Vector-valued Lg-splines II. Smoothing splines

In the first paper of this series, Lg-spline theory was extended to the vector-valued interpolating case. Here this work is complemented by giving the extension for smoothing splines. The problem is formulated as a constrained minimum norm problem in a reproducing kernel Hilbert space, and solved recursively using a congruent stochastic estimation model.     

Halbruppen ohne Frattinische Unterhalbgruppe

Ohne ZusammenfassungVorgelegt vonL. Rédei     

A stochastic framework for recursive computation of spline functions: Part II,smoothing splines

Many of the optimal curve-fitting problems arising in approximation theory have the same structure as certain estimation problems involving random processes. We develop this structural correspondence for the problem of smoothing inaccurate data with splines and show that the smoothing spline is a sample function of a certain linear least-squares estimate. Estimation techniques are then used to derive a recursive algorithm for spline smoothing.     

Differentiable Distance Spaces

The distance function \({\varrho(p, q) ({\rm or} d(p, q))}\) of a distance space (general metric space) is not differentiable in general. We investigate such distance spaces over \({\mathbb{R}^n}\), whose distance functions are differentiable like in case of Finsler spaces. These spaces have several good properties, yet they are not Finsler spaces (which are special distance spaces). They are situated between general metric spaces (distance spaces) and Finsler spaces. We will investigate such curves of differentiable distance spaces, which possess the same properties as geodesics do in Finsler spaces. So these curves can be considered as forerunners of Finsler geodesics. They are in greater plenitude than Finsler geodesics, but they become geodesics in a Finsler space. We show some properties of these curves, as well as some relations between differentiable distance spaces and Finsler spaces. We arrive to these curves and to our results by using distance spheres, and using no variational calculus. We often apply direct geometric considerations.     

Bizarre alterations of the morphology of pineal synaptic bodies under constant light and an evaluation of suitable 3D-reconstruction software

Three dimensional (3D) reconstruction and modelling software was evaluated to find a procedure suitable for visualization of small subcellular structures in transmission electron microscope images. The method applied in this study demonstrates bizarre alterations of the structure of synaptic bodies (SBs) in pinealocytes of the guinea-pig pineal gland caused by constant illumination. It can, in general, be used for any 3D reconstruction from serial sections.

Pineal glands of five guinea-pigs (two kept under a LD cycle of 12:12 h; three kept in constant light, for 4 months) were investigated. SBs consist of an electron-dense centre with attached vesicles. Under normal lighting conditions most SBs are flat plates (about 35 nm in thickness), which eventually may be bent. The proteins comprising the molecular basis of SBs, mainly RIBEYE A and B are polymerised in a regular manner in these plates. This is not the case in other SBs, which appear as spheres or irregular lumps. SBs lie in groups in which usually some of the plates are arranged in parallel arrays

Constant illumination caused different changes in morphology: many of the SBs lie in ‘paired fields’, i.e. appear in groups attached to the cell membranes of two pinealocytes directly opposite to each other. Some of the SBs in such groups are strongly bent, showing blebs and irregular thickened areas, others seem to aggregate and show inclusions of cytoplasm. Further goblet-like, shield-like and other bizarre forms of SBs occurred and the relative number of spheroid and lump-like SBs increased. Protrusions on larger SBs suggest detachment or fusion of SB material to a greater extent than in the control animals. There is a reduction of areas in which the polymerisation of the SB proteins remains well ordered, i.e. where the typical thickness of 35 nm is maintained. It remains unclear why this polymerisation pattern is only partly affected by constant light.