Elliptic operators in even subspaces

In the paper we consider the theory of elliptic operators acting in subspaces defined by pseudodifferential projections. This theory on closed manifolds is connected with the theory of boundary value problems for operators violating Atiyah-Bott condition. We prove an index formula for elliptic operators in subspaces defined by even projections on odd-dimensional manifolds and for boundary value problems, generalizing the classical result of Atiyah-Bott. Besides a topological contribution of Atiyah-Singer type, the index formulas contain an invariant of subspaces defined by even projections. This homotopy invariant can be expressed in terms of the eta-invariant. The results also shed new light on P.Gilkey's work on eta-invariants of even-order operators.Comment: 39 pages, 2 figure

Thirty-five years of computerized cognitive assessment of aging — Where are we now?

Over the past 35 years, the proliferation of technology and the advent of the internet have resulted in many reliable and easy to administer batteries for assessing cognitive function. These approaches have great potential for affecting how the health care system monitors and screens for cognitive changes in the aging population. Here, we review these new technologies with a specific emphasis on what they offer over and above traditional ‘paper-and-pencil’ approaches to assessing cognitive function. Key advantages include fully automated administration and scoring, the interpretation of individual scores within the context of thousands of normative data points, the inclusion of ‘meaningful change’ and ‘validity’ indices based on these large norms, more efficient testing, increased sensitivity, and the possibility of characterising cognition in samples drawn from the general population that may contain hundreds of thousands of test scores. The relationship between these new computerized platforms and existing (and commonly used) paper-and-pencil tests is explored, with a particular emphasis on why computerized tests are particularly advantageous for assessing the cognitive changes associated with aging

Exceptional responders in conservation.

Conservation operates within complex systems with incomplete knowledge of the system and the interventions utilized. This frequently results in the inability to find generally applicable methods to alleviate threats to Earth's vanishing wildlife. One approach used in medicine and the social sciences has been to develop a deeper understanding of positive outliers. Where such outliers share similar characteristics, they may be considered exceptional responders. We devised a 4-step framework for identifying exceptional responders in conservation: identification of the study system, identification of the response structure, identification of the threshold for exceptionalism, and identification of commonalities among outliers. Evaluation of exceptional responders provides additional information that is often ignored in randomized controlled trials and before-after control-intervention experiments. Interrogating the contextual factors that contribute to an exceptional outcome allow exceptional responders to become valuable pieces of information leading to unexpected discoveries and novel hypotheses

Elliptic operators in odd subspaces

An elliptic theory is constructed for operators acting in subspaces defined via odd pseudodifferential projections. Subspaces of this type arise as Calderon subspaces for first order elliptic differential operators on manifolds with boundary, or as spectral subspaces for self-adjoint elliptic differential operators of odd order. Index formulas are obtained for operators in odd subspaces on closed manifolds and for general boundary value problems. We prove that the eta-invariant of operators of odd order on even-dimesional manifolds is a dyadic rational number.Comment: 27 page

Laplacian Growth, Elliptic Growth, and Singularities of the Schwarz Potential

The Schwarz function has played an elegant role in understanding and in generating new examples of exact solutions to the Laplacian growth (or "Hele- Shaw") problem in the plane. The guiding principle in this connection is the fact that "non-physical" singularities in the "oil domain" of the Schwarz function are stationary, and the "physical" singularities obey simple dynamics. We give an elementary proof that the same holds in any number of dimensions for the Schwarz potential, introduced by D. Khavinson and H. S. Shapiro [17] (1989). A generalization is also given for the so-called "elliptic growth" problem by defining a generalized Schwarz potential. New exact solutions are constructed, and we solve inverse problems of describing the driving singularities of a given flow. We demonstrate, by example, how \mathbb{C}^n - techniques can be used to locate the singularity set of the Schwarz potential. One of our methods is to prolong available local extension theorems by constructing "globalizing families". We make three conjectures in potential theory relating to our investigation

Uniformization and an Index Theorem for Elliptic Operators Associated with Diffeomorphisms of a Manifold

We consider the index problem for a wide class of nonlocal elliptic operators on a smooth closed manifold, namely differential operators with shifts induced by the action of an isometric diffeomorphism. The key to the solution is the method of uniformization: We assign to the nonlocal problem a pseudodifferential operator with the same index, acting in sections of an infinite-dimensional vector bundle on a compact manifold. We then determine the index in terms of topological invariants of the symbol, using the Atiyah-Singer index theorem.Comment: 16 pages, no figure

Assessing awareness in severe Alzheimer’s disease

There is an urgent need to understand the nature of awareness in people with severe Alzheimer’s disease (AD) to ensure effective person-centered care. Objective biomarkers of awareness validated in other clinical groups (e.g., anesthesia, minimally conscious states) offer an opportunity to investigate awareness in people with severe AD. In this article we demonstrate the feasibility of using Transcranial magnetic stimulation (TMS) combined with EEG, event related potentials (ERPs) and fMRI to assess awareness in severe AD. TMS-EEG was performed in six healthy older controls and three people with severe AD. The perturbational complexity index (PCIST) was calculated as a measure of capacity for conscious awareness. People with severe AD demonstrated a PCIST around or below the threshold for consciousness, suggesting reduced capacity for consciousness. ERPs were recorded during a visual perception paradigm. In response to viewing faces, two patients with severe AD provisionally demonstrated similar visual awareness negativity to healthy controls. Using a validated fMRI movie-viewing task, independent component analysis in two healthy controls and one patient with severe AD revealed activation in auditory, visual and fronto-parietal networks. Activation patterns in fronto-parietal networks did not significantly correlate between the patient and controls, suggesting potential differences in conscious awareness and engagement with the movie. Although methodological issues remain, these results demonstrate the feasibility of using objective measures of awareness in severe AD. We raise a number of challenges and research questions that should be addressed using these biomarkers of awareness in future studies to improve understanding and care for people with severe AD

Assessing awareness in severe Alzheimer's disease

There is an urgent need to understand the nature of awareness in people with severe Alzheimer's disease (AD) to ensure effective person-centered care. Objective biomarkers of awareness validated in other clinical groups (e.g., anesthesia, minimally conscious states) offer an opportunity to investigate awareness in people with severe AD. In this article we demonstrate the feasibility of using Transcranial magnetic stimulation (TMS) combined with EEG, event related potentials (ERPs) and fMRI to assess awareness in severe AD. TMS-EEG was performed in six healthy older controls and three people with severe AD. The perturbational complexity index (PCIST) was calculated as a measure of capacity for conscious awareness. People with severe AD demonstrated a PCIST around or below the threshold for consciousness, suggesting reduced capacity for consciousness. ERPs were recorded during a visual perception paradigm. In response to viewing faces, two patients with severe AD provisionally demonstrated similar visual awareness negativity to healthy controls. Using a validated fMRI movie-viewing task, independent component analysis in two healthy controls and one patient with severe AD revealed activation in auditory, visual and fronto-parietal networks. Activation patterns in fronto-parietal networks did not significantly correlate between the patient and controls, suggesting potential differences in conscious awareness and engagement with the movie. Although methodological issues remain, these results demonstrate the feasibility of using objective measures of awareness in severe AD. We raise a number of challenges and research questions that should be addressed using these biomarkers of awareness in future studies to improve understanding and care for people with severe AD

Noncommutative elliptic theory. Examples

We study differential operators, whose coefficients define noncommutative algebras. As algebra of coefficients, we consider crossed products, corresponding to action of a discrete group on a smooth manifold. We give index formulas for Euler, signature and Dirac operators twisted by projections over the crossed product. Index of Connes operators on the noncommutative torus is computed.Comment: 23 pages, 1 figur