Level sets of functions and symmetry sets of smooth surface sections

We prove that the level sets of a real C^s function of two variables near a non-degenerate critical point are of class C^[s/2] and apply this to the study of planar sections of surfaces close to the singular section by the tangent plane at hyperbolic points or elliptic points, and in particular at umbilic points. We also analyse the cases coming from degenerate critical points, corresponding to elliptic cusps of Gauss on a surface, where the differentiability is now reduced to C^[s/4]. However in all our applications to symmetry sets of families of plane curves, we assume the C^infty smoothness.Comment: 15 pages, Latex, 6 grouped figures. The final version will appear in Mathematics of Surfaces. Lecture Notes in Computer Science (2005

A Single Basis for Developmental Buffering of Drosophila Wing Shape

The nature of developmental buffering processes has been debated extensively, based on both theoretical reasoning and empirical studies. In particular, controversy has focused on the question of whether distinct processes are responsible for canalization, the buffering against environmental or genetic variation, and for developmental stability, the buffering against random variation intrinsic in developmental processes. Here, we address this question for the size and shape of Drosophila melanogaster wings in an experimental design with extensively replicated and fully controlled genotypes. The amounts of variation among individuals and of fluctuating asymmetry differ markedly among genotypes, demonstrating a clear genetic basis for size and shape variability. For wing shape, there is a high correlation between the amounts of variation among individuals and fluctuating asymmetry, which indicates a correspondence between the two types of buffering. Likewise, the multivariate patterns of shape variation among individuals and of fluctuating asymmetry show a close association. For wing size, however, the amounts of individual variation and fluctuating asymmetry are not correlated. There was a significant link between the amounts of variation between wing size and shape, more so for fluctuating asymmetry than for variation among individuals. Overall, these experiments indicate a considerable degree of shared control of individual variation and fluctuating asymmetry, although it appears to differ between traits

The genetic architecture of fluctuating asymmetry of mandible size and shape in a population of mice: Another look

Fluctuating asymmetry (FA), typically measured by variation in the differences between right and left sides of bilateral traits, is commonly used to assess developmental instability (DI) in populations. A previous quantitative trait locus (QTL) investigation using an F2 intercross mouse population found little evidence of individual loci affecting FA in mandible size, but an abundance of epistatic interactions between loci. Here we extend this work by testing whether these patterns replicate in an F3 population derived from the same intercross. Using a large number of molecular markers genotyped in over 1200 mice, we uncovered significant interactions between loci (QTLs) affecting FA in mandible size (and shape). Epistasis contributed roughly 20% of the variation in FASIZE and 19% of the variation in FASHAPE at the 0.0001 probability level alone, and was comparable to that previously estimated for the F2 mice, and much greater than that generated from the few single-locus QTLs affecting the mandible FA traits. The positions of the single-locus and epistatic QTLs for FA that we discovered suggested that logical candidate genes for DI are those controlling size or shape in the traits themselves, and that they may be interacting with genes for heat shock proteins

Bound States in Mildly Curved Layers

It has been shown recently that a nonrelativistic quantum particle constrained to a hard-wall layer of constant width built over a geodesically complete simply connected noncompact curved surface can have bound states provided the surface is not a plane. In this paper we study the weak-coupling asymptotics of these bound states, i.e. the situation when the surface is a mildly curved plane. Under suitable assumptions about regularity and decay of surface curvatures we derive the leading order in the ground-state eigenvalue expansion. The argument is based on Birman-Schwinger analysis of Schroedinger operators in a planar hard-wall layer.Comment: LaTeX 2e, 23 page

A Hardy inequality in twisted waveguides

We show that twisting of an infinite straight three-dimensional tube with non-circular cross-section gives rise to a Hardy-type inequality for the associated Dirichlet Laplacian. As an application we prove certain stability of the spectrum of the Dirichlet Laplacian in locally and mildly bent tubes. Namely, it is known that any local bending, no matter how small, generates eigenvalues below the essential spectrum of the Laplacian in the tubes with arbitrary cross-sections rotated along a reference curve in an appropriate way. In the present paper we show that for any other rotation some critical strength of the bending is needed in order to induce a non-empty discrete spectrum.Comment: LaTeX, 20 page

Searches for Stable Strangelets in Ordinary Matter: Overview and a Recent Example

Our knowledge on the possible existence in nature of stable exotic particles depends solely upon experimental observation. Guided by this general principle and motivated by theoretical hypotheses on the existence of stable particles of strange quark matter, a variety of experimental searches have been performed. We provide an introduction to the theoretical hypotheses, an overview of the past searches, and a more detailed description of a recent search for helium-like strangelets in the Earth's atmosphere using a sensitive laser spectroscopy method

Magnetized strangelets at finite temperature

The main properties of magnetized strangelets, namely, their energy per baryon, radius and electric charge, are studied. Temperature effects are also taken into account in order to study their stability compared to the 56Fe isotope and non-magnetized strangelets using the liquid drop model. Massive quarks are considered with the aim to have a more realistic description for strangelets in the astrophysical context and the environment of heavy ion colliders, playing also an important role in the thermodynamical quantities of the quark gas. It is concluded that the presence of a magnetic field tends to stabilize more the strangelets, even when temperature effects are taken into account. Magnetized strangelets in a paired superconductor phase (magnetized color flavor locked phase) are also discussed. It is shown that they are more stable than ordinary magnetized strangelets for typical gap values of the order of O(100) MeV.Comment: 10 pages, 10 figures, discussion extended, new references adde

Strangelets: Who is Looking, and How?

It has been over 30 years since the first suggestion that the true ground state of cold hadronic matter might be not nuclear matter but rather strange quark matter (SQM). Ever since, searches for stable SQM have been proceeding in various forms and have observed a handful of interesting events but have neither been able to find compelling evidence for stable strangelets nor to rule out their existence. I will survey the current status and near future of such searches with particular emphasis on the idea of SQM from strange star collisions as part of the cosmic ray flux.Comment: Talk given at International Conference on Strangeness in Quark Matter, 2006. 8 pages. 1 figur

The type numbers of closed geodesics

A short survey on the type numbers of closed geodesics, on applications of the Morse theory to proving the existence of closed geodesics and on the recent progress in applying variational methods to the periodic problem for Finsler and magnetic geodesicsComment: 29 pages, an appendix to the Russian translation of "The calculus of variations in the large" by M. Mors