Multi-Component Model Sets and Invariant Densities

Model sets (also called cut and project sets) are generalizations of lattices, and multi-component model sets are generalizations of lattices with colourings. In this paper, we study self-similarities of multi-component model sets. The main point may be simply summarized: whenever there is a self-similarity, there are also naturally related density functions. As in the case of ordinary model sets, we show that invariant densities exist and that they produce absolutely continuous invariant measures in internal space, these features now appearing in matrix form. We establish a close connection between the theory of invariant densities and the spectral theory of matrix continuous refinement operators.Comment: 12 pages, 2 figures, to appear in: Aperiodic 9

Self-Similarities and Invariant Densities for Model Sets

Model sets (also called cut and project sets) are generalizations of lattices. Here we show how the self-similarities of model sets are a natural replacement for the group of translations of a lattice. This leads us to the concept of averaging operators and invariant densities on model sets. We prove that invariant densities exist and that they produce absolutely continuous invariant measures in internal space. We study the invariant densities and their relationships to diffraction, continuous refinement operators, and Hutchinson measures.Comment: 15 pages, 2 figures, to appear in: Algebraic Methods and Theoretical Physics (ed. Y. St. Aubin

Invariant Submodules and Semigroups of Self-Similarities for Fibonacci Modules

The problem of invariance and self-similarity in Z-modules is investigated. For a selection of examples relevant to quasicrystals, especially Fibonacci modules, we determine the semigroup of self-similarities and encapsulate the number of similarity submodules in terms of Dirichlet series generating functions.Comment: 7 pages; to appear in: Aperiodic 97, eds. M. de Boissieu, J. L. Verger-Gaugry and R. Currat, World Scientific, Singapore (1998), in pres

Similar sublattices of the root lattice A4A_4

Similar sublattices of the root lattice $A_4$ are possible, according to a result of Conway, Rains and Sloane, for each index that is the square of a non-zero integer of the form $m^2 + mn - n^2$. Here, we add a constructive approach, based on the arithmetic of the quaternion algebra $\mathbb{H} (\mathbb{Q} (\sqrt{5}))$ and the existence of a particular involution of the second kind, which also provides the actual sublattices and the number of different solutions for a given index. The corresponding Dirichlet series generating function is closely related to the zeta function of the icosian ring.Comment: 17 pages, 1 figure; revised version with several additions and improvement

Characterizations of model sets by dynamical systems

It is shown how regular model sets can be characterized in terms of regularity properties of their associated dynamical systems. The proof proceeds in two steps. First, we characterize regular model sets in terms of a certain map $\beta$ and then relate the properties of $\beta$ to ones of the underlying dynamical system. As a by-product, we can show that regular model sets are, in a suitable sense, as close to periodic sets as possible among repetitive aperiodic sets.Comment: 41 pages, revised versio

Davis v. Department of Revenue of Kentucky: A Preliminary Impact Assessment

States with income taxes frequently exempt municipal bond interest from state taxation. Such exemptions, referred to as double exempts, are tax expenditures that reduce state revenues, but are viewed as a subsidy to the cost of capital for the state and its localities. All but a few states provide the income tax exemption for state based issues while taxing interest from municipal bonds issued by muni issuers in other states. A recent court case, Davis vs. Department of Revenue of Kentucky, declared state statutes limiting the state income tax exemptions to ”in-state” issues unconstitutional. This paper provides some legal background and context for the current case and addresses two key fiscal implications of this case. First, the paper presents a basic model that suggests that bonds issued by states with higher marginal tax rates would see the yields increase on their obligations while states with lower than average marginal tax rates would see their yields decline. The yields would converge at new market equilibrium due to the elimination of tax preferences across the states. Second, the preliminary estimates suggest a good deal of variance in how much tax revenue each state will lose if the case is upheld by the Supreme Court.

Philanthropy 19922022 What difference can 30 years make?

Thirty years is not a long time for some fields — geology, for instance. But for philanthropy and the nonprofit sector — and especially the study of and training in those fields — thirty years is a big deal. Consider that the premier industry publication, the Chronicle of Philanthropy, was only founded in 1988. Also, most institutions for research about this field, and nearly all university degree programs, have been established in the past three decades.The Center for Philanthropy and Nonprofit Leadership — now called the Dorothy A. Johnson Center for Philanthropy — was founded at Grand Valley State University in 1992. It seems fitting, then, on the occasion of our 30th anniversary, to look back at what things looked like at the beginning — as well as at the remarkable transformation and growth of the field since then.Who was giving in 1992 and in what ways? What was the foundation world like in 1992? What stories were on the front page of the Chronicle of Philanthropy? How many other university-based centers were there at the time? And how does this compare to the current landscape of giving, nonprofits, and philanthropic studies?This essay not only reminds us what things looked like in 1992, but reveals how the practices of giving, the makeup and number of institutions, and the intensity and breadth of research and teaching about philanthropy have all expanded considerably and changed in sometimes dramatic ways