ON THE ZEROS OF POLYNOMIALS OVER QUATERNIONS

ABSTRACT

We review and expand some results on the number of zeros of polynomials over Hamilton's quaternions, with particular emphasis on those polynomials with coefficients in a degree two subfield.     

Starshapedness of level sets of solutions to elliptic PDEs

We investigate how some geometric properties of the domain are inherited by level sets of solutions of elliptic equations. In particular we prove that, under suitable assumptions, solutions of elliptic Dirichlet problems in starshaped rings have starshaped level sets. Our results are applicable to a large class of operators, including fully-nonlinear ones.     

Two criteria for locally noetherian Grothendieck categories

We give two criteria for a Grothendieck category to be locally noetherian. The first of these is in terms of direct sum decompositions into uniform components. The second is related to the existence of finitely injective objects that cannot be expressed as a direct sum of injectives.     

Aeroelastic stability analysis considering a continuous flight envelope

This paper presents a new methodology to analyze aeroelastic stability in a continuous range of flight envelope with varying parameter of velocity and altitude. The focus of the paper is to demonstrate that linear matrix inequalities can be used to evaluate the aeroelastic stability in a region of flight envelope instead of a single point, like classical methods. The proposed methodology can also be used to study if a system remains stable during an arbitrary motion from one point to another in the flight envelope, i.e., when the problem becomes time-variant. The main idea is to represent the system as a polytopic differential inclusion system using rational function approximation to write the model in time domain. The theory is outlined and simulations are carried out on the benchmark AGARD 445.6 wing to demonstrate the method. The classical pk-method is used for comparing results and validating the approach. It is shown that this method is efficient to identify stability regions in the flight envelope.     

New detected CO(1-0) emission from planetary nebulae

We report the results of CO(1-0) (J=l-0) observations for 20 selected planetary nebulae (PNe), using the 13.7 m radio telescope of Purple Mountain Observatory at the Qinghai Station. Due to weak CO(1-0) emission, long integrated time observations have been carried out for most of sampled PNe. Among these PNe, nine are first detections in CO(1-0) named NGC 6445, Ml-59, M4-9, M2-51, M4-18, He2-459, Sh2-71, K3-31, M2-52, and one is the first possible detection named V-Vl-8, a probably misclassified PN. NGC 6445, Ml-59, M4-9 and M2-51 have been detected in CO(2-1)(J=2-1). Although having been observed in previous CO surveys, Sh2-71 and M4-18 were not detected in CO; M2-52, K3-31, He2-459 are the first detected PNe in CO emission and V-Vl-8 is given with a first completed spectrum detected in CO(1-0). Comparing these data with other previous observational ones, we discuss the relationships between CO(1-0) emission and evolution of PNe: While the nebular radii increase, the CO(l-0) line integrated intensities multiplied by distances have a decreasing trend. Also the masses of molecular envelopes (from 0.001 to 1 M⊙) for PNe are decreased with increasing nebular radii. All the results mean the CO(1-0) emission decreases dramatically with the nebular evolution. The detectability of CO restricted by different PNe which have different dust properties and different evolutionary stages is also analyzed.