Close-packed dimers on the line: diffraction versus dynamical spectrum

The translation action of \RR^{d} on a translation bounded measure $\omega$ leads to an interesting class of dynamical systems, with a rather rich spectral theory. In general, the diffraction spectrum of $\omega$, which is the carrier of the diffraction measure, live on a subset of the dynamical spectrum. It is known that, under some mild assumptions, a pure point diffraction spectrum implies a pure point dynamical spectrum (the opposite implication always being true). For other systems, the diffraction spectrum can be a proper subset of the dynamical spectrum, as was pointed out for the Thue-Morse sequence (with singular continuous diffraction) in \cite{EM}. Here, we construct a random system of close-packed dimers on the line that have some underlying long-range periodic order as well, and display the same type of phenomenon for a system with absolutely continuous spectrum. An interpretation in terms of `atomic' versus `molecular' spectrum suggests a way to come to a more general correspondence between these two types of spectra.Comment: 14 pages, with some additions and improvement

Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies

Delone sets of finite local complexity in Euclidean space are investigated. We show that such a set has patch counting and topological entropy 0 if it has uniform cluster frequencies and is pure point diffractive. We also note that the patch counting entropy is 0 whenever the repetitivity function satisfies a certain growth restriction.Comment: 16 pages; revised and slightly expanded versio

Random Tilings: Concepts and Examples

We introduce a concept for random tilings which, comprising the conventional one, is also applicable to tiling ensembles without height representation. In particular, we focus on the random tiling entropy as a function of the tile densities. In this context, and under rather mild assumptions, we prove a generalization of the first random tiling hypothesis which connects the maximum of the entropy with the symmetry of the ensemble. Explicit examples are obtained through the re-interpretation of several exactly solvable models. This also leads to a counterexample to the analogue of the second random tiling hypothesis about the form of the entropy function near its maximum.Comment: 32 pages, 42 eps-figures, Latex2e updated version, minor grammatical change

Institutionalising Kant's political philosophy: Foregrounding cosmopolitan right

There exists a longstanding debate over the global institutional implications of Immanuel Kant's political philosophy: does such a philosophy entail a federal world government, or instead only a confederal ‘league of nations’? However, while the systematic nature of Kant's tripartite ‘doctrine of right' is well recognised, this debate has been conducted with all but exclusive focus on ‘international right' in particular. This article, by contrast, brings ‘cosmopolitan right' firmly into view. It proceeds by way of engagement with the two Kantian arguments made in defence of a ‘league of nations’ in discussion of international right, each of which appeals to aspects of states’ supposed ‘personhood’: the first appeals to states’ distinctive moral personality; the second to states’ physical manifestation. The article considers what happens when we assess these arguments not just in light of the demands of international right, but also in light of cosmopolitan right, and thus in light of public right more comprehensively. The answer is that such arguments cannot succeed as full defences of a league of nations. Indeed, when we assess such arguments with cosmopolitan right in view, they point instead – either tentatively or definitively – in the direction of world government

The roots of romantic cognitivism:(post) Kantian intellectual intuition and the unity of creation and discovery

During the romantic period, various authors expressed the belief that through creativity, we can directly access truth. To modern ears, this claim sounds strange. In this paper, I attempt to render the position comprehensible, and to show how it came to seem plausible to the romantics. I begin by offering examples of this position as found in the work of the British romantics. Each thinks that the deepest knowledge can only be gained by an act of creativity. I suggest the belief should be seen in the context of the post-Kantian embrace of “intellectual intuition.” Unresolved tensions in Kant's philosophy had encouraged a belief that creation and discovery were not distinct categories. The post-Kantians held that in certain cases of knowledge (for Fichte, knowledge of self and world; for Schelling, knowledge of the Absolute) the distinction between discovering a truth and creating that truth dissolves. In this context, the cognitive role assigned to acts of creativity is not without its own appeal