Amended Formula for the Decay of Radioactive Material for Cosmic Times

An amended formula for the decay of radioactivematerial is presented. It is a modification of thestandard exponential formula. The new formula appliesfor long cosmic times comparable to the Hubble time. It reduces to the standard formula for shorttimes. It is shown that the material decays faster thanexpected. The application of the new formula to directmeasurements of the age of the universe and its implications are briefly discussed.     

Propylene polymerization by C1‐symmetric {ONNO′}‐type salan zirconium complexes

The activities of C₁‐symmetric dibenzyl zirconium complexes of Salan ligands that bear a halo‐substituted phenolate ring and an alkyl‐substituted phenolate ring in propylene polymerization with methylaluminoxane as cocatalyst were studied. These {ONNO′}ZrBn₂‐type catalysts exhibited moderate‐to‐high activities and yielded polypropylene of low molecular weight. The degree of tacticity was found to depend on the steric bulk of the substituents on both phenolate rings and ranged from practically atactic to substantially isotactic (74–78% [mmmm] for polymerizations at room temperature by Lig⁵ZrBn₂). Hemi‐isotactic polypropylene was not obtained, despite the diastereotopicity of the two positions. The pattern of stereo errors was consistent with the enantiomorphic site control of propylene insertion typically observed for C₂‐symmetric catalysts and implied a facile site‐averaging mechanism. A regular 1,2‐insertion and a β‐H transfer to an incoming monomer correspond to the main propagation and termination processes, respectively. © 2012 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2013     

Universal properties of stable states of a free energy model with small parameters

We consider the problem of minimizing functionals of the form , for small , subject to the constraint . (Here is typically a double well potential.) We study structural properties of minimizers that are shared by all minimizers of the problem, for sufficiently small values of . It is shown that the average ‘mass’ and ‘energy’ of minimizers, over intervals of length , is almost uniformly distributed (depending on ), for all sufficiently small . Here is a constant depending on the degree of ‘uniformity’, but independent of . Received May 5, 1996 / Accepted October 28, 1996     

On Sums of Units

It is shown that if R is a finitely generated integral domain of zero characteristic, then for every n there exist elements of R which are not sums of at most n units. This applies in particular to rings of integers in finite extensions of the rationals. On the other hand there are many infinite algebraic extensions of the rationals in which every integer is a sum of two units.     

On queueing delays of dispersed messages

We study the message queueing delays in a node of a communication system, where a message consists of a block of consecutive packets. The message delay is defined as the time elapsing between the arrival epoch of the first packet of the message to the system until after the transmission of the last packet of that message is completed. We distinguish between two types of message generation processes. The message can be generated as abatch or it can bedispersed over time. In this paper we focus on the dispersed generation model. The main difficulty in the analysis is due to the correlation between the system states observed by different packets of the same message. This paper introduces a new technique to analyze the message delay in such systems for different arrival models and different number of sessions. For anM/M/1 system with variable size messages and for the bursty traffic model, we obtain an explicit expression for the Laplace-Stieltjes transform (LST) of the message delay. Derivations are also provided for anM/G/1 system, for multiple session systems and for fixed message sizes. We show that the correlation has a strong effect on the performance of the system, and that the commonly usedindependence assumption, i.e., the assumption that the delays of packets are independent from packet to packet, can lead to wrong conclusions.     

Characterization of Powder Beds by Thermal Conductivity: Effect of Gas Pressure on the Thermal Resistance of Particle Contact Points

The thermal conductivity of ceramic powder packed beds was measured at temperatures below 100 °C for various powder sizes and compositions and under different gas atmospheres. Measurements at low pressures (down to 10 Pa) combined with a theoretical model allowed the elucidation of geometrical and thermal resistance parameters for the contact points between granules. The gap thickness and contact point size were found to be well correlated with the mean particle size. The thermal conductivities of all powders at low pressure were found to differ at most by a factor of two, whereas the solid‐phase conductivities of the powder materials differed by more than one order of magnitude. A theoretical model accounting for the size‐dependence of contact point conductivity is incorporated to rationalize this trend.     

A continuous–discontinuous second‐order transition in the satisfiability of random Horn‐SAT formulas

We compute the probability of satisfiability of a class of random Horn‐SAT formulae, motivated by a connection with the nonemptiness problem of finite tree automata. In particular, when the maximum clause length is three, this model displays a curve in its parameter space, along which the probability of satisfiability is discontinuous, ending in a second‐order phase transition where it is continuous but its derivative diverges. This is the first case in which a phase transition of this type has been rigorously established for a random constraint satisfaction problem. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007