Topological transversals to a family of convex sets

Let $\mathcal F$ be a family of compact convex sets in $\mathbb R^d$. We say that $\mathcal F$ has a \emph{topological $\rho$-transversal of index $(m,k)$} ($\rho<m$, $0<k\leq d-m$) if there are, homologically, as many transversal $m$-planes to $\mathcal F$ as $m$-planes containing a fixed $\rho$-plane in $\mathbb R^{m+k}$. Clearly, if $\mathcal F$ has a $\rho$-transversal plane, then $\mathcal F$ has a topological $\rho$-transversal of index $(m,k),$ for $\rho<m$ and $k\leq d-m$. The converse is not true in general. We prove that for a family $\mathcal F$ of $\rho+k+1$ compact convex sets in $\mathbb R^d$ a topological $\rho$-transversal of index $(m,k)$ implies an ordinary $\rho$-transversal. We use this result, together with the multiplication formulas for Schubert cocycles, the Lusternik-Schnirelmann category of the Grassmannian, and different versions of the colorful Helly theorem by B\'ar\'any and Lov\'asz, to obtain some geometric consequences

CYP17A1 deficient XY mice display susceptibility to atherosclerosis, altered lipidomic profile and atypical sex development

CYP17A1 is a cytochrome P450 enzyme with 17-alpha-hydroxylase and C17,20-lyase activities. CYP17A1 genetic variants are associated with coronary artery disease, myocardial infarction and visceral and subcutaneous fat distribution; however, the underlying pathological mechanisms remain unknown. We aimed to investigate the function of CYP17A1 and its impact on atherosclerosis in mice. At 4-6 months, CYP17A1-deficient mice were viable, with a KO:Het:WT ratio approximating the expected Mendelian ratio of 1:2:1. All Cyp17a1 knockout (KO) mice were phenotypically female; however, 58% were Y chromosome-positive, resembling the phenotype of human CYP17A1 deficiency, leading to 46,XY differences/disorders of sex development (DSD). Both male and female homozygous KO mice were infertile, due to abnormal genital organs. Plasma steroid analyses revealed a complete lack of testosterone in XY-KO mice and marked accumulation of progesterone in XX-KO mice. Elevated corticosterone levels were observed in both XY and XX KO mice. In addition, Cyp17a1 heterozygous mice were also backcrossed onto an Apoe KO atherogenic background and fed a western-type diet (WTD) to study the effects of CYP17A1 on atherosclerosis. Cyp17a1 x Apoe double KO XY mice developed more atherosclerotic lesions than Apoe KO male controls, regardless of diet (standard or WTD). Increased atherosclerosis in CYP17A1 XY KO mice lacking testosterone was associated with altered lipid profiles. In mice, CYP17A1 deficiency interferes with sex differentiation. Our data also demonstrate its key role in lipidomic profile, and as a risk factor in the pathogenesis of atherosclerosis

Further solutions of critical ABF RSOS models

The restricted SOS model of Andrews, Baxter and Forrester has been studied. The finite size corrections to the eigenvalue spectra of the transfer matrix of the model with a more general crossing parameter have been calculated. Therefore the conformal weights and the central charges of the non-unitary or unitary minimal conformal field have been extracted from the finite size corrections.Comment: Pages 11; revised versio

Electrochemical behaviour of gamma hydroxybutyric acid at a platinum electrode in acidic medium

The electrooxidation of Gamma Hydroxybutyric Acid (GHB) on a polycrystalline platinum electrode is studied by cyclic voltammetry in acidic medium. Two oxidation peaks, A and B, are obtained in the positive scan within the potential range of the double layer region and of the platinum oxide region, respectively. In the negative going potential sweep an inverted oxidation peak with an onset partially overlapping with the tail of the cathodic peak for the reduction of the platinum oxide formed during the anodic scan is obtained (peak C). This inverted peak can be observed at a potential close to +0.2 V (vs Ag/AgCl at pH 2) and separated 0.4 and 0.8 V from the two other oxidation peaks obtained during the anodic scan and in such conditions that the surface is particularly activated to favour this electrochemical process. The response obtained in the electronic current for the different peaks when GHB concentration and scan rate were changed to allows inferring that these are the result of a potential dependent mechanism. The behaviour observed is according with the oxidation of the alcohol group to the corresponding aldehyde and carboxylic acid (succinic acid) as main products

Hirzebruch-Milnor classes and Steenbrink spectra of certain projective hypersurfaces

We show that the Hirzebruch-Milnor class of a projective hypersurface, which gives the difference between the Hirzebruch class and the virtual one, can be calculated by using the Steenbrink spectra of local defining functions of the hypersurface if certain good conditions are satisfied, e.g. in the case of projective hyperplane arrangements, where we can give a more explicit formula. This is a natural continuation of our previous paper on the Hirzebruch-Milnor classes of complete intersections.Comment: 15 pages, Introduction is modifie

Bifurcation of critical points for continuous families of C^2 functionals of Fredholm type

Given a continuous family of C^2 functionals of Fredholm type, we show that the non-vanishing of the spectral flow for the family of Hessians along a known (trivial) branch of critical points not only entails bifurcation of nontrivial critical points but also allows to estimate the number of bifurcation points along the branch. We use this result for several parameter bifurcation, estimating the number of connected components of the complement of the set of bifurcation points and apply our results to bifurcation of periodic orbits of Hamiltonian systems. By means of a comparison principle for the spectral flow, we obtain lower bounds for the number of bifurcation points of periodic orbits on a given interval in terms of the coefficients of the linearization

The Maslov index in weak symplectic functional analysis

We recall the Chernoff-Marsden definition of weak symplectic structure and give a rigorous treatment of the functional analysis and geometry of weak symplectic Banach spaces. We define the Maslov index of a continuous path of Fredholm pairs of Lagrangian subspaces in continuously varying Banach spaces. We derive basic properties of this Maslov index and emphasize the new features appearing.Comment: 34 pages, 13 figures, 45 references, to appear in Ann Glob Anal Geom. The final publication will be available at http://www.springerlink.com. arXiv admin note: substantial text overlap with arXiv:math/040613

Exotic group actions on simply connected smooth 4-manifolds

We produce infinite families of exotic actions of finite cyclic groups on simply connected smooth 4-manifolds with nontrivial Seiberg-Witten invariants.Comment: 13 pages. Minor changes, some references added. To appear in Journal of Topolog

General Spectral Flow Formula for Fixed Maximal Domain

We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous family of (unbounded) self-adjoint Fredholm operators in terms of the Maslov index of two related curves of Lagrangian spaces. One curve is given by the varying domains, the other by the Cauchy data spaces. We provide rigorous definitions of the underlying concepts of spectral theory and symplectic analysis and give a full (and surprisingly short) proof of our General Spectral Flow Formula for the case of fixed maximal domain. As a side result, we establish local stability of weak inner unique continuation property (UCP) and explain its role for parameter dependent spectral theory.Comment: 22 page