Non-Nudgable Subgroups of Permutations

Motivated by a problem from behavioral economics, we study subgroups of permutation groups that have a certain strong symmetry. Given a fixed permutation, consider the set of all permutations with disjoint inversion sets. The group is called non-nudgable, if the cardinality of this set always remains the same when replacing the initial permutation with its inverse. It is called nudgable otherwise. We show that all full permutation groups, standard dihedral groups, half of the alternating groups, and any abelian subgroup are non-nudgable. In the right probabilistic sense, it is thus quite likely that a randomly generated subgroup is non-nudgable. However, the other half of the alternating groups are nudgable. We also construct a smallest possible nudgable group, a 6-element subgroup of the permutation group on 4 elements.Comment: new version contains some simplifications and extension

Length-based cryptanalysis: The case of Thompson's Group

The length-based approach is a heuristic for solving randomly generated equations in groups which possess a reasonably behaved length function. We describe several improvements of the previously suggested length-based algorithms, that make them applicable to Thompson's group with significant success rates. In particular, this shows that the Shpilrain-Ushakov public key cryptosystem based on Thompson's group is insecure, and suggests that no practical public key cryptosystem based on this group can be secure.Comment: Final version, to appear in JM

Generic Subgroups of Group Amalgams

For many groups the structure of finitely generated subgroups is generically simple. That is with asymptotic density equal to one a randomly chosen finitely generated subgroup has a particular well-known and easily analyzed structure. For example a result of D.B.A.Epstein says that a finitely generated subgroup of GL(n;R) is generically a free group. We say that a group G has the generic free group property if any finitely generated subgroup is generically a free group. Further G has the strong generic free group property if given randomly chosen elements g_1, ...,g_n in G then generically they are a free basis for the free subgroup they generate. In this paper we show that for any arbitrary free product of finitely generated infinite groups satisfies the strong generic free group property. There are also extensions to more general amalgams - free products with amalgamation and HNN groups. These results have implications in cryptography. In particular several cryptosystems use random choices of subgroups as hard cryptographic problems. In groups with the generic free group property any such cryptosystem may be attackable by a length based attack

Estimating parameters of a multipartite loglinear graph model via the EM algorithm

We will amalgamate the Rash model (for rectangular binary tables) and the newly introduced $\alpha$-$\beta$ models (for random undirected graphs) in the framework of a semiparametric probabilistic graph model. Our purpose is to give a partition of the vertices of an observed graph so that the generated subgraphs and bipartite graphs obey these models, where their strongly connected parameters give multiscale evaluation of the vertices at the same time. In this way, a heterogeneous version of the stochastic block model is built via mixtures of loglinear models and the parameters are estimated with a special EM iteration. In the context of social networks, the clusters can be identified with social groups and the parameters with attitudes of people of one group towards people of the other, which attitudes depend on the cluster memberships. The algorithm is applied to randomly generated and real-word data

Discovery of new stellar groups in the Orion complex

We test the ability of two unsupervised machine learning algorithms, \textit{EnLink} and Shared Nearest Neighbour (SNN), to identify stellar groupings in the Orion star-forming complex as an application to the 5-dimensional astrometric data from \textit{Gaia} DR2. The algorithms represent two distinct approaches to limiting user bias when selecting parameter values and evaluating the relative weights among astrometric parameters. \textit{EnLink} adopts a locally adaptive distance metric and eliminates the need of parameter tuning through automation. The original SNN relies only on human input for parameter tuning so we modified SNN to run in two stages. We first ran the original SNN 7,000 times, each with a randomly generated sample according to within-source co-variance matrices provided in \textit{Gaia} DR2 and random parameter values within reasonable ranges. During the second stage, we modified SNN to identify the most repeating stellar groups from 25,798 we obtained in the first stage. We reveal 21 spatially- and kinematically-coherent groups in the Orion complex, 12 of which previously unknown. The groups show a wide distribution of distances extending as far as about 150 pc in front of the star-forming Orion molecular clouds, to about 50 pc beyond them where we find, unexpectedly, several groups. Our results expose to view the wealth of sub-structure in the OB association, within and beyond the classical Blaauw Orion OBI sub-groups. A full characterization of the new groups is of the essence as it offers the potential to unveil how star formation proceeds globally in large complexes such as Orion. The data and code that generated the groups in this work as well as the final table can be found at \protect\url{ https://github.com/BoquanErwinChen/GaiaDR2_Orion_Dissection}.Comment: 9 pages, 4 figures. Accepted by A&A. Comments welcom