Realization of analytic moduli for parabolic Dulac germs

In a previous paper [P. Mardešić and M. Resman. Analytic moduli for parabolic Dulac germs. Russian Math. Surveys, to appear, 2021, arXiv:1910.06129v2.] we determined analytic invariants, that is, moduli of analytic classification, for parabolic generalized Dulac germs. This class contains parabolic Dulac (almost regular) germs, which appear as first-return maps of hyperbolic polycycles. Here we solve the problem of realization of these moduli.

Some properties of an odd-dimensional space

In this paper, we investigate the absolute convergence of Fourier series of functions in several variables for an odd-dimensional space when these functions have continuous partial derivatives. It should be noted that similar properties for an even-dimensional space were given in [L. D. Gogoladze and V. S. Tsagareishvili, On absolute convergence of multiple Fourier series (in Russian), Izv. Vyssh. Uchebn. Zaved. Mat. 2015, 9, 12–21; translation in Russian Math. (Iz. VUZ) 59 (2015), no. 9, 9–17]. The obtained results are the best possible in a certain sense.

Realizing exterior Cromwell moves on rectangular diagrams by Reidemeister moves

If a rectangular diagram represents the trivial knot, then it can be deformed into the trivial rectangular diagram with only four edges by a finite sequence of merge operations and exchange operations, without increasing the number of edges, which was shown by Dynnikov in [Arc-presentations of links: Monotone simplification, Fund. Math. 190 (2006) 29–76; Recognition algorithms in knot theory, Uspekhi Mat. Nauk 58 (2003) 45–92. Translation in Russian Math. Surveys 58 (2003) 1093–1139]. Using this, Henrich and Kauffman gave in [Unknotting unknots, preprint (2011), arXiv:1006.4176v4 [math.GT]] an upper bound for the number of Reidemeister moves needed for unknotting a knot diagram of the trivial knot. However, exchange or merge moves on the top and bottom pairs of edges of rectangular diagrams are not considered in [Unknotting unknots, preprint (2011), arXiv:1006.4176v4 [math.GT]]. In this paper, we show that there is a rectangular diagram of the trivial knot which needs such an exchange move for being unknotted, and study upper bound of the number of Reidemeister moves needed for realizing such an exchange or merge move.