let $g$ be a non-abelian group of order $p^n$‎, ‎where $nleq 5$ in which $g$ is not extra special of order $p^5$‎. ‎in this paper we determine the maximal size of subsets $x$ of $g$‎ ‎with the property that $xyneq yx$ for any $x,y$ in $x$ with‎ ‎$xneq y$‎.

Two elements in an oriented matroid are inseparable if they have either the same sign in every signed circuit containing them both or opposite signs in every signed circuit containing them both. Two elements of a matroid are adjacent if there is no M(K4)-minor using them both, and in which they correspond to a matching of K4. We prove that two elements e, { of an oriented matroid are inseparabl...

The lottery preparation, a new general kind of Laver preparation, works uniformly with supercompact cardinals, strongly compact cardinals, strong cardinals, measurable cardinals, or what have you. And like the Laver preparation, the lottery preparation makes these cardinals indestructible by various kinds of further forcing. A supercompact cardinal κ, for example, becomes fully indestructible b...

To prove 1.1, one first assumes X is compact and G is a Lie group. In this case, X is "measure-theoretically" the product Y x G; this follows from the existence of local cross-sections to the projection n [6]. Let n2 : X ~ Y x G —> G, and define a map £ from L(Y, v) to the space of Radon measures on G as follows: £(ƒ) = TÏ2 [if ° n) ' M] • Apply the Dunford-Pettis Theorem [3] to ? to obtain a m...

When equipped with the natural topology first defined by Chabauty, the closed subgroups of a locally compact group G form a compact space C(G). We analyse the structure of C(G) for some low-dimensional Lie groups, concentrating mostly on the 3-dimensional Heisenberg group H. We prove that C(H) is a 6-dimensional space that is path–connected but not locally connected. The lattices in H form a de...

Various processes can generate associations between the larvae of different helminth species in their fish intermediate or paratenic host. We investigated the pairwise associations among larval helminth species in eight different fish populations, using two different coefficients of associations, in order to determine in what situations they are strongest. All helminth species included use the ...

The lottery preparation, a new general kind of Laver preparation, works uniformly with supercompact cardinals, strongly compact cardinals, strong cardinals, measurable cardinals, or what have you. And like the Laver preparation, the lottery preparation makes these cardinals indestructible by various kinds of further forcing. A supercompact cardinal κ, for example, becomes fully indestructible b...

This paper presents a compact pairwise model describing the spread of multi-stage epidemics on networks. The multi-stage model corresponds to a gamma-distributed infectious period which interpolates between the classical Markovian models with exponentially distributed infectious period and epidemics with a constant infectious period. We show how the compact approach leads to a system of equatio...

A graph is strongly perfect if every induced subgraph H of it has a stable set that meets maximal clique H. claw-free no vertex three pairwise nonadjacent neighbors. The characterization graphs are by forbidden subgraphs was conjectured Ravindra in 1990 and proved Wang 2006. Here we give shorter proof this characterization.

We call a topological space X locally compact with defects if all points in possess neighborhoods except for some points. investigate this weaker version of local compactness. show that x ∈ X• the partition singletons X\(X• ∪ (U\U)) is finite, where U ̸= an open neighborhood x, then Tychonoff space. Let be T1c such each has union pairwise disjoint subsets S s∈S Fs. Then, we family {Fs}s∈S finite...