Time reversal for obstacle location in elastodynamics from acoustic recording

The Note is concerned with a feasibility study of time reversal in a non-homogeneous elastic medium, from data recorded in an acoustic medium. Our aim here is to determine the presence and some physical properties of elastic “inclusions” (unknown, not observable solid objects, characterized by their elastic properties) from partial observations of acoustic waves scattered by these inclusions. A finite element numerical method, based on a variational acousto-elastodynamics formulation, is derived and used to solve the forward, and then, the time-reversed problem. A criterion, derived from the reverse time migration framework, is introduced, to help construct images of the inclusions to be determined. Numerical illustrations on configurations that mimic the breast cancer configuration are proposed, and show that one can differentiate between two inclusions, even with different properties.     

Various solution methods,accompanied by dynamic investigation,for the same problem as a means for enriching the mathematical toolbox

A geometrical task is presented with multiple solutions using different methods, in order to show the connection between various branches of mathematics and to highlight the importance of providing the students with an extensive ‘mathematical toolbox’. Investigation of the property that appears in the task was carried out using a computerized tool.     

Mechanistic Insight into the Stereochemical Control of Lactide Polymerization by Salan–Aluminum Catalysts

Alkyl aluminum complexes of chiral salan ligands assembled around the 2,2′‐bipyrrolidine core form as single diastereomers that have identical configurations of the N donors. Active catalysts for the polymerization of lactide were formed upon the addition of benzyl alcohol. Polymeryl exchange between enantiomorphous aluminum species had a dramatic effect on the tacticity of the poly(lactic acid) (PLA) in the polymerization of racemic lactide (rac‐LA): The enantiomerically pure catalyst of the nonsubstituted salan ligand led to isotactic PLA, and the racemic catalyst exhibited lower stereocontrol. The enantiomerically pure catalyst of the chloro‐substituted salan ligand led to PLA with a slight tendency toward heterotacticity, whereas the racemic catalyst led to PLA of almost perfect heterotacticity following an insertion/auto‐inhibition/exchange mechanism.     

Dynamic investigation of triangles inscribed in a circle,which tend to an equilateral triangle

We present a geometrical investigation of the process of creating an infinite sequence of triangles inscribed in a circle, whose areas, perimeters and lengths of radii of the inscribed circles tend to a limit in a monotonous manner.

First, using geometrical software, we investigate four theorems that represent interesting geometrical properties, after which we present formal proofs that rest on a combination between different fields of mathematics: trigonometry, algebra and geometry, and the use of the concept of standard deviation that is taken from statistics.     

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     

Stable Subnorms II

In this paper we continue our study of stability properties of subnorms on subsets of finite-dimensional, power-associative algebras over the real or the complex numbers.     

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.