Numerical assessment of stability of interface discontinuous finite element pressure spaces

The stability of two recently developed pressure spaces has been assessed numerically: The space proposed by Ausas et al. [R.F. Ausas, F.S. Sousa, G.C. Buscaglia, An improved finite element space for discontinuous pressures, Comput. Methods Appl. Mech. Engrg. 199 (2010) 1019-1031], which is capable of representing discontinuous pressures, and the space proposed by Coppola-Owen and Codina [A.H. Coppola-Owen, R. Codina, Improving Eulerian two-phase flow finite element approximation with discontinuous gradient pressure shape functions, Int. J. Numer. Methods Fluids, 49 (2005) 1287-1304], which can represent discontinuities in pressure gradients. We assess the stability of these spaces by numerically computing the inf-sup constants of several meshes. The inf-sup constant results as the solution of a generalized eigenvalue problems. Both spaces are in this way confirmed to be stable in their original form. An application of the same numerical assessment tool to the stabilized equal-order P-1/P-1 formulation is then reported. An interesting finding is that the stabilization coefficient can be safely set to zero in an arbitrary band of elements without compromising the formulation's stability. An analogous result is also reported for the mini-element P-1(+)/P-1 when the velocity bubbles are removed in an arbitrary band of elements. (C) 2012 Elsevier B.V. All rights reserved.

Reproducing kernel Hilbert spaces associated with kernels on topological spaces

We analyze reproducing kernel Hilbert spaces of positive definite kernels on a topological space X being either first countable or locally compact. The results include versions of Mercer's theorem and theorems on the embedding of these spaces into spaces of continuous and square integrable functions.

Three new species of Lebiasina (Characiformes: Lebiasinidae) from the Brazilian shield border at Serra do Cachimbo, Pará, Brazil

Lebiasina marilynae n. sp., L. melanoguttata n. sp., and L. minuta n. sp. are described from the headwaters of the rio Curuá, in Serra do Cachimbo, Pará, Brazil, and represent the only members of the Lebiasininae in the Brazilian Shied, so far. A close relationship among these species is proposed based on: 1) the presence of a pair of foramina through which the ramus palatinus of the facial nerve passes, a modification unique in Lebiasinidae and apparently in the Characiformes, 2) the enlargement of the extrascapular bone, 3) the absence of the secondary stripe, and 4) the nearly equal length of caudal-fin lobes. Lebiasina marilynae additionally differs from all congeners in having the primary stripe extending from the tip of the snout to the distal border of the caudal-fin peduncle, the possession of two series of dark blotches parallel to the primary stripe, and a rounded dorsal surface of the mesethmoid. Lebiasina melanoguttata and Lebiasina minuta additionally differ from all congeners in the absence of the primary stripe and the caudal blotch, and the presence of three longitudinal series of dark blotches at the base of the scales of series 3-5. Lebiasina melanoguttata differs from Lebiasina minuta in the absence of a dark blotch at the base of the median rays of the dorsal fin, second infrapharyngobranchial bearing conical teeth, the reddish overall coloration of the eye and fins, and the dark blotches never coalescing (vs. dark dorsal-fin blotch present; the second infrapharyngobranchial being edentulous; dark, olive green eyes, and the yellowish overall color of body and fins; and the dark blotches of longitudinal series 3 and 4 coalescing where scales of adjacent longitudinal series overlap). The occurrence of species of the Lebiasininae on the Brazilian Shield is discussed, and the distribution pattern of the species described herein is compared to that of other endemic species of the Serra do Cachimbo, a highly biodiverse area isolated from the rest of the Amazon basin.

Weighted Fourier–Laplace transforms in reproducing kernel Hilbert spaces on the sphere

We study the action of a weighted Fourier–Laplace transform on the functions in the reproducing kernel Hilbert space (RKHS) associated with a positive definite kernel on the sphere. After defining a notion of smoothness implied by the transform, we show that smoothness of the kernel implies the same smoothness for the generating elements (spherical harmonics) in the Mercer expansion of the kernel. We prove a reproducing property for the weighted Fourier–Laplace transform of the functions in the RKHS and embed the RKHS into spaces of smooth functions. Some relevant properties of the embedding are considered, including compactness and boundedness. The approach taken in the paper includes two important notions of differentiability characterized by weighted Fourier–Laplace transforms: fractional derivatives and Laplace–Beltrami derivatives.

Pullback exponential attractors for evolution processes in Banach spaces: properties and applications

This article is a continuation of our previous work [5], where we formulated general existence theorems for pullback exponential attractors for asymptotically compact evolution processes in Banach spaces and discussed its implications in the autonomous case. We now study properties of the attractors and use our theoretical results to prove the existence of pullback exponential attractors in two examples, where previous results do not apply.