Activities of the Lithuanian Mathematical Society in 2018–2021

The article gives a survey of mathematical life and activities of the Lithuanian Mathematical Society during the past four years.

Locally compact flows on connected manifolds

In this paper, we completely characterize locally compact flows G G of homeomorphisms of connected manifolds M M by proving that they are either circle groups or real groups. For M = R m M = \mathbb R^m , we prove that every recurrent element in G G is periodic, and we obtain a generalization of the result of Yang [Hilbert’s fifth problem and related problems on transformation groups, American Mathematical Society, Providence, RI, 1976, pp. 142–146.] by proving that there is no nontrivial locally compact flow on R m \mathbb R^m in which all elements are recurrent.

Places of Near-Fields: To Heinrich Wefelscheid

Wefelscheid (Untersuchungen über Fastkörper und Fastbereiche, Habilitationsschrift, Hamburg, 1971) generalised the well-known Theorem of Artin/Schreier about the characterization of formally real fields and the fundamental result of Baer/Krull to near-fields. In the last fifty years arose from the Theorem of Baer/Krull a theory, which analyses the entirety of the orderings of a field (E. Becker, L. Bröcker, M. Marshall et al.), as presented e.g. in the book by Lam (Orderings, valuations and quadratic forms, American Mathematical Society, Providence, 1983). At the centre of this theory are preorders and their compatibility with valuations or places. We develop some essential results of this theory for the near-field case. In particular, we derive the Brown/Marshall’s inequalities and Bröcker’s Theorem on the trivialisation of fans in the near-field case.

Ramanujan's Beautiful Integrals

International audience Throughout his entire mathematical life, Ramanujan loved to evaluate definite integrals. One can find them in his problems submitted to the Journal of the Indian Mathematical Society, notebooks, Quarterly Reports to the University of Madras, letters to Hardy, published papers and the Lost Notebook. His evaluations are often surprising, beautiful, elegant, and useful in other mathematical contexts. He also discovered general methods for evaluating and approximating integrals. A survey of Ramanujan's contributions to the evaluation of integrals is given, with examples provided from each of the above-mentioned sources.

Subsystems of transitive subshifts with linear complexity

We bound the number of distinct minimal subsystems of a given transitive subshift of linear complexity, continuing work of Ormes and Pavlov [On the complexity function for sequences which are not uniformly recurrent. Dynamical Systems and Random Processes (Contemporary Mathematics, 736). American Mathematical Society, Providence, RI, 2019, pp. 125--137]. We also bound the number of generic measures such a subshift can support based on its complexity function. Our measure-theoretic bounds generalize those of Boshernitzan [A unique ergodicity of minimal symbolic flows with linear block growth. J. Anal. Math.44(1) (1984), 77–96] and are closely related to those of Cyr and Kra [Counting generic measures for a subshift of linear growth. J. Eur. Math. Soc.21(2) (2019), 355–380].