A priori bounds for co-dimension one isometric embeddings

We prove a priori bounds for the trace of the second fundamental form of a $C^4$ isometric embedding into $R^{n+1}$ of a metric $g$ of non-negative sectional curvature on $S^n$, in terms of the scalar curvature, and the diameter of $g$. These estimates give a bound on the extrinsic geometry in terms of intrinsic quantities. They generalize estimates originally obtained by Weyl for the case $n=2$ and positive curvature, and then by P. Guan and the first author for non-negative curvature and $n=2$. Using $C^{2,\alpha}$ interior estimates of Evans and Krylov for concave fully nonlinear elliptic partial differential equations, these bounds allow us to obtain the following convergence theorem: For any $\epsilon>0$, the set of metrics of non-negative sectional curvature and scalar curvature bounded below by $\epsilon$ which are isometrically embedable in Euclidean space $R^{n+1}$ is closed in the H\"older space $C^{4,\alpha}$, $0<\alpha<1$. These results are obtained in an effort to understand the following higher dimensional version of the Weyl embedding problem which we propose: \emph{Suppose that $g$ is a smooth metric of non-negative sectional curvature and positive scalar curvature on \S^n$which is locally isometrically embeddable in$R^{n+1}$. Does$(S^n,g)$then admit a smooth global isometric embedding into$R^{n+1}$?

Approximate Sparse Recovery: Optimizing Time and Measurements

An approximate sparse recovery system consists of parameters $k,N$, an $m$-by-$N$ measurement matrix, $\Phi$, and a decoding algorithm, $\mathcal{D}$. Given a vector, $x$, the system approximates $x$ by $\widehat x =\mathcal{D}(\Phi x)$, which must satisfy $\| \widehat x - x\|_2\le C \|x - x_k\|_2$, where $x_k$ denotes the optimal $k$-term approximation to $x$. For each vector $x$, the system must succeed with probability at least 3/4. Among the goals in designing such systems are minimizing the number $m$ of measurements and the runtime of the decoding algorithm, $\mathcal{D}$. In this paper, we give a system with $m=O(k \log(N/k))$ measurements--matching a lower bound, up to a constant factor--and decoding time $O(k\log^c N)$, matching a lower bound up to $\log(N)$ factors. We also consider the encode time (i.e., the time to multiply $\Phi$ by $x$), the time to update measurements (i.e., the time to multiply $\Phi$ by a 1-sparse $x$), and the robustness and stability of the algorithm (adding noise before and after the measurements). Our encode and update times are optimal up to $\log(N)$ factors

Circadian rest-activity rhythms predict cognitive function in early Parkinson's disease independently of sleep

BACKGROUND: Cognitive impairment is a common and debilitating symptom of Parkinson's disease (PD), and its etiology is likely multifactorial. One candidate mechanism is circadian disruption. Although there is evidence of circadian abnormalities in PD, no studies have directly assessed their association with cognitive impairment. OBJECTIVES: Investigate whether circadian rest-activity rhythm is associated with cognitive function in PD independently of sleep. METHODS: Thirty-five participants with PD wore wrist actigraph monitors and completed sleep diaries for 7 to 10 days, then underwent neuropsychological testing. Rest-activity rhythm was characterized using nonparametric circadian rhythm analysis of actigraphy data. Objective sleep parameters were also estimated using actigraphy data. Hierarchical regression models assessed the independent contributions of sleep and rest-activity rhythm to cognitive performance. RESULTS: Less stable day-to-day rest-activity rhythm was associated with poorer executive, visuospatial, and psychomotor functioning, but not with memory. Hierarchical regressions showed that interdaily stability's contribution to cognitive performance was independent of sleep's contributions. Whereas sleep contributed to executive function, but not psychomotor or visuospatial performance, rest-activity rhythm stability significantly contributed to variance in all three of these domains, uniquely accounting for 14.4% to 17.6% of their performance variance. CONCLUSIONS: Our findings indicate that circadian rest-activity rhythm is associated with cognitive impairment independently of sleep. This suggests the possible utility of rest-activity rhythm as a biomarker for circadian function in PD. Future research should explore interventions to stabilize behavioral rhythms in order to strengthen circadian function, which, in turn, may reduce cognitive impairment in PD.R00 HL102241 - NHLBI NIH HHS; R01 AG048108 - NIA NIH HHSAccepted manuscrip

Linking Aboveground Traits to Root Traits and Local Environment: Implications of the Plant Economics Spectrum.

The plant economics spectrum proposes that ecological traits are functionally coordinated and adapt along environmental gradients. However, empirical evidence is mixed about whether aboveground and root traits are consistently linked and which environmental factors drive functional responses. Here we measure the strength of relationships between aboveground and root traits, and examine whether community-weighted mean trait values are adapted along gradients of light and soil fertility, based on the seedling censuses of 57 species in a subtropical forest. We found that aboveground traits were good predictors of root traits; specific leaf area, dry matter, nitrogen and phosphorus content were strongly correlated with root tissue density and specific root length. Traits showed patterns of adaptation along the gradients of soil fertility and light; species with fast resource-acquisitive strategies were more strongly associated with high soil phosphorus, potassium, openness, and with low nitrogen, organic matter conditions. This demonstrates the potential to estimate belowground traits from known aboveground traits in seedling communities, and suggests that soil fertility is one of the main factors driving functional responses. Our results extend our understanding of how ecological strategies shape potential responses of plant communities to environmental change

An immune system based genetic algorithm using permutation-based dualism for dynamic traveling salesman problems

Copyright @ Springer-Verlag Berlin Heidelberg 2009.In recent years, optimization in dynamic environments has attracted a growing interest from the genetic algorithm community due to the importance and practicability in real world applications. This paper proposes a new genetic algorithm, based on the inspiration from biological immune systems, to address dynamic traveling salesman problems. Within the proposed algorithm, a permutation-based dualism is introduced in the course of clone process to promote the population diversity. In addition, a memory-based vaccination scheme is presented to further improve its tracking ability in dynamic environments. The experimental results show that the proposed diversification and memory enhancement methods can greatly improve the adaptability of genetic algorithms for dynamic traveling salesman problems.This work was supported by the Key Program of National Natural Science Foundation (NNSF) of China under Grant No. 70431003 and Grant No. 70671020, the Science Fund for Creative Research Group of NNSF of China under GrantNo. 60521003, the National Science and Technology Support Plan of China under Grant No. 2006BAH02A09 and the Engineering and Physical Sciences Research Council (EPSRC) of UK under Grant No. EP/E060722/1

On a common circle: natural scenes and Gestalt rules

To understand how the human visual system analyzes images, it is essential to know the structure of the visual environment. In particular, natural images display consistent statistical properties that distinguish them from random luminance distributions. We have studied the geometric regularities of oriented elements (edges or line segments) present in an ensemble of visual scenes, asking how much information the presence of a segment in a particular location of the visual scene carries about the presence of a second segment at different relative positions and orientations. We observed strong long-range correlations in the distribution of oriented segments that extend over the whole visual field. We further show that a very simple geometric rule, cocircularity, predicts the arrangement of segments in natural scenes, and that different geometrical arrangements show relevant differences in their scaling properties. Our results show similarities to geometric features of previous physiological and psychophysical studies. We discuss the implications of these findings for theories of early vision.Comment: 3 figures, 2 large figures not include

Gasoline Prices and Their Relationship to Drunk-Driving Crashes

his study investigates the relationship between changing gasoline prices and drunk-driving crashes. Specifically, we examine the effects of gasoline prices on drunk-driving crashes in Mississippi by age, gender, and race from 2004Ð2008, a period experiencing great fluctuation in gasoline prices. An exploratory visualization by graphs shows that higher gasoline prices are generally associated with fewer drunk-driving crashes. Higher gasoline prices depress drunk- driving crashes among younger and older drivers, among male and female drivers, and among white, black, and Hispanic drivers. The statistical results suggest that higher gasoline prices lead to lower drunk-driving crashes for female and black drivers. However, alcohol consumption is a better predictor of drunk-driving crashes, especially for male, white, and older drivers.Drunk-driving crashes, gasoline prices, alcohol consumption, Mississippi

Network of two-Chinese-character compound words in Japanese language

Some statistical properties of a network of two-Chinese-character compound words in Japanese language are reported. In this network, a node represents a Chinese character and an edge represents a two-Chinese-character compound word. It is found that this network has properties of "small-world" and "scale-free." A network formed by only Chinese characters for common use ({\it joyo-kanji} in Japanese), which is regarded as a subclass of the original network, also has small-world property. However, a degree distribution of the network exhibits no clear power law. In order to reproduce disappearance of the power-law property, a model for a selecting process of the Chinese characters for common use is proposed

Magnetic rogue wave in a perpendicular anisotropic ferromagnetic nanowire with spin-transfer torque

We present the current controlled motion of dynamic soliton embedded in spin wave background in ferromagnetic nanowire. With the stronger breather character we get the novel magnetic rogue wave and clarify its formation mechanism. The generation of magnetic rogue wave is mainly arose from the accumulation of energy and magnons toward to its central part. We also observe that the spin-polarized current can control the exchange rate of magnons between envelope soliton and background, and the critical current condition is obtained analytically. Even more interesting is that the spin-transfer torque plays the completely opposite role for the cases of below and above the critical value.Comment: 5 figure