On Houseswapping, the Strict Core, Segmentation, and Linear Programming

We consider the n-player houseswapping game of Shapley-Scarf (1974), with indifferences in preferences allowed. It is well-known that the strict core of such a game may be empty, single-valued, or multivalued. We define a condition on such games called “segmentability”, which means that the set of players can be partitioned into a “top trading segmentation.” It generalizes Gale’s well-known idea of the partition of players into “top trading cycles” (which is used to find the unique strict core allocation in the model with no indifference). We prove that a game has a nonempty strict core if and only if it is segmentable. We then use this result to devise an O(n 3 ) algorithm which takes as input any houseswapping game, and returns either a strict core allocation or a report that the strict core is empty. Finally, we are also able to construct a linear inequality system whose feasible region’s extreme points precisely correspond to the allocations of the strict core. This last result parallels the results of Vande Vate (1989) and Rothblum (1991) for the marriage game of Gale and Shapley (1962)

Human metaphase chromosome consists of randomly arranged chromatin fibres with up to 30-nm diameter

Wako, T., Yoshida, A., Kato, J. et al. Human metaphase chromosome consists of randomly arranged chromatin fibres with up to 30-nm diameter. Sci Rep 10, 8948 (2020). https://doi.org/10.1038/s41598-020-65842-z

Cerium features in kilonova near-infrared spectra: implication from a chemically peculiar star

Observations of the kilonova from a neutron star merger event GW170817 opened a way to directly study r-process nucleosynthesis by neutron star mergers. It is, however, challenging to identify the individual elements in the kilonova spectra due to lack of complete atomic data, in particular, at near-infrared wavelengths. In this paper, we demonstrate that spectra of chemically peculiar stars with enhanced heavy element abundances can provide us with an excellent astrophysical laboratory for kilonova spectra. We show that the photosphere of a late B-type chemically peculiar star HR 465 has similar lanthanide abundances and ionization degrees with those in the line forming region in a kilonova at $\sim 2.5$ days after the merger. The near-infrared spectrum of HR 465 taken with Subaru/IRD indicates that Ce III lines give the strongest absorption features around 16,000 A and there are no other comparably strong transitions around these lines. The Ce III lines nicely match with the broad absorption features at 14,500 A observed in GW170817 with a blueshift of v=0.1c, which supports recent identification of this feature as Ce III by Domoto et al. (2022).Comment: 12 pages, 6 figures, accepted for publication in Ap

On Houseswapping, the Strict Core, Segmentation, and Linear Programming

We consider the n-player houseswapping game of Shapley-Scarf (1974), with indfferences in preferences allowed. It is well-known that the strict core of such a game may be empty, single-valued, or multi-valued. We define a condition on such games called "segmentability", which means that the set of players can be partitioned into a "top trading segmentation". It generalizes Gale's well-known idea of the partition of players into "top trading cycles" (which is used to find the unique strict core allocation in the model with no indifference). We prove that a game has a nonempty strict core if and only if it is segmentable. We then use this result to devise and O(n^3) algorithm which takes as input any houseswapping game, and returns either a strict core allocation or a report that the strict core is empty. Finally, we are also able to construct a linear inequality system whose feasible region's extreme points precisely correspond to the allocations of the strict core. This last result parallels the results of Vande Vate (1989) and Rothbum (1991) for the marriage game of Gale and Shapley (1962).Shapley-Scarf Economy, Strict Core, Linear Inequality System, Extreme Points