37 research outputs found
Computing the Rank Profile Matrix
The row (resp. column) rank profile of a matrix describes the staircase shape
of its row (resp. column) echelon form. In an ISSAC'13 paper, we proposed a
recursive Gaussian elimination that can compute simultaneously the row and
column rank profiles of a matrix as well as those of all of its leading
sub-matrices, in the same time as state of the art Gaussian elimination
algorithms. Here we first study the conditions making a Gaus-sian elimination
algorithm reveal this information. Therefore, we propose the definition of a
new matrix invariant, the rank profile matrix, summarizing all information on
the row and column rank profiles of all the leading sub-matrices. We also
explore the conditions for a Gaussian elimination algorithm to compute all or
part of this invariant, through the corresponding PLUQ decomposition. As a
consequence, we show that the classical iterative CUP decomposition algorithm
can actually be adapted to compute the rank profile matrix. Used, in a Crout
variant, as a base-case to our ISSAC'13 implementation, it delivers a
significant improvement in efficiency. Second, the row (resp. column) echelon
form of a matrix are usually computed via different dedicated triangular
decompositions. We show here that, from some PLUQ decompositions, it is
possible to recover the row and column echelon forms of a matrix and of any of
its leading sub-matrices thanks to an elementary post-processing algorithm
Solid 4He and the Supersolid Phase: from Theoretical Speculation to the Discovery of a New State of Matter? A Review of the Past and Present Status of Research
The possibility of a supersolid state of matter, i.e., a crystalline solid
exhibiting superfluid properties, first appeared in theoretical studies about
forty years ago. After a long period of little interest due to the lack of
experimental evidence, it has attracted strong experimental and theoretical
attention in the last few years since Kim and Chan (Penn State, USA) reported
evidence for nonclassical rotational inertia effects, a typical signature of
superfluidity, in samples of solid 4He. Since this "first observation", other
experimental groups have observed such effects in the response to the rotation
of samples of crystalline helium, and it has become clear that the response of
the solid is extremely sensitive to growth conditions, annealing processes, and
3He impurities. A peak in the specific heat in the same range of temperatures
has been reported as well as anomalies in the elastic behaviour of solid 4He
with a strong resemblance to the phenomena revealed by torsional oscillator
experiments. Very recently, the observation of unusual mass transport in hcp
solid 4He has also been reported, suggesting superflow. From the theoretical
point of view, powerful simulation methods have been used to study solid 4He,
but the interpretation of the data is still rather difficult; dealing with the
question of supersolidity means that one has to face not only the problem of
the coexistence of quantum coherence phenomena and crystalline order, exploring
the realm of spontaneous symmetry breaking and quantum field theory, but also
the problem of the role of disorder, i.e., how defects, such as vacancies,
impurities, dislocations, and grain boundaries, participate in the phase
transition mechanism.Comment: Published on J. Phys. Soc. Jpn., Vol.77, No.11, p.11101