The radial curvature of an end that makes eigenvalues vanish in the essential spectrum II

Under the quadratic-decay-conditions of the radial curvatures of an end, we shall derive growth estimates of solutions to the eigenvalue equation and show the absence of eigenvalues.Comment: "$\ge$" in the conditions $(*_4)$ and $(*_5)$ should be replaced by "$>$". $\gamma \ge \frac{n-1}{2}(b-a)$ in the conclusion of Theorem 1.3 should be replaced by $\gamma > \frac{n-1}{2}(b-a)$; trivial miss-calculatio

On the essential spectrum of Nadirashvili-Martin-Morales minimal surfaces

We show that the spectrum of a complete submanifold properly immersed into a ball of a Riemannian manifold is discrete, provided the norm of the mean curvature vector is sufficiently small. In particular, the spectrum of a complete minimal surface properly immersed into a ball of $\mathbb{R}^{3}$ is discrete. This gives a positive answer to a question of Yau.Comment: This article is an improvement of an earlier version titled On the spectrum of Martin-Morales minimal surfaces. 7 page

Functional imaging of cognition in an old-old population: A case for portable functional near-infrared spectroscopy

In this study, functional near-infrared spectroscopy (fNIRS) was used to record brain activa- tion during cognitive testing in older individuals (88±6yo; N = 19) living in residential care communities. This population, which is often associated with loss of personal independence due to physical or cognitive decline associated with aging, is also often under-represented in neuroscience research because of a limited means to participate in studies which often take place in large urban or university centers. In this study, we demonstrate the feasibility and initial results using a portable 8-source by 4-detector fNIRS system to measure brain activity from participants within residential care community centers. Using fNIRS, brain sig- nals were recorded during a series of computerized cognitive tests, including a Symbol Digit Coding test (SDC), Stroop Test (ST), and Shifting Attention Test (SAT). The SDC and SAT elicited greater activity in the left middle frontal region of interest. Three components of the ST produced increases in the right middle frontal and superior frontal, and left superior frontal regions. An association between advanced age and increased activation in the right middle frontal region was observed during the incongruent ST. Although none of the partici- pants had clinical dementia based on the short portable mental status questionnaire, the group performance was slightly below age-normed values on these cognitive tests. These results demonstrate the capability for obtaining functional neuroimaging measures in resi- dential settings, which ultimately may aid in prognosis and care related to dementia in older adults