Thin subsets of groups

For a group $G$ and a natural number $m$, a subset $A$ of $G$ is called $m$-thin if, for each finite subset $F$ of $G$, there exists a finite subset $K$ of $G$ such that $|Fg\cap A|\leqslant m$ for every $g\in G\setminus K$. We show that each $m$-thin subset of a group $G$ of cardinality $\aleph_n$, $n= 0,1,...$ can be partitioned into $\leqslant m^{n+1}$ 1-thin subsets. On the other side, we construct a group $G$ of cardinality $\aleph_\omega$ and point out a 2-thin subset of $G$ which cannot be finitely partitioned into 1-thin subsets

Experimental constraints for additional short-range forces from neutron experiments

We present preliminary results on sensitivity of experiments with slow neutrons to constrain additional forces in a wide distance range: from picometers to micrometers. In the sub-nanometer range, available data on lengths of neutron scattering at nuclei provide the most competitive constraint. We show that it can be improved significantly in a dedicated measurement of asymmetry of neutron scattering at noble gases. In the micrometer range, we present sensitivity of the future GRANIT experiment. Further analysis will be presented in following publications.Comment: presented in "les rencontres de Moriond" 2007 conferenc

The comb-like representations of cellular ordinal balleans

Given two ordinal $\lambda$ and $\gamma$, let $f:[0,\lambda) \rightarrow [0,\gamma)$ be a function such that, for each $\alpha<\gamma$, $\sup\{f(t): t\in[0, \alpha]\}<\gamma.$ We define a mapping $d_{f}: [0,\lambda)\times [0,\lambda) \longrightarrow [0,\gamma)$ by the rule: if $x<y$ then $d_{f}(x,y)= d_{f}(y,x)= \sup\{f(t): t\in(x,y]\}$, $d(x,x)=0$. The pair $([0,\lambda), d_{f})$ is called a $\gamma-$comb defined by $f$. We show that each cellular ordinal ballean can be represented as a $\gamma-$comb. In {\it General Asymptology}, cellular ordinal balleans play a part of ultrametric spaces.Comment: 5 pages, preprin

Orbitally discrete coarse spaces

[EN] Given a coarse space (X, E), we endow X with the discrete topology and denote X ♯ = {p ∈ βG : each member P ∈ p is unbounded }. For p, q ∈ X ♯ , p||q means that there exists an entourage E ∈ E such that E[P] ∈ q for each P ∈ p. We say that (X, E) is orbitally discrete if, for every p ∈ X ♯ , the orbit p = {q ∈ X ♯ : p||q} is discrete in βG. We prove that every orbitally discrete space is almost finitary and scattered.Protasov, IV. (2021). Orbitally discrete coarse spaces. Applied General Topology. 22(2):303-309. https://doi.org/10.4995/agt.2021.13874OJS303309222T. Banakh and I. Protasov, Set-theoretical problems in Asymptology,arXiv: 2004.01979.T. Banakh, I. Protasov and S. Slobodianiuk, Scattered subsets of groups, Ukr. Math. J. 67 (2015), 347-356. https://doi.org/10.1007/s11253-015-1084-2T. Banakh and I. Zarichnyi, Characterizing the Cantor bi-cube in asymptotic categories, Groups Geom. Dyn. 5, no. 4 (2011), 691-728. https://doi.org/10.4171/GGD/145Ie. Lutsenko and I. Protasov, Space, thin and other subsets of groups, Intern. J. Algebra Comput. 19 (2009), 491-510. https://doi.org/10.1142/S0218196709005135Ie. Lutsenko and I. V. Protasov, Thin subset of balleans, Appl. Gen. Topology 11 (2010), 89-93. https://doi.org/10.4995/agt.2010.1710O. Petrenko and I. V. Protasov, Balleans and filters, Mat. Stud. 38 (2012), 3-11. https://doi.org/10.1007/s11253-012-0653-xI. V. Protasov, Normal ball structures, Mat. Stud. 20 (2003), 3-16.I. V. Protasov, Balleans of bounded geometry and G-spaces, Algebra Dicrete Math. 7, no. 2 (2008), 101-108.I. V. Protasov, Sparse and thin metric spaces, Mat. Stud. 41 (2014), 92-100.I. Protasov, Decompositions of set-valued mappings, Algebra Discrete Math. 30, no. 2 (2020), 235-238. https://doi.org/10.12958/adm1485I. Protasov, Coarse spaces, ultrafilters and dynamical systems, Topol. Proc. 57 (2021), 137-148.I. Protasov and T. Banakh, Ball Structures and Colorings of Groups and Graphs, Mat. Stud. Monogr. Ser. Vol. 11, VNTL, Lviv, 2003.I. Protasov and K. Protasova, Lattices of coarse structures, Math. Stud. 48 (2017), 115-123.I. V. Protasov and S. Slobodianiuk, Thin subsets of groups, Ukrain. Math. J. 65 (2013), 1245-1253. https://doi.org/10.1007/s11253-014-0866-2I. Protasov and S. Slobodianiuk, On the subset combinatorics of $G$-spaces, Algebra Dicrete Math. 17, no. 1 (2014), 98-109.I. Protasov and S. Slobodianiuk, Ultracompanions of subsets of a group, Comment. Math. Univ. Carolin. 55, no. 1 (2014), 257-265.I. Protasov and S. Slobodianiuk, The dynamical look at the subsets of groups, Appl. Gen. Topology 16 (2015), 217-224. https://doi.org/10.4995/agt.2015.3584I. Protasov and M. Zarichnyi, General Asymptology, Math. Stud. Monogr. Ser., Vol. 12, VNTL, Lviv, 2007.J. Roe, Lectures on Coarse Geometry, Univ. Lecture Ser., vol. 31, American Mathematical Society, Providence RI, 2003. https://doi.org/10.1090/ulect/03

Antideuteron production in proton-proton and proton-nucleus collisions

The experimental data of the antideuteron production in proton-proton and proton-nucleus collisions are analyzed within a simple model based on the diagrammatic approach to the coalescence model. This model is shown to be able to reproduce most of existing data without any additional parameter.Comment: To appear in Eur. Phys. J A (2002

Selective survey on spaces of closed subgroups of topological groups

We survey different topologizations of the set $\mathcal{S}(G)$ of all closed subgroups of a topological group $G$ and demonstrate some applications in Topological Grous, Model Theory, Geometric Group Theory, Topological Dynamics.Comment: Keywords: space of closed subgroups, Chabauty topology, Vietoris topology, Bourbaki uniformit

Asymptotic proximities

[EN] A ballean is a set endowed with some family of subsets which are called the balls. The properties of the family of balls are postulated in such a way that the balleans can be considered as a natural asymptotic counterparts of the uniform topological spaces. We introduce and study an asymptotic proximity as a counterpart of proximity relation for uniform topological space.Protasov, IV. (2008). Asymptotic proximities. Applied General Topology. 9(2):189-195. doi:10.4995/agt.2008.1799.SWORD1891959

Extraresolvability of balleans

summary:A ballean is a set endowed with some family of balls in such a way that a ballean can be considered as an asymptotic counterpart of a uniform topological space. We introduce and study a new cardinal invariant of a ballean, the extraresolvability, which is an asymptotic reflection of the corresponding invariant of a topological space