The zeros of random polynomials cluster uniformly near the unit circle

In this paper we deduce a universal result about the asymptotic distribution of roots of random polynomials, which can be seen as a complement to an old and famous result of Erdos and Turan. More precisely, given a sequence of random polynomials, we show that, under some very general conditions, the roots tend to cluster near the unit circle, and their angles are uniformly distributed. The method we use is deterministic: in particular, we do not assume independence or equidistribution of the coefficients of the polynomial.Comment: Corrects some typos and strengthens Theorem

The Co-ordination of Swimmeret Movements in the Crayfish, Procambarus Clarkii (Girard)

1. Electrical activity recorded in the first roots of the abdominal nerve cord show bursts of activity which accompany rhythmic movements of the swimmerets. These bursts persist when a root is cut distal to the electrodes. 2. Proprioceptive impulses have been recorded from the distal portion of these nerves during passive movements of the swimmerets. They have also been recorded in the connectives during both passive and active movements. 3. Cutting all second and third roots of the abdominal ganglia does not interfere with the rhythmic swimmeret movements of the isolated abdomen. Rhythmic efferent bursts persist in first roots even when only a single first root remains intact in such a preparation. 4. Intermittent bursts are also found in the first roots of a completely isolated cord but their pattern, frequency and phase relationships differ from those recorded in the same roots before isolation was completed. Such bursts are very rare in second and third roots. 5. Stimulation of small bundles of fibres in the circumoesophageal commissure produces well-defined inhibitory and excitatory effects on the discharge of single units in the first roots. 6. One specific bundle found in many preparations, when stimulated at 50/sec., caused a tonic retraction of all the swimmerets which then began to beat rhythmically and continued to do so when stimulation was discontinued. 7. Intermittent bursts in the first roots took place with correct phasing during this stimulation even when the abdominal ganglia were isolated except for their connexion with the last thoracic ganglion. Again, such bursts were absent in the second and third roots. 8. It is concluded that both the inflow from peripheral proprioceptors and intrinsic properties of the central ganglia play essential parts in the co-ordination of the metachronal movements of the swimmerets

Neuronal Pathways and Synaptic Connexions in the Abdominal Cord of the Crayfish

1. An investigation has been made into the function and distribution of nerve fibres in the abdominal ganglion chain and its roots in the crayfish, Procambarus clarkii, by leading off action potentials from small prepared bundles following sensory stimulation. 2 .The sensory fields belonging to the first and second roots of each abdominal ganglion were determined, and the antero-posterior pathway of sensory fibres within the cord noted. It was found that the primary sensory fibres of the dorsal muscle receptor organs, entering through the second root, send out an anterior branch to the brain and a posterior one to the last ganglion. For most other sensory fibres much shorter intracentral branches are indicated, though some of them extend for two ganglia in the anterior direction and for one posteriorly. All sensory fibres in the connectives run on the same side as they enter. 3. The segmental divisions of the external skeleton and of the nervous system do not coincide, the neural segment slants in a posterior dorsal direction with respect to the skeletal one. 4. For the majority of the interneurones which innervate more than two abdominal segments it has been proved that they synapse with primary sensory fibres in each of the ganglia that these enter. Depending on the segment stimulated with respect to the leading-off position, both ascending and descending impulses are obtained in such interneurones and collision of the impulses has been observed. Some consequences of this type of integration are discussed. 5. For interneurones responding to bilateral or heterolateral stimulation the course of the impulses proved to be of at least two types. In some, cutting the fibre prevents the arrival of impulses except those set up on the side of the cut from which the recording is made. In others, recording from either side of the cut fibre does not exclude any of the sensory fields to which the fibre normally responded. 6. At least one interneurone is present in which all primary sensory fibres from the different segments to whose activity it responds collect in one ganglion

Mock-Gaussian Behaviour for Linear Statistics of Classical Compact Groups

We consider the scaling limit of linear statistics for eigenphases of a matrix taken from one of the classical compact groups. We compute their moments and find that the first few moments are Gaussian, whereas the limiting distribution is not. The precise number of Gaussian moments depends upon the particular statistic considered

Pre-design study for a modern four-bladed rotor for the Rotor System Research Aircraft (RSRA)

Various candidate rotor systems were compared in an effort to select a modern four-bladed rotor for the RSRA. The YAH-64 rotor system was chosen as the candidate rotor system for further development for the RSRA. The process used to select the rotor system, studies conducted to mate the rotor with the RSRA and provide parametric variability, and the development plan which would be used to implement these studies are presented. Drawings are included