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Let $G$ be a nonabelian group, $Asubseteq G$ an abelian subgroup and $ngeqslant 2$ an integer. We say that $G$ has an $n$-abelian partition with respect to $A$, if there exists a partition of $G$ into $A$ and $n$ disjoint commuting subsets $A_1, A_2, ldots, A_n$ of $G$, such that $|A_i|>1$ for each $i=1, 2, ldots, n$. We first classify all nonabelian groups, up to isomorphism, which have an $n$-abelian partition for $n=2, 3$. Then, we provide some formulas concerning the number of spanning trees of commuting graphs associated with certain finite groups. Finally, we point out some ways to finding the number of spanning trees of the commuting graphs of some specific groups.
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In this article we give an expository account of the holomorphic motion theorem based on work of M`a~ne-Sad-Sullivan, Bers-Royden, and Chirka. After proving this theorem, we show that tangent vectors to holomorphic motions have $|epsilon log epsilon|$ moduli of continuity and then show how this type of continuity for tangent vectors can be combined with Schwarzs lemma and integration over the holomorphic variable to produce Holder continuity on the mappings. We also prove, by using holomorphic motions, that Kobayashis and Teichmullers metrics on the Teichmuller space of a Riemann surface coincide. Finally, we present an application of holomorphic motions to complex dynamics, that is, we prove the Fatou linearization theorem for parabolic germs by involving holomorphic motions.
We study a curious class of partitions, the parts of which obey an exceedingly strict congruence condition we refer to as sequential congruence: the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part congruent to zero modulo the length of the partition. It turns out these obscure-seeming objects are embedded in a natural way in partition theory. We show that sequentially congruent partitions with largest part $n$ are in bijection with the partitions of $n$. Moreover, we show sequentially congruent partitions induce a bijection between partitions of $n$ and partitions of length $n$ whose parts obey a strict frequency congruence condition -- the frequency (or multiplicity) of each part is divisible by that part -- and prove families of similar bijections, connecting with G. E. Andrewss theory of partition ideals.
We study partial homology and cohomology from ring theoretic point of view via the partial group algebra $mathbb{K}_{par}G$. In particular, we link the partial homology and cohomology of a group $G$ with coefficients in an irreducible (resp. indecomposable) $mathbb{K}_{par}G$-module with the ordinary homology and cohomology groups of $G$ with in general non-trivial coefficients. Furthermore, we compare the standard cohomological dimension $cd_{ mathbb{K}}(G)$ (over a field $mathbb{K}$) with the partial cohomological dimension $cd_{ mathbb{K}}^{par}(G)$ (over $mathbb{K}$) and show that $cd_{ mathbb{K}}^{par}(G) geq cd_{ mathbb{K}}(G)$ and that there is equality for $G = mathbb{Z}$.
In this article, we shall explore the constructions of Bernstein sets, and prove that every Bernstein set is nonmeasurable and doesnt have the property of Baire. We shall also prove that Bernstein sets dont have the perfect set property.
Due to the discovery of the hidden-charm pentaquark $P_c$ states by the LHCb collaboration, the interests on the candidates of hidden-bottom pentaquark $P_b$ states are increasing. They are anticipated to exist as the analogues of the $P_c$ states in the bottom sector and predicted by many models. We give an exploration of searching for a typical $P_b$ in the $gamma p to Upsilon p$ reaction, which shows a promising potential to observe it at an electron-ion collider. The possibility of searching for $P_b$ in open-bottom channels are also briefly discussed. Meanwhile, the $t$-channel non-resonant contribution, which in fact covers several interesting topics at low energies, is systematically investigated.
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