Marcinkiewicz--Zygmund measures on manifolds

Let ${\mathbb X}$ be a compact, connected, Riemannian manifold (without boundary), $\rho$ be the geodesic distance on ${\mathbb X}$, $\mu$ be a probability measure on ${\mathbb X}$, and $\{\phi_k\}$ be an orthonormal system of continuous functions, $\phi_0(x)=1$ for all $x\in{\mathbb X}$, $\{\ell_k\}_{k=0}^\infty$ be an nondecreasing sequence of real numbers with $\ell_0=1$, $\ell_k\uparrow\infty$ as $k\to\infty$, $\Pi_L:={\mathsf {span}}\{\phi_j : \ell_j\le L\}$, $L\ge 0$. We describe conditions to ensure an equivalence between the $L^p$ norms of elements of $\Pi_L$ with their suitably discretized versions. We also give intrinsic criteria to determine if any system of weights and nodes allows such inequalities. The results are stated in a very general form, applicable for example, when the discretization of the integrals is based on weighted averages of the elements of $\Pi_L$ on geodesic balls rather than point evaluations.Comment: 28 pages, submitted for publicatio

Locally Learning Biomedical Data Using Diffusion Frames

Diffusion geometry techniques are useful to classify patterns and visualize high-dimensional datasets. Building upon ideas from diffusion geometry, we outline our mathematical foundations for learning a function on high-dimension biomedical data in a local fashion from training data. Our approach is based on a localized summation kernel, and we verify the computational performance by means of exact approximation rates. After these theoretical results, we apply our scheme to learn early disease stages in standard and new biomedical datasets

Splines and Wavelets on Geophysically Relevant Manifolds

Analysis on the unit sphere $\mathbb{S}^{2}$ found many applications in seismology, weather prediction, astrophysics, signal analysis, crystallography, computer vision, computerized tomography, neuroscience, and statistics. In the last two decades, the importance of these and other applications triggered the development of various tools such as splines and wavelet bases suitable for the unit spheres $\mathbb{S}^{2}$, $\>\>\mathbb{S}^{3}$ and the rotation group $SO(3)$. Present paper is a summary of some of results of the author and his collaborators on generalized (average) variational splines and localized frames (wavelets) on compact Riemannian manifolds. The results are illustrated by applications to Radon-type transforms on $\mathbb{S}^{d}$ and $SO(3)$.Comment: The final publication is available at http://www.springerlink.co

A mathematical model of mitochondrial swelling

<p>Abstract</p> <p>Background</p> <p>The <it>permeabilization </it>of mitochondrial membranes is a decisive event in apoptosis or necrosis culminating in cell death. One fundamental mechanism by which such permeabilization events occur is the calcium-induced mitochondrial permeability transition. Upon Ca²⁺-uptake into mitochondria an increase in inner membrane permeability occurs by a yet unclear mechanism. This leads to a net water influx in the mitochondrial matrix, mitochondrial swelling, and finally the rupture of the outer membrane. Although already described more than thirty years ago, many unsolved questions surround this important biological phenomenon. Importantly, theoretical modeling of the mitochondrial permeability transition has only started recently and the existing mathematical models fail to characterize the swelling process throughout the whole time range.</p> <p>Results</p> <p>We propose here a new mathematical approach to the mitochondrial permeability transition introducing a specific delay equation and resulting in an optimized representation of mitochondrial swelling. Our new model is in accordance with the experimentally determined course of volume increase throughout the whole swelling process, including its initial lag phase as well as its termination. From this new model biological consequences can be deduced, such as the confirmation of a positive feedback of mitochondrial swelling which linearly depends on the Ca²⁺-concentration, or a negative exponential dependence of the average swelling time on the Ca²⁺-concentration. Finally, our model can show an initial shrinking phase of mitochondria, which is often observed experimentally before the actual swelling starts.</p> <p>Conclusions</p> <p>We present a model of the mitochondrial swelling kinetics. This model may be adapted and extended to diverse other inducing/inhibiting conditions or to mitochondria from other biological sources and thus may benefit a better understanding of the mitochondrial permeability transition.</p

Scattered data approximation on the bisphere and application to texture analysis.

The paper deals with the approximation and optimal interpolation of functions defined on the bisphere from scattered data. We demonstrate how the least square approximation to the function can be computed in a stable and efficient manner. The analysis of this problem is based on Marcinkiewicz-Zygmund inequalities for scattered data which we present here for the bisphere. The complementary problem of optimal interpolation is also solved by using well-localized kernels for our setting. Finally, we discuss the application of the developed methods to problems of texture analysis in material science