An improved finite element space for discontinuous pressures

We consider incompressible Stokes flow with an internal interface at which the pressure is discontinuous, as happens for example in problems involving surface tension. We assume that the mesh does not follow the interface, which makes classical interpolation spaces to yield suboptimal convergence rates (typically, the interpolation error in the L(2)(Omega)-norm is of order h(1/2)). We propose a modification of the P(1)-conforming space that accommodates discontinuities at the interface without introducing additional degrees of freedom or modifying the sparsity pattern of the linear system. The unknowns are the pressure values at the vertices of the mesh and the basis functions are computed locally at each element, so that the implementation of the proposed space into existing codes is straightforward. With this modification, numerical tests show that the interpolation order improves to O(h(3/2)). The new pressure space is implemented for the stable P(1)(+)/P(1) mini-element discretization, and for the stabilized equal-order P(1)/P(1) discretization. Assessment is carried out for Poiseuille flow with a forcing surface and for a static bubble. In all cases the proposed pressure space leads to improved convergence orders and to more accurate results than the standard P(1) space. In addition, two Navier-Stokes simulations with moving interfaces (Rayleigh-Taylor instability and merging bubbles) are reported to show that the proposed space is robust enough to carry out realistic simulations. (c) 2009 Elsevier B.V. All rights reserved.

Finite-dimensional global attractors in Banach spaces

We provide bounds on the upper box-counting dimension of negatively invariant subsets of Banach spaces, a problem that is easily reduced to covering the image of the unit ball under a linear map by a collection of balls of smaller radius. As an application of the abstract theory we show that the global attractors of a very broad class of parabolic partial differential equations (semilinear equations in Banach spaces) are finite-dimensional. (C) 2010 Elsevier Inc. All rights reserved.

Lack of photoreceptor signaling alters the expression of specific synaptic proteins in the retina

Synaptic modulation by activity-dependent changes constitutes a cellular mechanism for neuronal plasticity. However, it is not clear how the complete lack of neuronal signaling specifically affects elements involved in the communication between neurons. In the retina, it is now well established that both chemical and electrical synapses are essential to mediate the transmission of visual signaling triggered by the photoreceptors. In this study, we compared the expression of synaptic proteins in the retinas of wild-type (WT) vs. rd/rd mice, an animal model that displays inherited and specific ablation of photoreceptors caused by a mutation in the gene encoding the beta-subunit of rod cGMP-phosphodiesterase (Pde6b(rd1)). We specifically examined the expression of connexins (Cx), the proteins that form the gap junction channels of electrical synapses, in addition to synaptophysin and synapsin 1, which are involved in the release of neurotransmitters at chemical synapses. Our results revealed that Cx36 gene expression levels are lower in the retinas of rd/rd when compared with WT. Confocal analysis indicated that Cx36 immunolabeling almost disappeared in the outer plexiform layer without significant changes in protein distribution within the inner plexiform layer of rd/rd retinas. Likewise, synaptophysin expression remarkably decreased in the outer plexiform layer of rd/rd retinas, and this down-regulation was also associated with diminished transcript levels. Furthermore, we observed down-regulation of Cx57 gene expression in rd/rd retinas when compared with WT and also changes in protein distribution. Interestingly, Cx45 and synapsin I expression in rd/rd retinas showed no noticeable changes when compared with WT. Taken together, our results revealed that the loss of photoreceptors leads to decreased expression of some synaptic proteins. More importantly, this study provides evidence that neuronal activity regulates, but is not essential to maintain, the expression of synaptic elements. (c) 2008 IBRO. Published by Elsevier Ltd. All rights reserved.

Equivariant Nielsen root theory for G-maps

Let X be a compact Hausdorff space, Y be a connected topological manifold, f : X -> Y be a map between closed manifolds and a is an element of Y. The vanishing of the Nielsen root number N(f; a) implies that f is homotopic to a root free map h, i.e., h similar to f and h(-1) (a) = empty set. In this paper, we prove an equivariant analog of this result for G-maps between G-spaces where G is a finite group. (C) 2010 Elsevier B.V. All rights reserved.

Finite determinacy and Whitney equisingularity of map germs from C(n) to C(2n-1)

We show that a holomorphic map germ f : (C(n), 0) -> (C(2n-1), 0) is finitely determined if and only if the double point scheme D(f) is a reduced curve. If n >= 3, we have that mu(D(2)(f)) = 2 mu(D(2)(f)/S(2))+C(f)-1, where D(2)(f) is the lifting of the double point curve in (C(n) x C(n), 0), mu(X) denotes the Milnor number of X and C(f) is the number of cross-caps that appear in a stable deformation of f. Moreover, we consider an unfolding F(t, x) = (t, f(t)(x)) of f and show that if F is mu-constant, then it is excellent in the sense of Gaffney. Finally, we find a minimal set of invariants whose constancy in the family f(t) is equivalent to the Whitney equisingularity of F. We also give an example of an unfolding which is topologically trivial, but it is not Whitney equisingular.