Electrical Transitions/Memristors

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Periodic intermediate ββ-expansions of Pisot numbers

The subshift of finite type property (also known as the Markov property) is ubiquitous in dynamical systems and the simplest and most widely studied class of dynamical systems are $\beta$-shifts, namely transformations of the form $T_{\beta, \alpha} \colon x \mapsto \beta x + \alpha \bmod{1}$ acting on $[-\alpha/(\beta - 1), (1-\alpha)/(\beta - 1)]$, where $(\beta, \alpha) \in \Delta$ is fixed and where $\Delta = \{ (\beta, \alpha) \in \mathbb{R}^{2} \colon \beta \in (1,2) \; \text{and} \; 0 \leq \alpha \leq 2-\beta \}$. Recently, it was shown, by Li et al. (Proc. Amer. Math. Soc. 147(5): 2045-2055, 2019), that the set of $(\beta, \alpha)$ such that $T_{\beta, \alpha}$ has the subshift of finite type property is dense in the parameter space $\Delta$. Here, they proposed the following question. Given a fixed $\beta \in (1, 2)$ which is the $n$-th root of a Perron number, does there exists a dense set of $\alpha$ in the fiber $\{\beta\} \times (0, 2- \beta)$, so that $T_{\beta, \alpha}$ has the subshift of finite type property? We answer this question in the positive for a class of Pisot numbers. Further, we investigate if this question holds true when replacing the subshift of finite type property by the property of beginning sofic (that is a factor of a subshift of finite). In doing so we generalise, a classical result of Schmidt (Bull. London Math. Soc., 12(4): 269-278, 1980) from the case when $\alpha = 0$ to the case when $\alpha \in (0, 2 - \beta)$. That is, we examine the structure of the set of eventually periodic points of $T_{\beta, \alpha}$ when $\beta$ is a Pisot number and when $\beta$ is the $n$-th root of a Pisot number.Comment: 13 pages, 1 figur

Brief for the United States as Amicus Curiae in Support of Neither Party

Amicus ("friend of the court") brief written by the United States in support of neither party in AMP v. Myriad Genetics (No. 2010-1406)

Multivariate patterns of brain-behavior associations across the adult lifespan

The nature of brain-behavior covariations with increasing age is poorly understood. In the current study, we used a multivariate approach to investigate the covariation between behavioral-health variables and brain features across adulthood. We recruited healthy adults aged 20–73 years-old (29 younger, mean age = 25.6 years; 30 older, mean age = 62.5 years), and collected structural and functional MRI (s/fMRI) during a resting-state and three tasks. From the sMRI, we extracted cortical thickness and subcortical volumes; from the fMRI, we extracted activation peaks and functional network connectivity (FNC) for each task. We conducted canonical correlation analyses between behavioral-health variables and the sMRI, or the fMRI variables, across all participants. We found significant covariations for both types of neuroimaging phenotypes (ps = 0.0004) across all individuals, with cognitive capacity and age being the largest opposite contributors. We further identified different variables contributing to the models across phenotypes and age groups. Particularly, we found behavior was associated with different neuroimaging patterns between the younger and older groups. Higher cognitive capacity was supported by activation and FNC within the executive networks in the younger adults, while it was supported by the visual networks’ FNC in the older adults. This study highlights how the brain-behavior covariations vary across adulthood and provides further support that cognitive performance relies on regional recruitment that differs between older and younger individuals