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.

     

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...     

Cooperative Multicomponent Self‐Assembly of Nucleic Acid Structures for the Activation of DNAzyme Cascades: A Paradigm for DNA Sensors and Aptasensors

Self‐assembly line : In the presence of a DNA analyte or low‐molecular‐weight substrates, multicomponent nucleic acids self‐assemble into cooperatively stabilized functional nanostructures (see scheme) that activate DNAzyme cascades.

     

Coherent Activation of DNA Tweezers: A “SET–RESET” Logic System

Open sesame : Aptamer–substrate complexes activate the coherent operation of two tweezers that act as a “SET–RESET” logic system. Each tweezer cycles between a fluorescent open state and a closed quenched state (Q=quencher, F=fluorophore) when triggered by adenosine monophosphate (AMP) and adenosine deaminase (AD).

     

Availability of inspected systems subject to shocks – A matrix algorithmic approach

We examine the limiting average availability of a maintained system that deteriorates due to random shock process and as a response to its usage (wear out). System’s failures are not self-announcing, hence, failures must be detected via inspection. We consider randomly occurring shocks that arrive according to a Poisson process and cumulatively damage the system. Two models are considered: in Model 1 the shock and wear out processes are independent of the external environment and in Model 2, the shocks arrival rate, the shock magnitudes and the wear out rate are governed by a random environment which evolves as a Markov process. We obtain the system’s availability for both models.     

Predicting solute adsorption on activated carbon: phenol

Activated carbon (AC), the most widely used adsorbent in water and in wastewater treatment, comprises a high surface area of very small, convoluted and interconnected pores. Despite the wide use of AC, there is little fundamental atomic-level understanding of its adsorption capacity and selectivity as well as its pore structure. The purpose of this work is to suggest the methodology for calculation of equilibrium adsorption capacity of common water organic pollutants and use it for phenol as a model. The effects of various functional groups, pore size, and coverage on thermodynamics of phenol adsorption from the gas phase and from water media are calculated using molecular mechanics (MM) and density functional theory (DFT) approaches.     

Analysis and design of nonlinear fiber Bragg gratings and their application for optical compression of reflected pulses

We demonstrate a novel split-step solution for analyzing nonlinear fiber Bragg gratings. The solution is used for designing nonlinear fiber Bragg gratings with a low reflectivity. The structure of the grating is designed according to the profiles of the incident and reflected pulses. We demonstrate our method for nonlinear compression of a pulse reflected from a fiber Bragg grating. The method allows us to obtain compressed pulses with a very low wing intensity.     

Optimization of Fabry-Perot ring resonator embedding a grating based mirror

We study a triangular arrangement of two flat mirrors and a grating mirror for coupling a \(-1{\mathrm{st}}\) diffracted order light beam into a traveling-wave intra-cavity light recirculation and out-of-cavity output, combining the \(0\,{\mathrm{th}}\) and the repeatedly diffracted secondary \(+1{\mathrm{st}}\) order beams. The thus designed and then optimized grating Fabry-Perot ring cavity demonstrates high output contrast and detection resolution for wavelength, cavity length and incidence angle.