Minimal polynomials and radii of elements in finite-dimensional power-associative algebras

In the first section of this paper we revisit the definition and some of the properties of the minimal polynomial of an element of a finite-dimensional power-associative algebra over an arbitrary field . Our main observation is that , the minimal polynomial of , may depend not only on , but also on the underlying algebra. More precisely, if is a subalgebra of , and if is the minimal polynomial of in , then may differ from , in which case we have .

In the second section we restrict attention to the case where is either the real or the complex numbers, and define , the radius of an element in , to be the largest root in absolute value of the minimal polynomial of . We show that possesses some of the familiar properties of the classical spectral radius. In particular, we prove that is a continuous function on .

In the third and last section, we deal with stability of subnorms acting on subsets of finite-dimensional power-associative algebras. Following a brief survey, we enhance our understanding of the subject with the help of our findings of the previous section. Our main new result states that if , a subset of an algebra , satisfies certain assumptions, and is a continuous subnorm on , then is stable on if and only if majorizes the radius defined above.

     

On balking from an empty queue

The intuition while observing the economy of queueing systems, is that one’s motivation to join the system, decreases with its level of congestion. Here we present a queueing model where sometimes the opposite is the case. The point of departure is the standard first-come first-served single server queue with Poisson arrivals. Customers commence service immediately if upon their arrival the server is idle. Otherwise, they are informed if the queue is empty or not. Then, they have to decide whether to join or not. We assume that the customers are homogeneous and when they consider whether to join or not, they assess their queueing costs against their reward due to service completion. As the whereabouts of customers interact, we look for the (possibly mixed) join/do not join Nash equilibrium strategy, a strategy that if adopted by all, then under the resulting steady-state conditions, no one has any incentive not to follow it oneself. We show that when the queue is empty then depending on the service distribution, both ‘avoid the crowd’ (ATC) and ‘follow the crowd’ (FTC) scenarios (as well as none-of-the-above) are possible. When the queue is not empty, the situation is always that of ATC. Also, we show that under Nash equilibrium it is possible (depending on the service distribution) that the joining probability when the queue is empty is smaller than it is when the queue is not empty. This research was supported by The Israel Science Foundation Grant No. 237/02.     

Cosmological Relativity: A New Theory ofCosmology1

A new general-relativistic theory of cosmology, the dynamical variables of whichare those of Hubble's, namely distances and redshifts, is presented. The theorydescribes the universe as having a three-phase evolution with a deceleratingexpansion followed by a constant and an accelerating expansion, and it predictsthat the universe is now in the latter phase. The theory is actually a generalizationof Hubble's law taking gravity into account by means of Einstein's theory ofgeneral relativity. The equations obtained for the universe expansion are elegantand very simple. It is shown, assuming ₀ = 0.24, that the time at which theuniverse goes over from a decelerating to an accelerating expansion, i.e., theconstant expansion phase, occurs at 0.03 from the big bang, where is theHubble time in vacuum. Also, at that time the cosmic radiation temperature was11 K. Recent observations of distant supernovae imply, in defiance of expectations,that the universe's growth is accelerating, contrary to what has always beenassumed, that the expansion is slowing down due to gravity. Our theory confirmsthese recent experimental results by showing that the universe now is definitelyin a stage of accelerating expansion.     

Finite generation of powers of ideals

Suppose is a maximal ideal of a commutative integral domain and that some power of is finitely generated. We show that is finitely generated in each of the following cases: (i) is of height one, (ii) is integrally closed and , (iii) is a monoid domain over a field , where is a cancellative torsion-free monoid such that , and is the maximal ideal . We extend the above results to ideals of a reduced ring such that is Noetherian. We prove that a reduced ring is Noetherian if each prime ideal of has a power that is finitely generated. For each with , we establish existence of a -dimensional integral domain having a nonfinitely generated maximal ideal of height such that is -generated.

     

Preconditioning spectral element schemes for definite and indefinite problems

Spectral element schemes for the solution of elliptic boundary value problems are considered. Preconditioning methods based on finite difference and finite element schemes are implemented. Numerical experiments show that inverting the preconditioner by a single multigrid iteration is most efficient and that the finite difference preconditioner is superior to the finite element one for both definite and indefinite problems. A multigrid preconditioner is also derived from the finite difference preconditioner and is found suitable for the CGS acceleration method. It is pointed out that, for the finite difference and finite element preconditioners, CGS does not always converge to the accurate algebraic solution. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 535–543, 1999     

Reactions along the astrophysical s-process path and prospects for neutron radiotherapy with the Liquid-Lithium Target (LiLiT) at the Soreq Applied Research Accelerator Facility (SARAF)

The European Physical Journal A - An empirical formula is proposed for the two-proton decay half-lives. This study is an extension of the empirical formula reported recently by us for calculating...     

The numerical radius and specttural matrices

In this paper we investigate spectral matrices, i.e., matrices with equal spectral and numerical radii. Various characterizations and properties of these matrices are given.