H\"{o}lder Continuity of the Spectral Measures for One-Dimensional Schr\"{o}dinger Operator in Exponential Regime

Avila and Jitomirskaya prove that the spectral measure $\mu_{\lambda v, \alpha,x}^f$ of quasi-periodic Schr\"{o}dinger operator is $1/2$-H\"{o}lder continuous with appropriate initial vector $f$, if $\alpha$ satisfies Diophantine condition and $\lambda$ is small. In the present paper, the conclusion is extended to that for all $\alpha$ with $\beta(\alpha)<\infty$, the spectral measure $\mu_{\lambda v, \alpha,x}^f$ is $1/2$-H\"{o}lder continuous with small $\lambda$, if $v$ is real analytic in a neighbor of $\{|\Im x|\leq C\beta\}$, where $C$ is a large absolute constant. In particular, the spectral measure $\mu_{\lambda, \alpha,x}^f$ of almost Mathieu operator is $1/2$-H\"{o}lder continuous if $|\lambda|<e^{-C\beta}$ with $C$ a large absolute constant

Spectral Gaps of Almost Mathieu Operator in Exponential Regime

For almost Mathieu operator $(H_{\lambda,\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+2\lambda \cos2\pi(\theta+n\alpha)u_n$, the dry version of Ten Martini problem predicts that the spectrum $\Sigma_{\lambda,\alpha}$ of $H_{\lambda,\alpha,\theta}$ has all gaps open for all $\lambda\neq 0$ and $\alpha \in \mathbb{R}\backslash \mathbb{Q}$. Avila and Jitomirskaya prove that $\Sigma_{\lambda,\alpha}$ has all gaps open for Diophantine $\alpha$ and $0<|\lambda|<1$. In the present paper, we show that $\Sigma_{\lambda,\alpha}$ has all gaps open for all $\alpha \in \mathbb{R}\backslash \mathbb{Q}$ with small $\lambda$

Life Equations for the Senescence Process

The Gompertz law of mortality quantitatively describes the mortality rate of humans and almost all multicellular animals. However, its underlying kinetic mechanism is unclear. The Gompertz law cannot explain the effect of temperature on lifespan and the mortality plateau at advanced ages. In this study a reaction kinetics model with a time dependent rate coefficient is proposed to describe the survival and senescence processes. A temperature-dependent mortality function was derived. The new mortality function becomes the Gompertz mortality function with the same relationship of parameters prescribed by the Strehler-Mildvan correlation when age is smaller than a characteristic value {\delta}, and reaches the mortality plateau when age is greater than {\delta}. A closed-form analytical expression for describing the relationship of average lifespan with temperature and other equations are derived from the new mortality function. The derived equations can be used to estimate the limit of average lifespan, predict the maximal longevity, calculate the temperature coefficient of lifespan, and explain the tendency of survival curve. This prediction is consistent with the most recently reported mortality trajectories for single-year birth cohorts. This study suggests that the senescence process results from the imbalance between damaging energy and protecting energy for the critical chemical substance in the body. The rate of senescence of the organism increases while the protecting energy decreases. The mortality plateau is reached when the protecting energy decreases to its minimal levels. The decreasing rate of the protecting energy is temperature-dependent. This study is exploring the connection between biochemical mechanism and demography.Comment: 14 pages. 4 figure