We study locally compact groups having all subgroups minimal. We call such groups hereditarily minimal. In 1972 Prodanov proved that the infinite hereditarily minimal compact abelian groups are precisely the groups Zp of p-adic integers. We extend Prodanov's theorem to the non-abelian case at several levels. For infinite hypercentral (in particular, nilpotent) locally compact groups we show that the hereditarily minimal ones remain the same as in the abelian case. On the other hand, we classify completely the locally compact solvable hereditarily minimal groups, showing that in particular they are always compact and metabelian.The proofs involve the (hereditarily) locally minimal groups, introduced similarly. In particular, we prove a conjecture by He, Xiao and the first two authors, showing that the group Qp ⋊ Q * p is hereditarily locally minimal, where Q * p is the multiplicative group of non-zero p-adic numbers acting on the first component by multiplication. Furthermore, it turns out that the locally compact solvable hereditarily minimal groups are closely related to this group.

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In this paper we propose a methodology based on Topogical\ud Data Analysis (TDA) for capturing when a complex system,\ud represented by a multivariate time series, changes its inter-\ud nal organization. The modication of the inner organization\ud among the entities belonging to a complex system can induce\ud a phase transition of the entire system. In order to identify\ud these reorganizations, we designed a new methodology that\ud is based on the representation of time series by simplicial\ud complexes. The topologization of multivariate time series\ud successfully pinpoints out when a complex system evolves.\ud Simplicial complexes are characterized by persistent homo-\ud logy techniques, such as the clique weight rank persistent\ud homology and the topological invariants are used for com-\ud puting a new entropy measure, the so-called weighted per-\ud sistent entropy. With respect to the global invariants, e.g.\ud the Betti numbers, the entropy takes into account also the\ud topological noise and then it captures when a phase transi-\ud tion happens in a system. In order to verify the reliability of\ud the methodology, we have analyzed the EEG signals of Phy-\ud sioNet database and we have found numerical evidences that\ud the methodology is able to detect the transition between the\ud pre-ictal and ictal states

We study the algebraic entropy of continuous endomorphisms of compactly covered, locally compact, topologically quasihamiltonian groups. We provide a Limit-free formula which helps us to simplify the computations of this entropy. Moreover, several Addition Theorems are given. In particular, we prove that the Addition Theorem holds for endomorphisms of quasihamiltonian torsion FC-groups (e.g., Hamiltonian groups).2010 Mathematics Subject Classification. 37A35, 22D40, 28D20, 20K35.

We study the Zariski topology Z_G, the Markov topology M_G and the precompact Markov topology P_G of an infinite group G, introduced in byDikranjan and Shakhmatov (2007--2010). We prove that P_G is discrete for a non-abelian divisible solvable group G, concluding that a countable divisible solvable group G is abelian if and only if M_G = P_G if and only if P_G is non-discrete. This answers a question of Dikranjan and Shakhmatov (2010). We study in detail the space (G,Z_G) for two types of linear groups, obtaining a complete description of various topological properties (as dimension, Noetherianity, etc.). This allows us to distinguish, in the case of linear groups, the Zariski topology defined via words (i.e., the verbal topology in terms of Bryant) from the affine topology usually considered in algebraic geometry. We compare the properties of the Zariski topology of these linear groups with the corresponding ones obtained in Dikranjan and Shakhmatov (2010) in the case of abelian groups