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Register a new userThe nonmeasurability of Bernstein sets and related topics
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Cheng Hao
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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.
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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.
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.
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.
Linear spaces with an Euclidean metric are ubiquitous in mathematics, arising both from quadratic forms and inner products. Operators on such spaces also occur naturally. In recent years, the study of multivariate operator theory has made substantial progress. Although the study of self-adjoint operators goes back a few decades, the non self-adjoint theory has developed at a slower pace. While several approaches to this topic has been developed, the one that has been most fruitful is clearly the study of Hilbert spaces that are modules over natural function algebras like $mathcal A({Omega})$, where $Omega subseteq mathbb C^m$ is a bounded domain, consisting of complex valued functions which are holomorphic on some open set $U$ containing $overline{Omega}$, the closure of $Omega$. The book, Hilbert Modules over function algebra, R. G. Douglas and V. I. Paulsen showed how to recast many of the familiar theorems of operator theory in the language of Hilbert modules. The book, Spectral decomposition of analytic sheaves, J. Eschmeier and M. Putinar and the book, Analytic Hilbert modules, X. Chen and K. Guo, provide an account of the achievements from the recent past. The impetus for much of what is described below comes from the interplay of operator theory with other areas of mathematics like complex geometry and representation theory of locally compact groups.
In this paper, we give a necessary and sufficient condition for a graphical strip in the Heisenberg group $mathbb{H}$ to be area-minimizing in the slab ${-1<x<1}$. We show that our condition is necessary by introducing a family of deformations of graphical strips based on varying a vertical curve. We show that it is sufficient by showing that strips satisfying the condition have monotone epigraphs. We use this condition to show that any area-minimizing ruled entire intrinsic graph in the Heisenberg group is a vertical plane and to find a boundary curve that admits uncountably many fillings by area-minimizing surfaces.
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