Generalized Dirichlet to Neumann operator on invariant differential forms and equivariant cohomology

In a recent paper, Belishev and Sharafutdinov consider a compact Riemannian manifold $M$ with boundary $\partial M$. They define a generalized Dirichlet to Neumann (DN) operator $\Lambda$ on all forms on the boundary and they prove that the real additive de Rham cohomology structure of the manifold in question is completely determined by $\Lambda$. This shows that the DN map $\Lambda$ inscribes into the list of objects of algebraic topology. In this paper, we suppose $G$ is a torus acting by isometries on $M$. Given $X$ in the Lie algebra of $G$ and the corresponding vector field $X_M$ on $M$, one defines Witten's inhomogeneous coboundary operator $d_{X_M} = d+\iota_{X_M}$ on invariant forms on $M$. The main purpose is to adapt Belishev and Sharafutdinov's boundary data to invariant forms in terms of the operator $d_{X_M}$ and its adjoint $\delta_{X_M}$. In other words, we define an operator $\Lambda_{X_M}$ on invariant forms on the boundary which we call the $X_M$-DN map and using this we recover the long exact $X_M$-cohomology sequence of the topological pair $(M,\partial M)$ from an isomorphism with the long exact sequence formed from our boundary data. We then show that $\Lambda_{X_M}$ completely determines the free part of the relative and absolute equivariant cohomology groups of $M$ when the set of zeros of the corresponding vector field $X_M$ is equal to the fixed point set $F$ for the $G$-action. In addition, we partially determine the mixed cup product (the ring structure) of $X_M$-cohomology groups from $\Lambda_{X_M}$. These results explain to what extent the equivariant topology of the manifold in question is determined by the $X_M$-DN map $\Lambda_{X_M}$. Finally, we illustrate the connection between Belishev and Sharafutdinov's boundary data on $\partial F$ and ours on $\partial M$.Comment: 21 page

Detection of scene-relative object movement and optic flow parsing across the adult lifespan

Moving around safely relies critically on our ability to detect object movement. This is made difficult because retinal motion can arise from object movement or our own movement. Here we investigate ability to detect scene-relative object movement using a neural mechanism called optic flow parsing. This mechanism acts to subtract retinal motion caused by self-movement. Because older observers exhibit marked changes in visual motion processing, we consider performance across a broad age range (N = 30, range: 20–76 years). In Experiment 1 we measured thresholds for reliably discriminating the scene-relative movement direction of a probe presented among three-dimensional objects moving onscreen to simulate observer movement. Performance in this task did not correlate with age, suggesting that ability to detect scene-relative object movement from retinal information is preserved in ageing. In Experiment 2 we investigated changes in the underlying optic flow parsing mechanism that supports this ability, using a well-established task that measures the magnitude of globally subtracted optic flow. We found strong evidence for a positive correlation between age and global flow subtraction. These data suggest that the ability to identify object movement during self-movement from visual information is preserved in ageing, but that there are changes in the flow parsing mechanism that underpins this ability. We suggest that these changes reflect compensatory processing required to counteract other impairments in the ageing visual system

Critical points of higher order for the normal map of immersions in R^d

We study the critical points of the normal map v : NM -> Rk+n, where M is an immersed k-dimensional submanifold of Rk+n, NM is the normal bundle of M and v(m, u) = m + u if u is an element of NmM. Usually, the image of these critical points is called the focal set. However, in that set there is a subset where the focusing is highest, as happens in the case of curves in R-3 with the curve of the centers of spheres with contact of third order with the curve. We give a definition of r-critical points of a smooth map between manifolds, and apply it to study the 2 and 3-critical points of the normal map in general and the 2-critical points for the case k = n = 2 in detail. In the later case we analyze the relation with the strong principal directions of Montaldi (1986) [2]. (C) 2011 Elsevier B.V. All rights reserved.Work partially supported by CAPES (BEX 4533/06-2).Monera, M.; Montesinos-Amilibia, A.; Moraes, S.; Sanabria Codesal, E. (2012). Critical points of higher order for the normal map of immersions in R^d. Topology and its Applications. 159:537-544. https://doi.org/10.1016/j.topol.2011.09.029S53754415

Pattern separation underpins expectation-modulated memory

Pattern separation and completion are fundamental hippocampal computations supporting memory encoding and retrieval. However, despite extensive exploration of these processes, it remains unclear whether and how top-down processes adaptively modulate the dynamics between these computations. Here we examine the role of expectation in shifting the hippocampus to perform pattern separation. In a behavioral task, 29 participants (7 males) learned a cue-object category contingency. Then, at encoding, one-third of the cues preceding the to-be-memorized objects, violated the studied rule. At test, participants performed a recognition task with old objects (targets) and a set of parametrically manipulated (very similar to dissimilar) foils for each object. Accuracy was found to be better for foils of high similarity to targets that were contextually unexpected at encoding compared with expected ones. Critically, there were no expectation-driven differences for targets and low similarity foils. To further explore these effects, we implemented a computational model of the hippocampus, performing the same task as the human participants. We used representational similarity analysis to examine how top-down expectation interacts with bottom-up perceptual input, in each layer. All subfields showed more dissimilar representations for unexpected items, with dentate gyrus (DG) and CA3 being more sensitive to expectation violation than CA1. Again, representational differences between expected and unexpected inputs were prominent for moderate to high levels of input similarity. This effect diminished when inputs from DG and CA3 into CA1 were lesioned. Overall, these novel findings strongly suggest that pattern separation in DG/CA3 underlies the effect that violation of expectation exerts on memory

Stability of relative equilibria with singular momentum values in simple mechanical systems

A method for testing $G_\mu$-stability of relative equilibria in Hamiltonian systems of the form "kinetic + potential energy" is presented. This method extends the Reduced Energy-Momentum Method of Simo et al. to the case of non-free group actions and singular momentum values. A normal form for the symplectic matrix at a relative equilibrium is also obtained.Comment: Partially rewritten. Some mistakes fixed. Exposition improve

Golden gaskets: variations on the Sierpi\'nski sieve

We consider the iterated function systems (IFSs) that consist of three general similitudes in the plane with centres at three non-collinear points, and with a common contraction factor \la\in(0,1). As is well known, for \la=1/2 the invariant set, \S_\la, is a fractal called the Sierpi\'nski sieve, and for \la<1/2 it is also a fractal. Our goal is to study \S_\la for this IFS for 1/2<\la<2/3, i.e., when there are "overlaps" in \S_\la as well as "holes". In this introductory paper we show that despite the overlaps (i.e., the Open Set Condition breaking down completely), the attractor can still be a totally self-similar fractal, although this happens only for a very special family of algebraic \la's (so-called "multinacci numbers"). We evaluate \dim_H(\S_\la) for these special values by showing that \S_\la is essentially the attractor for an infinite IFS which does satisfy the Open Set Condition. We also show that the set of points in the attractor with a unique ``address'' is self-similar, and compute its dimension. For ``non-multinacci'' values of \la we show that if \la is close to 2/3, then \S_\la has a nonempty interior and that if \la<1/\sqrt{3} then \S_\la$ has zero Lebesgue measure. Finally we discuss higher-dimensional analogues of the model in question.Comment: 27 pages, 10 figure