Cohomogeneity one manifolds and selfmaps of nontrivial degree

We construct natural selfmaps of compact cohomgeneity one manifolds with finite Weyl group and compute their degrees and Lefschetz numbers. On manifolds with simple cohomology rings this yields in certain cases relations between the order of the Weyl group and the Euler characteristic of a principal orbit. We apply our construction to the compact Lie group SU(3) where we extend identity and transposition to an infinite family of selfmaps of every odd degree. The compositions of these selfmaps with the power maps realize all possible degrees of selfmaps of SU(3).Comment: v2, v3: minor improvement

Exact Equal Time Statistics of Orszag-McLaughlin Dynamics By The Hopf Characteristic Functional Approach

By employing Hopf's functional method, we find the exact characteristic functional for a simple nonlinear dynamical system introduced by Orszag. Steady-state equal-time statistics thus obtained are compared to direct numerical simulation. The solution is both non-trivial and strongly non-Gaussian.Comment: 6 pages and 2 figure

Quasi-Gaussian Statistics of Hydrodynamic Turbulence in 3/4+\epsilon dimensions

The statistics of 2-dimensional turbulence exhibit a riddle: the scaling exponents in the regime of inverse energy cascade agree with the K41 theory of turbulence far from equilibrium, but the probability distribution functions are close to Gaussian like in equilibrium. The skewness \C S \equiv S_3(R)/S^{3/2}_2(R) was measured as \C S_{\text{exp}}\approx 0.03. This contradiction is lifted by understanding that 2-dimensional turbulence is not far from a situation with equi-partition of enstrophy, which exist as true thermodynamic equilibrium with K41 exponents in space dimension of $d=4/3$. We evaluate theoretically the skewness \C S(d) in dimensions ${4/3}\le d\le 2$, show that \C S(d)=0 at $d=4/3$, and that it remains as small as \C S_{\text{exp}} in 2-dimensions.Comment: PRL, submitted, REVTeX 4, 4 page

Cutting the same fraction of several measures

We study some measure partition problems: Cut the same positive fraction of $d+1$ measures in $\mathbb R^d$ with a hyperplane or find a convex subset of $\mathbb R^d$ on which $d+1$ given measures have the same prescribed value. For both problems positive answers are given under some additional assumptions.Comment: 7 pages 2 figure

Convergence of Quantum Annealing with Real-Time Schrodinger Dynamics

Convergence conditions for quantum annealing are derived for optimization problems represented by the Ising model of a general form. Quantum fluctuations are introduced as a transverse field and/or transverse ferromagnetic interactions, and the time evolution follows the real-time Schrodinger equation. It is shown that the system stays arbitrarily close to the instantaneous ground state, finally reaching the target optimal state, if the strength of quantum fluctuations decreases sufficiently slowly, in particular inversely proportionally to the power of time in the asymptotic region. This is the same condition as the other implementations of quantum annealing, quantum Monte Carlo and Green's function Monte Carlo simulations, in spite of the essential difference in the type of dynamics. The method of analysis is an application of the adiabatic theorem in conjunction with an estimate of a lower bound of the energy gap based on the recently proposed idea of Somma et. al. for the analysis of classical simulated annealing using a classical-quantum correspondence.Comment: 6 pages, minor correction

Changes in temperature of heel skin under pressure in hip surgery patients

OBJECTIVE: The aim of the study was to examine the effect of external pressure of the bed surface on heel skin temperature in adults in the first 3 days after hip surgery. DESIGN: A quasi-experimental study in a prospective, within-subjects, repeated-measures design. SETTING: This study was performed at 2 acute-care hospitals. PARTICIPANTS: Eighteen subjects (9 men and 9 women) with a mean age of 58.3 (±16.1) years were recruited after hip surgery at the 2 hospitals. METHODS: Temperature sensors were placed on the plantar surface of each foot, close to the heels. Measures were taken when the heels were (1) suspended above the bed surface for 20 minutes (preload), (2) on the bed surface for 15 minutes (loading), and (3) suspended again above the bed surface for 15 minutes (unloading). MAIN OUTCOME MEASURES: Heel skin temperature and demographic data. RESULTS: Heel temperature increased during loading and unloading in both legs on postoperative days 1 (P = .003) and 3 (P = .04) but not on postoperative day 2. Heel temperature in the nonoperative leg decreased in the first 3 minutes of unloading on postoperative days 2 (P = .02) and 3 (P .01). CONCLUSION: Heel temperature increased with loading and unloading on postoperative days 1 and 3. Upon immediate unloading, hyperemic response was present only in the nonoperative leg. Keeping the heels off the bed surface at all times may avoid heel skin temperature changes and prevent tissue damage. Further research is needed to identify the mechanisms that explain the effect of external pressure on heel temperature

Familiäre Kavernome des Zentralnervensystems: Eine klinische und genetische Studie an 15 deutsche Familien

Zusammenfassung: 1928 beschrieb Hugo Friedrich Kufs erstmalig eine Familie mit zerebralen, retinalen und kutanen Kavernomen. Mittlerweile wurden über 300 weitere Familien beschrieben. Ebenfalls wurden drei Genloci 7q21-q22 (mit dem Gen CCM1), 7p15-p13 (Gen CCM2) und 3q25.2-q27 (Gen CCM3) beschrieben, in denen Mutationen zu Kavernomen führen. Das Genprodukt von CCM1 ist das Protein Krit1 (Krev Interaction Trapped 1), das über verschiedene Mechanismen mit der Angiogenese interagiert. Das neu entdeckte CCM2-Gen enkodiert ein Protein, das möglicherweise eine dem Krit1 ähnliche Funktion in der Regulation der Angiogenese hat. Das CCM3-Gen wurde noch nicht beschrieben. In dieser Arbeit werden sowohl die klinischen und genetischen Befunde bei 15 deutschen Familien beschriebe

On the structure of the set of bifurcation points of periodic solutions for multiparameter Hamiltonian systems

This paper deals with periodic solutions of the Hamilton equation with many parameters. Theorems on global bifurcation of solutions with periods $2\pi/j,$ $j\in\mathbb{N},$ from a stationary point are proved. The Hessian matrix of the Hamiltonian at the stationary point can be singular. However, it is assumed that the local topological degree of the gradient of the Hamiltonian at the stationary point is nonzero. It is shown that (global) bifurcation points of solutions with given periods can be identified with zeros of appropriate continuous functions on the space of parameters. Explicit formulae for such functions are given in the case when the Hessian matrix of the Hamiltonian at the stationary point is block-diagonal. Symmetry breaking results concerning bifurcation of solutions with different minimal periods are obtained. A geometric description of the set of bifurcation points is given. Examples of constructive application of the theorems proved to analytical and numerical investigation and visualization of the set of all bifurcation points in given domain are provided. This paper is based on a part of the author's thesis [W. Radzki, ``Branching points of periodic solutions of autonomous Hamiltonian systems'' (Polish), PhD thesis, Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, Toru\'{n}, 2005].Comment: 35 pages, 4 figures, PDFLaTe

Evolution of the Chern-Simons Vortices

Based on the gauge potential decomposition theory and the $\phi$-mapping theory, the topological inner structure of the Chern-Simons-Higgs vortex has been showed in detail. The evolution of CSH vortices is studied from the topological properties of the Higgs scalar field. The vortices are found generating or annihilating at the limit points and encountering, splitting or merging at the bifurcation points of the scalar field $\phi .$Comment: 10 pages, 10 figure