Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials

Nonsymmetric Askey-Wilson polynomials are usually written as Laurent polynomials. We write them equivalently as 2-vector-valued symmetric Laurent polynomials. Then the Dunkl-Cherednik operator of which they are eigenfunctions, is represented as a 2x2 matrix-valued operator. As a new result made possible by this approach we obtain positive definiteness of the inner product in the orthogonality relations, under certain constraints on the parameters. A limit transition to nonsymmetric little q-Jacobi polynomials also becomes possible in this way. Nonsymmetric Jacobi polynomials are considered as limits both of the Askey-Wilson and of the little q-Jacobi case.Comment: 16 pages. Dedicated to Paul Butzer on the occasion of his 80th birthday. v4: minor correction in (4.14

The Bernoulli sieve revisited

We consider an occupancy scheme in which "balls" are identified with $n$ points sampled from the standard exponential distribution, while the role of "boxes" is played by the spacings induced by an independent random walk with positive and nonlattice steps. We discuss the asymptotic behavior of five quantities: the index $K_n^*$ of the last occupied box, the number $K_n$ of occupied boxes, the number $K_{n,0}$ of empty boxes whose index is at most $K_n^*$, the index $W_n$ of the first empty box and the number of balls $Z_n$ in the last occupied box. It is shown that the limiting distribution of properly scaled and centered $K_n^*$ coincides with that of the number of renewals not exceeding $\log n$. A similar result is shown for $K_n$ and $W_n$ under a side condition that prevents occurrence of very small boxes. The condition also ensures that $K_{n,0}$ converges in distribution. Limiting results for $Z_n$ are established under an assumption of regular variation.Comment: Published in at http://dx.doi.org/10.1214/08-AAP592 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Corticospinal output and loss of force during motor fatigue

The objective of this study was to analyze central motor output changes in relation to contraction force during motor fatigue. The triple stimulation technique (TST, Magistris et al. in Brain 121(Pt 3):437-450, 1998) was used to quantify a central conduction index (CCI=amplitude ratio of central conduction response and peripheral nerve response, obtained simultaneously by the TST). The CCI removes effects of peripheral fatigue from the quantification. It allows a quantification of the percentage of the entire target muscle motor unit pool driven to discharge by a transcranial magnetic stimulus. Subjects (n=23) performed repetitive maximal voluntary contractions (MVC) of abductor digiti minimi (duration 1s, frequency 0.5Hz) during 2min. TST recordings were obtained every 15s, using stimulation intensities sufficient to stimulate all cortical motor neurons (MNs) leading to the target muscle, and during voluntary contractions of 20% of the MVC to facilitate the responses. TST was also repetitively recorded during recovery. This basic exercise protocol was modified in a number of experiments to further characterize influences on CCI of motor fatigue (4min exercise at 50% MVC; delayed fatigue recovery during local hemostasis, "stimulated exercise” by 20Hz trains of 1s duration at 0.5Hz during 2min). In addition, the cortical silent period was measured during the basic exercise protocol. Force fatigued to approximately 40% of MVC in all experiments and in all subjects. In all subjects, CCI decreased during exercise, but this decrease varied markedly between subjects. On average, CCI reductions preceded force reductions during exercise, and CCI recovery preceded force recovery. Exercising at 50% for 4min reduced muscle force more markedly than CCI. Hemostasis induced by a cuff delayed muscle force recovery, but not CCI recovery. Stimulated exercise reduced force markedly, but CCI decreased only marginally. Summarized, force reduction and reduction of the CCI related poorly quantitatively and in time, and voluntary drive was particularly critical to reduce the CCI. The fatigue induced reduction of CCI may result from a central inhibitory phenomenon. Voluntary muscle activation is critical for the CCI reduction, suggesting a primarily supraspinal mechanis

A Limit Relation for Dunkl-Bessel Functions of Type A and B

We prove a limit relation for the Dunkl-Bessel function of type BN with multiplicity parameters k₁ on the roots ±ei and k₂ on ±ei±ej where k₁ tends to infinity and the arguments are suitably scaled. It gives a good approximation in terms of the Dunkl-type Bessel function of type An₋₁orresponding limit relation for Bessel functions on matrix cones

A clinical study of motor evoked potentials using a triple stimulation technique

Amplitudes of motor evoked potentials (MEPs) are usually much smaller than those of motor responses to maximal peripheral nerve stimulation, and show marked variation between normal subjects and from one stimulus to another. Consequently, amplitude measurements have low sensitivity to detect central motor conduction failures due to the broad range of normal values. Since these characteristics are mostly due to varying desynchronization of the descending action potentials, causing different degrees of phase cancellation, we applied the recently developed triple stimulation technique (TST) to study corticospinal conduction to 489 abductor digiti minimi muscles of 271 unselected patients referred for possible corticospinal dysfunction. The TST allows resynchronization of the MEP, and thereby a quantification of the proportion of motor units activated by the transcranial stimulus. TST results were compared with those of conventional MEPs. In 212 of 489 sides, abnormal TST responses suggested conduction failure of various degrees. By contrast, conventional MEPs detected conduction failures in only 77 of 489 sides. The TST was therefore 2.75 times more sensitive than conventional MEPs in disclosing corticospinal conduction failures. When the results of the TST and conventional MEPs were combined, 225 sides were abnormal: 145 sides showed central conduction failure, 13 sides central conduction slowing and 67 sides both conduction failure and slowing. It is concluded that the TST is a valuable addition to the study of MEPs, since it improves detection and gives quantitative information on central conduction failure, an abnormality which appears to be much more frequent than conduction slowing. This new technique will be useful in following the natural course and the benefit of treatments in disorders affecting central motor conductio

Repetitive spinal motor neuron discharges following single transcranial magnetic stimulation: relation to dexterity

Transcranial magnetic stimulation allows to study the properties of the human corticospinal tract non-invasively. After a single transcranial magnetic stimulus, spinal motor neurons (MNs) sometimes fire not just once, but repetitively. The biological significance of such repetitive MN discharges (repMNDs) is unknown. To study the relation of repMNDs to other measures of cortico-muscular excitability and to physiological measures of the skill for finely tuned precision movements, we used a previously described quadruple stimulation (QuadS) technique (Z'Graggen et al. 2005) to quantify the amount of repMNDs in abductor digiti minimi muscles (ADMs) on both sides of 20 right-handed healthy subjects. Skillfulness for finger precision movements of both hands was assessed using a finger tapping task. In 16 subjects, a follow-up examination was performed after training of either precision movements (n=8) or force (n=8) of the left ADM. The size of the QuadS response (amplitude and area ratios) was greater in the dominant right hand than in the left hand (QuadS amplitude ratio: 47.1±18.1 versus 37.7±22.0%, Wilcoxon test: P<0.05; QuadS area ratio: 49.7±16.2% versus 36.9±23.0%, Wilcoxon test: P<0.05), pointing to a greater amount of repMNDs. Moreover, the QuadS amplitude and area increased significantly after finger precision training, but not after force training. This increase of repMNDs correlated significantly with the increase in performance in the finger tapping task. Our results demonstrate that repMNDs are related to handedness and therefore probably reflect supraspinal excitability differences. The increase of repMNDs after skills training but not after force training supports the hypothesis of a supraspinal origin of repMND

Experimental and Numerical Study of the Dispersion and Transport of Automobile Exhaust Gases from Highways

This paper describes examples of modelling and of measurements of the dispersion and transport of exhaust gases from automobiles on a highway. Model runs were performed by a large-eddy-simulation model. The measurements were carried through by the DLR environmental research aircraft lee-side of the highway between München and Augsburg

The discretised harmonic oscillator: Mathieu functions and a new class of generalised Hermite polynomials

We present a general, asymptotical solution for the discretised harmonic oscillator. The corresponding Schr\"odinger equation is canonically conjugate to the Mathieu differential equation, the Schr\"odinger equation of the quantum pendulum. Thus, in addition to giving an explicit solution for the Hamiltonian of an isolated Josephon junction or a superconducting single-electron transistor (SSET), we obtain an asymptotical representation of Mathieu functions. We solve the discretised harmonic oscillator by transforming the infinite-dimensional matrix-eigenvalue problem into an infinite set of algebraic equations which are later shown to be satisfied by the obtained solution. The proposed ansatz defines a new class of generalised Hermite polynomials which are explicit functions of the coupling parameter and tend to ordinary Hermite polynomials in the limit of vanishing coupling constant. The polynomials become orthogonal as parts of the eigenvectors of a Hermitian matrix and, consequently, the exponential part of the solution can not be excluded. We have conjectured the general structure of the solution, both with respect to the quantum number and the order of the expansion. An explicit proof is given for the three leading orders of the asymptotical solution and we sketch a proof for the asymptotical convergence of eigenvectors with respect to norm. From a more practical point of view, we can estimate the required effort for improving the known solution and the accuracy of the eigenvectors. The applied method can be generalised in order to accommodate several variables.Comment: 18 pages, ReVTeX, the final version with rather general expression

Licófitas paleozóicas do Chile

Impressões caulinares de licófitas da Formação Arrayan, do Neopaleozóico da Cordilheira Chilena são aqui estudadas. Seu caráter fragmentário e sua má preservação não permitem uma identificação genérica segura. Entretanto foram reconhecidos 5 tipos básicos de cicatrizes os quais apresentam relações possíveis com Haplostigma Seward, Cyclostigma pacífica Jongmans, Bergeria Presl e mais distantemente Protolepidodendron e Sigillaria do tipo Eusigillaria. São feitas algumas reflexões em relação a uma possível idade D superior a C inferior