417,084 research outputs found

    A scattering of orders

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    A linear ordering is scattered if it does not contain a copy of the rationals. Hausdorff characterised the class of scattered linear orderings as the least family of linear orderings that includes the class B \mathcal B of well-orderings and reversed well-orderings, and is closed under lexicographic sums with index set in B \mathcal B. More generally, we say that a partial ordering is κ \kappa -scattered if it does not contain a copy of any κ \kappa -dense linear ordering. We prove analogues of Hausdorff's result for κ \kappa -scattered linear orderings, and for κ \kappa -scattered partial orderings satisfying the finite antichain condition. We also study the Qκ \mathbb{Q}_\kappa -scattered partial orderings, where Qκ \mathbb{Q}_\kappa is the saturated linear ordering of cardinality κ \kappa , and a partial ordering is Qκ \mathbb{Q}_\kappa -scattered when it embeds no copy of Qκ \mathbb{Q}_\kappa . We classify the Qκ \mathbb{Q}_\kappa -scattered partial orderings with the finite antichain condition relative to the Qκ \mathbb{Q}_\kappa -scattered linear orderings. We show that in general the property of being a Qκ \mathbb{Q}_\kappa -scattered linear ordering is not absolute, and argue that this makes a classification theorem for such orderings hard to achieve without extra set-theoretic assumptions

    Optimal Composition Ordering Problems for Piecewise Linear Functions

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    In this paper, we introduce maximum composition ordering problems. The input is nn real functions f1,,fn:RRf_1,\dots,f_n:\mathbb{R}\to\mathbb{R} and a constant cRc\in\mathbb{R}. We consider two settings: total and partial compositions. The maximum total composition ordering problem is to compute a permutation σ:[n][n]\sigma:[n]\to[n] which maximizes fσ(n)fσ(n1)fσ(1)(c)f_{\sigma(n)}\circ f_{\sigma(n-1)}\circ\dots\circ f_{\sigma(1)}(c), where [n]={1,,n}[n]=\{1,\dots,n\}. The maximum partial composition ordering problem is to compute a permutation σ:[n][n]\sigma:[n]\to[n] and a nonnegative integer k (0kn)k~(0\le k\le n) which maximize fσ(k)fσ(k1)fσ(1)(c)f_{\sigma(k)}\circ f_{\sigma(k-1)}\circ\dots\circ f_{\sigma(1)}(c). We propose O(nlogn)O(n\log n) time algorithms for the maximum total and partial composition ordering problems for monotone linear functions fif_i, which generalize linear deterioration and shortening models for the time-dependent scheduling problem. We also show that the maximum partial composition ordering problem can be solved in polynomial time if fif_i is of form max{aix+bi,ci}\max\{a_ix+b_i,c_i\} for some constants ai(0)a_i\,(\ge 0), bib_i and cic_i. We finally prove that there exists no constant-factor approximation algorithm for the problems, even if fif_i's are monotone, piecewise linear functions with at most two pieces, unless P=NP.Comment: 19 pages, 4 figure

    Commensurate-Incommensurate transition in the melting process of the orbital ordering in Pr0.5Ca0.5MnO3: neutron diffraction study

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    The melting process of the orbital order in Pr0.5Ca0.5MnO3 single crystal has been studied in detail as a function of temperature by neutron diffraction. It is demonstrated that a commensurate-incommensurate (C-IC) transition of the orbital ordering takes place in a bulk sample, being consistent with the electron diffraction studies. The lattice structure and the transport properties go through drastic changes in the IC orbital ordering phase below the charge/orbital ordering temperature Tco/oo, indicating that the anomalies are intimately related to the partial disordering of the orbital order, unlike the consensus that it is related to the charge disordering process. For the same T range, partial disorder of the orbital ordering turns on the ferromagnetic spin fluctuations which were observed in a previous neutron scattering study.Comment: 5 pages, 2 figures, REVTeX, to be published in Phys. Rev.

    Partial Kekule Ordering of Adatoms on Graphene

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    Electronic and transport properties of Graphene, a one-atom thick crystalline material, are sensitive to the presence of atoms adsorbed on its surface. An ensemble of randomly positioned adatoms, each serving as a scattering center, leads to the Bolzmann-Drude diffusion of charge determining the resistivity of the material. An important question, however, is whether the distribution of adatoms is always genuinely random. In this Article we demonstrate that a dilute adatoms on graphene may have a tendency towards a spatially correlated state with a hidden Kekule mosaic order. This effect emerges from the interaction between the adatoms mediated by the Friedel oscillations of the electron density in graphene. The onset of the ordered state, as the system is cooled below the critical temperature, is accompanied by the opening of a gap in the electronic spectrum of the material, dramatically changing its transport properties

    A Higher Bachmann-Howard Principle

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    We present a higher well-ordering principle which is equivalent (over Simpson's set theoretic version of ATR0\text{ATR}_0) to the existence of transitive models of Kripke-Platek set theory, and thus to Π11\Pi^1_1-comprehension. This is a partial solution to a conjecture of Montalb\'an and Rathjen: partial in the sense that our well-ordering principle is less constructive than demanded in the conjecture.Comment: This paper is no longer up to date: It is superseded by the author's PhD thesis (available at http://etheses.whiterose.ac.uk/20929/) and the streamlined presentation in arXiv:1809.06759. In contrast to the present abstract, we have now found a computable version of our well-ordering principle. Thus the conjecture by Montalb\'an and Rathjen can be considered as fully solve