including quantitative trait loci. <it>Mus spretus </it>diverged from <it>Mus musculus </it>around 2.5 million years ago and exhibits on average a single nucleotide polymorphism (SNP) in every 100 base pairs when compared with any of the classical laboratory strains. A genoproteomic approach was used to assess polymorphism of the major milk proteins between SEG/Pas and C57BL/6J, two inbred strains of mice representative of <it>Mus spretus </it>and <it>Mus musculus </it>species, respectively.</p> <p>Results</p> <p>The milk protein concentration was dramatically reduced in the SEG/Pas strain by comparison with the C57BL/6J strain (34 ± 9 g/L <it>vs</it>. 125 ± 12 g/L, respectively). Nine major proteins were identified in both milks using RP-HPLC, bi-dimensional electrophoresis and MALDI-Tof mass spectrometry. Two caseins (β and α_s1) and the whey acidic protein (WAP), showed distinct chromatographic and electrophoresis behaviours. These differences were partly explained by the occurrence of amino acid substitutions and splicing variants revealed by cDNA sequencing. A total of 34 SNPs were identified in the coding and 3'untranslated regions of the SEG/Pas <it>Csn1s1 </it>(11), <it>Csn2 </it>(7) and <it>Wap </it>(8) genes. In addition, a 3 nucleotide deletion leading to the loss of a serine residue at position 93 was found in the SEG/Pas <it>Wap </it>gene.</p> <p>Conclusion</p> <p>SNP frequencies found in three milk protein-encoding genes between <it>Mus spretus </it>and <it>Mus musculus </it>is twice the values previously reported at the whole genome level. However, the protein structure and post-translational modifications seem not to be affected by SNPs characterized in our study. Splicing mechanisms (cryptic splice site usage, exon skipping, error-prone junction sequence), already identified in casein genes from other species, likely explain the existence of multiple α_s1-casein isoforms both in SEG/Pas and C57BL/6J strains. Finally, we propose a possible mechanism by which the hallmark tandem duplication of a 18-nt exon (14 copies) may have occurred in the mouse genome.</p

Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman–Kac semigroups

We present an interacting particle system methodology for the numerical solving of the Lyapunov exponent of Feynman–Kac semigroups and for estimating the principal eigenvalue of Schrödinger generators. The continuous or discrete time models studied in this work consists of N interacting particles evolving in an environment with soft obstacles related to a potential function V. These models are related to genetic algorithms and Moran type particle schemes. Their choice is not unique. We will examine a class of models extending the hard obstacle model of K. Burdzy, R. Holyst and P. March and including the Moran type scheme presented by the authors in a previous work. We provide precise uniform estimates with respect to the time parameter and we analyze the fluctuations of continuous time particle models