Instability of semi-Riemannian closed geodesics

A celebrated result due to Poincar\'e affirms that a closed non-degenerate minimizing geodesic $\gamma$ on an oriented Riemannian surface is hyperbolic. Starting from this classical theorem, our first main result is a general instability criterion for timelike and spacelike closed semi-Riemannian geodesics on a (non)oriented manifold. A key role is played by the spectral index, a new topological invariant that we define through the spectral flow (being the Morse index truly infinite) of a path of selfadjoint Fredholm operators. A major step in the proof of this result is a em new spectral flow formula. Bott's iteration formula, introduced by author in 1956, relates in a clear way the Morse index of an iterated closed Riemannian geodesic and the so-called $\omega$-Morse indices. Our second result is a semi-Riemannian generalization of the famous Bott-type iteration formula in the case of closed (resp. timelike closed) Riemannian (resp. Lorentzian) geodesics. Our last result is a strong instability result obtained by controlling the Morse index of the geodesic and of all of its iterations.Comment: 33 pages, 2 figures. Fixed some typos and updated references. arXiv admin note: text overlap with arXiv:1705.0917

Entropy stabilizes floppy crystals of mobile DNA-coated colloids

Grafting linkers with open ends of complementary single-stranded DNA makes a flexible tool to tune interactions between colloids,which facilitates the design of complex self-assembly structures. Recently, it has been proposed to coat colloids with mobile DNA linkers, which alleviates kinetic barriers without high-density grafting, and also allows the design of valency without patches.However, the self-assembly mechanism of this novel system is poorly understood.Using a combination of theory and simulation, we obtain phase diagrams forthe system in both two and three dimensional spaces, and find stable floppy squareand CsCl crystals when the binding strength is strong, even in the infinite bindingstrength limit. We demonstrate that these floppy phases are stabilized by vibrational entropy, and "floppy" modes play an important role in stabilizing the floppy phases for the infinite binding strength limit. This special entropic effect in the self-assembly of mobile DNA-coated colloids is very different from conventional molecular self-assembly, and it offers new axis to help design novel functional materials using mobile DNA-coated colloids.Comment: Accepted in Physical Review Letter

ASSESSING SLEEP EFFICIENCY OF NULLIPAROUS WOMEN IN THE THIRD TRIMESTER OF PREGNANCY

Objectives: Pregnant women in the third trimester are more likely to experience sleep disorders, resulting in adverse effects on fetal health. This study aimed to evaluate the sleep efficiency (SE) of nulliparous women in their late pregnancy. Methods: Data was extracted from the National Sleep Research Resource focusing on nulliparous women (n = 1,803) in their third trimester of pregnancy. The two-sample Wilcoxon test for continuous variables and the Chi-square test for categorical variables were used based on a normal SE cut-off of 0.85. The least absolute shrinkage and selection operator (Lasso) was used to find the risk factors of the nulliparous women affecting the SE. Model results and predictive performance were evaluated and compared using Lasso and machine learning methods. Results: A total of 13.5 % of pregnant women in the study had poor sleep efficiency. Among them, 43.9% were White, 21.3% were Black, 24.6% were Hispanic, and 3.7% were Asian. The Lasso model selected total sleep duration with \u3e92% oxygen saturation, number of sleep-to-awake shifts per hour, and number of desaturations with \u3e= 2% oxygen desaturation in total sleep period as the most contributing variables. Random forest feature importance selected the same result as the Lasso feature selection on the top three variables. Conclusions: The study showed that the sleep efficiency of nulliparous women in the third trimester was significantly associated with oxygen saturation and desaturation status as well as change in sleeping patterns after sleep onset. Reduced time spent supine may be favourable to the breathing condition of pregnant women, thus improving their sleep efficiency. The Lasso model can be powerful in narrowing down useful information from a large number of variables