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Soliton automata are a mathematical model for molecular switching devices. For even the simple case of soliton automata based on trees the behaviour is not known. For example, only two examples of such soliton automata were known the transition monoid of which is not the symmetric group on the set of states; and in these cases the transition monoid is the corresponding alternating group.

We establish new bounds on the number of states of a tree-based soliton automaton and a sufficient condition for when the transition monoid of such a soliton automaton consists only of even permutations of the set of states. We also summarize the results of a systematic enumeration of tree-based soliton automata by which additional exceptions were found, each an alternating group in its natural representation.

While formal rules of group representation in China and most other countries address underrepresented groups concerning gender, race, and ethnicity, China’s practice extends to encompassing occupation and profession. The speciality highlights an unstudied question in terms of the widespread profession-oriented group representation (POGR) concept among the general public. We apply a mixed approach based on our own dataset containing the text from 45 professional internet forums and reveal that the public’s POGR concept arises from two mechanisms. The first, known as the disadvantaged groups with shared demand, leads individuals to perceive their own profession as a disadvantaged group with shared interests, prompting the call for professional representation in the legislature to champion these interests. The second mechanism, referred to as misperceptions about representation, involves the public’s misperceptions regarding the electoral system, the model of representation, and the deputies’ professional qualifications required to hold such a position. The findings contribute to a deeper understanding of group representation concept and practice in China.

In this paper, we discuss the McKay correspondence as a relation between geometry of a resolution of quotient singularity and algebra of the group which acts. To see an example of the generalized McKay correspondence, we consider concrete constructions of crepant resolutions for any finite subgroup of SL(3, ℂ). Moreover, we draw an atlas of the generalized McKay correspondence to look around inside this world.

This article proposes a unified method to estimation of group action by using Fourier analysis.

We developed a group representation theory and calculation method based on quantum representation theory. All the calculations, such as Character, Irreducible bases and Matrices, Projection operators, Clebsch-Gordan series and coefficients are reduced to solve a set of eigen-equations of a complete set of commuting operators consisting of the class operators of the group. This approach had been successfully applied to all of the finite groups and compact Lie groups and various useful coefficients for physical application of group representation theory had been calculated and tabulated.