On the Mellin Transforms of Dirac’S Delta Function, The Hausdorff Dimension Function, and The Theorem by Mellin

Mathematics Subject Classification: 44A05, 46F12, 28A78We prove that Dirac’s (symmetrical) delta function and the Hausdorff dimension function build up a pair of reciprocal functions. Our reasoning is based on the theorem by Mellin. Applications of the reciprocity relation demonstrate the merit of this approach

Collocation Method to Solve Elliptic Equations, Bivariate Poly-Sinc Approximation

The paper proposes a collocation method to solve bivariate elliptic partial differential equations. The method uses Lagrange approximation based on Sinc point collocations. The proposed approximation is collocating on non-equidistant interpolation points generated by conformal maps, called Sinc points. We prove the upper bound of the error for the bivariate Lagrange approximation at these Sinc points. Then we define a collocation algorithm using this approximation to solve elliptic PDEs. We verify the Poly-Sinc technique for different elliptic equations and compare the approximate solutions with exact solutions

Fluorescence Molecule Counting for Single-Molecule Studies in Crowded Environment of Living Cells without and with Broken Ergodicity

We present a new approach to distinguish between non-ergodic and ergodic behavior. Performing ensemble averaging in a subpopulation of individual molecules leads to a mean value that can be similar to the mean value obtained in an ergodic system. The averaging is carried out by minimizing the variation between the sum of the temporal averaged mean square deviation of the simulated data with respect to the logarithmic scaling behavior of the subpopulation. For this reason, we first introduce a kind of Continuous Time Random Walks (CTRW), which we call Limited Continuous Time Random Walks (LCTRW) on fractal support. The random waiting time distributions are sampled at points which fulfill the condition N < 1, where N is the Poisson probability of finding a single molecule in the femtoliter-sized observation volume ΔV at the single-molecule level. Given a subpopulation of different single molecules of the same kind, the ratio T/ Tm between the measurement time T and the meaningful time Tm, which is the time for observing just one and the same single molecule, is the experimentally accessible quantity that allows to compare different molecule numbers in the subpopulation. In addition, the mean square displacement traveled by the molecule during the time t is determined by an upper limit of the geometric dimension of the living cell or its nucleus

If motion sounds: Movement sonification based on inertial sensor data

Within last years, movement sonification turned out to be an appropriate support for motor perception and motor control that can display physical motion in a very rich and direct way. But how should movement sonification be configured to support motor learning? The appropriate selection of movement parameters and their transformation into characteristic motion features is essential for an auditory display to become effective. In this paper, we introduce a real-time sonification framework for all common MIDI environments based on acceleration and orientation data from inertial sensors. Fundamental processing steps to transform motion information into meaningful sound will be discussed. The proposed framework of inertial motion capturing, kinematic parameter selection and possible kinematic acoustic mapping provides a basis for mobile real-time movement sonification which is a prospective powerful training tool for rehabilitation and sports and offers a broad variety of application possibilities.EU/EFRE/W2-8011866

SoundScript - Supporting the acquisition of character writing by multisensory integration

This work is introducing a new movement sonification method called 'SoundScript' to support the acquisition of character writing by children. SoundScript creates 'sound traces' from the writing trace in real-time during the process of handwriting. The structural correlation of both - optic and acoustic - traces leads to an integrated audio-visual perception of writing with the expected stimulation of multisensory integration sites of the CNS. Data of a pilot study are introduced indicating that the writing kinematics is reproduced more adequately if additional sound traces are available during writing. In the future SoundScript shall be applied to verify if the establishment of internal character representations can be accelerated, if the conciseness of the specific shape of the particular characters can be made stronger and if thereby the efficiency of the handwriting learning process can be enhanced

A proposal for a psychopharmacology-pharmacotherapy catalogue of learning objectives and a curriculum in Europe.

Objectives Post-graduate training for specialisation in psychiatry and psychotherapy is part of a 4-6-year programme. This paper aims to inform on the general situation of teaching and training of psychopharmacology-psychopharmacotherapy in Europe. It presents the need for a psychopharmacotherapy education in psychiatric training programmes. Arguments as well as a proposal for a catalogue of learning objectives and an outline of a psychopharmacology curriculum are presented. Methods Based on their experience and on an analysis of the literature, the authors, experts in psychopharmacology-pharmacotherapy teaching, critically analyse the present situation and propose the development of a curriculum at the European level. Results Teaching programmes vary widely between European countries and, generally, teaching of psychopharmacology and pharmacotherapy does not exceed two-dozen hours. This is insufficient if one considers the central importance of psychopharmacology. A psychopharmacology-psychopharmacotherapy curriculum for the professional training of specialists in psychiatry and psychotherapy is proposed. Conclusions As the number of hours of theoretical teaching and practical training is insufficient, a catalogue of learning objectives should be established, which would then be part of a comprehensive curriculum at the European level. It could be inspired partly by those few previously proposed by other groups of authors and organisations