417,084 research outputs found

### A scattering of orders

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 $\mathcal B$ of well-orderings and reversed well-orderings, and is closed under lexicographic sums with index set in $\mathcal B$. More generally, we say that a partial ordering is $\kappa$-scattered if it does not contain a copy of any $\kappa$-dense linear ordering. We prove analogues of Hausdorff's result for $\kappa$-scattered linear orderings, and for $\kappa$-scattered partial orderings satisfying the finite antichain condition. We also study the $\mathbb{Q}_\kappa$-scattered partial orderings, where $\mathbb{Q}_\kappa$ is the saturated linear ordering of cardinality $\kappa$, and a partial ordering is $\mathbb{Q}_\kappa$-scattered when it embeds no copy of $\mathbb{Q}_\kappa$. We classify the $\mathbb{Q}_\kappa$-scattered partial orderings with the finite antichain condition relative to the $\mathbb{Q}_\kappa$-scattered linear orderings. We show that in general the property of being a $\mathbb{Q}_\kappa$-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

In this paper, we introduce maximum composition ordering problems. The input
is $n$ real functions $f_1,\dots,f_n:\mathbb{R}\to\mathbb{R}$ and a constant
$c\in\mathbb{R}$. We consider two settings: total and partial compositions. The
maximum total composition ordering problem is to compute a permutation
$\sigma:[n]\to[n]$ which maximizes $f_{\sigma(n)}\circ
f_{\sigma(n-1)}\circ\dots\circ f_{\sigma(1)}(c)$, where $[n]=\{1,\dots,n\}$.
The maximum partial composition ordering problem is to compute a permutation
$\sigma:[n]\to[n]$ and a nonnegative integer $k~(0\le k\le n)$ which maximize
$f_{\sigma(k)}\circ f_{\sigma(k-1)}\circ\dots\circ f_{\sigma(1)}(c)$.
We propose $O(n\log n)$ time algorithms for the maximum total and partial
composition ordering problems for monotone linear functions $f_i$, 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 $f_i$ is of form
$\max\{a_ix+b_i,c_i\}$ for some constants $a_i\,(\ge 0)$, $b_i$ and $c_i$. We
finally prove that there exists no constant-factor approximation algorithm for
the problems, even if $f_i$'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

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

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

We present a higher well-ordering principle which is equivalent (over
Simpson's set theoretic version of $\text{ATR}_0$) to the existence of
transitive models of Kripke-Platek set theory, and thus to
$\Pi^1_1$-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

- âŠ