Cubic Hermite Splines

Abstract. Starting with Hermite cubic splines as the primal multigenerator, first a dual multigenerator on R is constructed that consists of continuous functions, has small support, and is exact of order 2. We then derive multiresolution sequences on the interval while retaining the polynomial exactness on the primal and dual sides. This guarantees moment conditions of the corresponding wavelets. The concept of stable completions [CDP] is then used to construct the corresponding primal and dual multiwavelets on the interval as follows. An appropriate variation of what is known as a hierarchical basis in finite element methods is shown to be an initial completion. This is then, in a second step, projected into the desired complements spanned by compactly supported biorthogonal multiwavelets. The masks of all multigenerators and multiwavelets are finite so that decomposition and reconstruction algorithms are simple and efficient. Furthermore, in addition to the Jackson estimates which follow from the exactness, one can also show Bernstein inequalities for the primal and dual multiresolutions. Consequently, sequence norms for the coefficients based on such multiwavelet expansions characterize Sobolev norms �·�H s ([0,1]) for s ∈ (−0.824926, 2.5). In particular, the multiwavelets form Riesz bases for L2([0, 1]). 1

Adaptive Multiscale Methods for the Numerical Treatment of Systems of PDEs

These notes are concerned with numerical analysis issues arising in the solution of certain systems involving stationary and instationary linear variational problems. Standard examples are second order elliptic boundary value problems, where particular emphasis is placed on the treatment of essential boundary conditions, and linear parabolic equations. These operator equations serve as a core ingredient for control problems where in addition to the state, the solution of the PDE, a control is to be determined which together with the state minimizes a certain tracking-type objective functional. Having assured that the variational problems are well-posed, we discuss numerical schemes based on B-splines and B-spline-type wavelets as a particular multiresolution discretization methodology. The guiding principle is to devise fast and efficient solution schemes which are optimal in the number of arithmetic unknowns. We discuss optimal conditioning of the system matrices, numerical stability of discrete formulations, and adaptive approximations

The impact of feral camels (Camelus dromedarius) on remote waterholes in central Australia

The Katiti and Petermann Aboriginal Land Trusts (KPALT) in central Australia contain significant biological and cultural assets, including the World Heritage-listed Uluu-Kata Tjua National Park. Until relatively recently, waterbodies in this remote region were not well studied, even though most have deep cultural and ecological significance to local Aboriginal people. The region also contains some of the highest densities of feral dromedary camels (Camelus dromedarius) in the nation, and was a focus area for the recently completed Australian Feral Camel Management Project. Within the project, the specific impacts of feral camels on waterholes were assessed throughout the KPALT. We found that aquatic macroinvertebrate biodiversity was significantly lower at camel-accessible sites, and fewer aquatic taxa considered 'sensitive' to habitat degradation were found at sites when or after camels were present. Water quality at camel-accessible sites was also significantly poorer (e.g. more turbid) than at sites inaccessible to camels. These results, in combination with emerging research and anecdotal evidence, suggest that large feral herbivores, such as feral camels and feral horses, are the main immediate threat to many waterbodies in central Australia. Management of large feral herbivores will be a key component in efforts to maintain and improve the health of waterbodies in central Australia, especially those not afforded protection within the national park system

Finding new communities: A principle of neuronal network reorganization in Alzheimer’s disease.

Background: Graph-theoretical analyses have been previously used to investigate changes in the functional connectome in patients with Alzheimer's disease (AD). However, these analyses generally assume static organizational principles, thereby neglecting a fundamental reconfiguration of functional connections in the face of neurodegeneration.Methods: Here, we focus on differences in the community structure of the functional connectome in young and old individuals and patients with AD. Patients with AD, moreover, underwent molecular imaging positron emission tomography by using [18F]AV1451 to measure tau burden, a major hallmark of AD.Results: Although the overall organizational principles of the community structure of the human functional connectome were preserved even in advanced healthy aging, they were considerably changed in AD. We discovered that the communities in AD are re-organized, with nodes changing their allegiance to communities, thus resulting in an overall less efficient re-organized community structure. We further discovered that nodes with a tendency to leave the communities displayed a relatively higher tau pathology burden.Discussion: Together, this study suggests that local tau pathology in AD is associated to fundamental changes in basic organizational principles of the human connectome. Our results shed new light on previous findings obtained by using the graph theory in AD and imply a general principle of the brain in response to neurodegeneration