Dear Editor,Space flight(SF)is substantially increasing at present.The emergence of commercial suborbital SF,such as the Virgin Galactic with VSS Unity and VMS Eve spacecraft,is extending to civilians,being previously...Dear Editor,Space flight(SF)is substantially increasing at present.The emergence of commercial suborbital SF,such as the Virgin Galactic with VSS Unity and VMS Eve spacecraft,is extending to civilians,being previously confined to military and/or professional astronauts only.This new evidence offers additional opportunities for better characterizing the impact that the transition from Earth’s 1G to microgravity in space could have on the astronauts’health while comparing well-trained subjects such as the latt er to space newcomers[1].展开更多
A group action on a set is a process of developing an algebraic structure through a relation defined by the permutations in the group and the elements of the set. The process suppresses most of the group properties, e...A group action on a set is a process of developing an algebraic structure through a relation defined by the permutations in the group and the elements of the set. The process suppresses most of the group properties, emphasizing the permutation aspect, so that the algebraic structure has a wider application among other algebras. Such structures not only reveal connections between different areas in Mathematics but also make use of results in one area to suggest conjectures and also prove results in a related area. The structure (G, X) is a transitive permutation group G acting on the set X. Investigations on the properties associated with various groups acting on various sets have formed a subject of recent study. A lot of investigations have been done on the action of the symmetric group Sn on various sets, with regard to rank, suborbits and subdegrees. However, the action of the dihedral group has not been thoroughly worked on. This study aims at investigating the properties of suborbits of the dihedral group Dn acting on ordered subsets of ?X={1,2,...,N}. The action of Dn on X[r], the set of all ordered r-element subsets of X, has been shown to be transitive if and only if n = 3. The number of self-paired suborbits of Dn acting on X[r] has been determined, amongst other properties. Some of the results have been used to determine graphical properties of associated suborbital graphs, which also reflect some group theoretic properties. It has also been proved that when G = Dn acts on ordered adjacent vertices of G, the number of self-paired suborbits is n + 1 if n is odd and n + 2 if n is even. The study has also revealed a conjecture that gives a formula for computing the self-paired suborbits of the action of Dn on its ordered adjacent vertices. Pro-perties of suborbits are significant as they form a link between group theory and graph theory.展开更多
基金supported by the Hyperbaric Med School of the Department of Biomedical Sciences at the University of Padova,the Italian Air Force,and the Institute of Clinical Physiology(Milan)-National Research Council(IFC-CNR).
文摘Dear Editor,Space flight(SF)is substantially increasing at present.The emergence of commercial suborbital SF,such as the Virgin Galactic with VSS Unity and VMS Eve spacecraft,is extending to civilians,being previously confined to military and/or professional astronauts only.This new evidence offers additional opportunities for better characterizing the impact that the transition from Earth’s 1G to microgravity in space could have on the astronauts’health while comparing well-trained subjects such as the latt er to space newcomers[1].
文摘A group action on a set is a process of developing an algebraic structure through a relation defined by the permutations in the group and the elements of the set. The process suppresses most of the group properties, emphasizing the permutation aspect, so that the algebraic structure has a wider application among other algebras. Such structures not only reveal connections between different areas in Mathematics but also make use of results in one area to suggest conjectures and also prove results in a related area. The structure (G, X) is a transitive permutation group G acting on the set X. Investigations on the properties associated with various groups acting on various sets have formed a subject of recent study. A lot of investigations have been done on the action of the symmetric group Sn on various sets, with regard to rank, suborbits and subdegrees. However, the action of the dihedral group has not been thoroughly worked on. This study aims at investigating the properties of suborbits of the dihedral group Dn acting on ordered subsets of ?X={1,2,...,N}. The action of Dn on X[r], the set of all ordered r-element subsets of X, has been shown to be transitive if and only if n = 3. The number of self-paired suborbits of Dn acting on X[r] has been determined, amongst other properties. Some of the results have been used to determine graphical properties of associated suborbital graphs, which also reflect some group theoretic properties. It has also been proved that when G = Dn acts on ordered adjacent vertices of G, the number of self-paired suborbits is n + 1 if n is odd and n + 2 if n is even. The study has also revealed a conjecture that gives a formula for computing the self-paired suborbits of the action of Dn on its ordered adjacent vertices. Pro-perties of suborbits are significant as they form a link between group theory and graph theory.
