suppose that $h$ is a subgroup of $g$‎, ‎then $h$ is said to be‎ ‎$s$-permutable in $g$‎, ‎if $h$ permutes with every sylow subgroup of‎ ‎$g$‎. ‎if $hp=ph$ hold for every sylow subgroup $p$ of $g$ with $(|p|‎, ‎|h|)=1$)‎, ‎then $h$ is called an $s$-semipermutable subgroup of $g$‎. ‎in this paper‎, ‎we say that $h$ is partially $s$-embedded in $g$ if‎ ‎$g$ has a normal subgroup $t$ such that $ht...

Japanese tatami mats are often arranged so that no four mats meet. This local restriction imposes a rich combinatorial structure when applied to monomino-domino coverings of rectilinear grids. We describe a modular, mechanical game board, prototyped with a desktop 3D printer, that enforces this restriction, and transforms tatami pen-andpaper puzzles into interactive sculptures. We review some r...

Self-similar groups and permutational bimodules are used to study combinatorics and symbolic dynamics of expanding self-coverings. We describe functors between the category of contracting self-similar groups and the category of expanding self-coverings (with appropriate morphisms). These functors transform some questions in dynamical systems to questions in algebra. As examples we show how some...

We construct a class of lattices in three and higher dimensions for which the number of dimer coverings can be determined exactly using elementary arguments. These lattices are a generalization of the two-dimensional kagome lattice, and the method also works for graphs without translational symmetry. The partition function for dimer coverings on these lattices can be determined also for a class...

Sense of direction is a property of labelled networks (i.e., arc-coloured graphs) that allows one to assign coherently local identifiers to other processors on the basis of the route followed by incoming messages. We prove that (weak) sense of direction is preserved by the construction of regular coverings (i.e., coverings induced by voltage assignments in a group) whose voltage assignment depe...

Every elliptic curve is isomorphic to the double covering of the projective line P branching at four points. The set of the isomorphism classes of elliptic curves with marking of branching points can be identified with the configuration space M4pts of ordered four points on P. By considering their periods, one obtains a morphism per from M4pts to the set Mct of the isomorphism classes of comple...

Enumerating the isomorphism classes of several types of graph coverings is one of the central research topics in enumerative topological graph theory (see [R. Feng, J.H. Kwak, J. Kim, J. Lee, Isomorphism classes of concrete graph coverings, SIAM J. Discrete Math. 11 (1998) 265–272; R. Feng, J.H. Kwak, Typical circulant double coverings of a circulant graph, Discrete Math. 277 (2004) 73–85; R. F...

Rough set theory, a mathematical tool to deal with inexact or uncertain knowledge in information systems, has originally described the indiscernibility of elements by equivalence relations. Covering rough sets are a natural extension of classical rough sets by relaxing the partitions arising from equivalence relations to coverings. Recently, some topological concepts such as neighborhood have b...

Following A.M. Vinogradov and I.S. Krasilshchik, we regard systems of PDEs as manifolds with integrable distributions and consider their special morphisms called differential coverings, which include constructions like Lax pairs and Bäcklund transformations (BT) in soliton theory. We show that, similarly to usual coverings in topology, at least for some soliton equations differential coverings ...

Lysis of the blastocyst coverings in the rabbit normally starts at the abembryonic hemisphere of the blastocyst facing the antimesometrial endometrium. Theoretically, this restricted effect could be caused by factors locally produced by abembryonic trophoblast, antimesometrial endometrium, or a combination of them. Blastocysts abnormally implanting with reversed orientation exhibit lysis of bla...