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.     

Linear covering codes and error-correcting codes for limited-magnitude errors

The concepts of a linear covering code and a covering set for the limited-magnitude-error channel are introduced. A number of covering-set constructions, as well as some bounds, are given. In particular, optimal constructions are given for some cases involving small-magnitude errors. A problem of Stein is partially solved for these cases. Optimal packing sets and the corresponding error-correcting codes are also considered for some small-magnitude errors.     

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.     

Block–Stereoblock Copolymers of Poly(ϵ‐Caprolactone) and Poly(Lactic Acid)

A magnesium complex of the type {ONNN}Mg‐HMDS wherein {ONNN} is a sequential tetradentate monoanionic ligand is introduced. In the presence of an alcohol initiator this complex catalyzes the living and immortal homopolymerization of the lactide enantiomers and ?‐caprolactone at room‐temperature with exceptionally high activities, as well as the precise block copolymerization of these monomers in a one‐pot synthesis by sequential monomer addition. Copolymers of unprecedented microstructures such as the PCL‐b‐PLLA‐b‐PDLA and PDLA‐b‐PLLA‐b‐PCL‐b‐PLLA‐b‐PDLA block–stereoblock microstructures that feature unique thermal properties are readily accessed.     

On the best constant for Hardy's inequality in

Let be a domain in and . We consider the (generalized) Hardy inequality , where . The inequality is valid for a large family of domains, including all bounded domains with Lipschitz boundary. We here explore the connection between the value of the Hardy constant and the existence of a minimizer for this Rayleigh quotient. It is shown that for all smooth -dimensional domains, , where is the one-dimensional Hardy constant. Moreover it is shown that for all those domains not possessing a minimizer for the above Rayleigh quotient. Finally, for , it is proved that if and only if the Rayleigh quotient possesses a minimizer. Examples show that strict inequality may occur even for bounded smooth domains, but for convex domains.

     

Repulsive interaction can be a key design element of molecular rotary motors

Low-barrier molecular rotary motors having rotaxane architecture can be constructed using a cucurbituril host and a polyyne guest serving as stator and rotator, respectively. The repulsive interaction between these components is supported by molecular mechanics calculations with model systems and experimentally verified by X-ray crystallography with several synthetic host-guest complexes, all suggesting that the diyne rod floats at the center of the macrocyclic host with no apparent van der Waals contacts between them. Further support for these interactions is suggested by microcalorimetry measurements.