A class of translation bounded complex measures, which have the form of weighted Dirac combs, on locally compact Abelian groups is investigated. Given such a Dirac comb, we are interested in its diffraction spectrum which emerges as the Fourier transform of the autocorrelation measure. We present a sufficient set of conditions to ensure that the diffraction measure is a pure point measure. Simultaneously, we establish a natural link to the cut and project formalism and to the theory of almost periodic measures. Our conditions are general enough to cover the known theory of model sets, but also to include examples such as the visible lattice points.

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It is shown how regular model sets can be characterized in terms of the 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 β and then relate the properties of β to those 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.Downloaded

We show that for multi-colored Delone point sets with finite local complexity and uniform cluster frequencies the notions of pure point diffraction and pure point dynamical spectrum are equivalent.

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Several members of the kinesin family of microtubule motor proteins play essential roles in mitotic spindle function and are potential targets for the discovery of novel antimitotic cancer therapies. KSP, also known as HsEg5, is a kinesin that plays an essential role in formation of a bipolar mitotic spindle and is required for cell cycle progression through mitosis. We identified a potent inhibitor of KSP, CK0106023, which causes mitotic arrest and growth inhibition in several human tumor cell lines. Here we show that CK0106023 is an allosteric inhibitor of KSP motor domain ATPase with a K i of 12 nM. Among five kinesins tested, CK0106023 was specific for KSP. In tumor-bearing mice, CK0106023 exhibited antitumor activity comparable to or exceeding that of paclitaxel and caused the formation of monopolar mitotic figures identical to those produced in cultured cells. KSP was most abundant in proliferating human tissues and was absent from cultured postmitotic neurons. These findings are the first to demonstrate the feasibility of targeting mitotic kinesins for the treatment of cancer.

There is a growing body of results in the theory of discrete point sets and tiling systems giving conditions under which such systems are pure point diffractive. Here we look at the opposite direction: what can we infer about a discrete point set or tiling, defined through a primitive substitution system, given that it is pure point diffractive? Our basic objects are Delone multisets and tilings, which are self-replicating under a primitive substitution system of affine mappings with a common expansive map Q. Our first result gives a partial answer to a question of Lagarias and Wang: we characterize repetitive substitution 1 RVM is grateful for the continuing support of an NSERC Operating Grant in this research. 2 BS acknowledges support from NSF grants DMS 9800786 and DMS 0099814. 1Delone multisets that can be represented by substitution tilings using a concept of "legal cluster." This allows us to move freely between both types of objects. Our main result is that for lattice substitution multiset systems (in arbitrary dimensions) being a regular model set is not only sufficient for having pure point spectrum-a known fact-but is also necessary.This completes a circle of equivalences relating pure point dynamical and diffraction spectra, modular coincidence, and model sets for lattice substitution systems begun by the first two authors of this paper.

This article surveys the mathematics of the cut and project method as applied to point sets, called here model sets. It covers the geometric, arithmetic, and analytical sides of this theory as well as diffraction and the connection with dynamical systems.Dedicated to the memory of Richard (Dick) Slansky The spirit of the universe is subtle and informs all life. Things live and die and change their forms, without knowing the root from which they come. Abundantly it multiplies; eternally it stands by itself. The greatest reaches of space do not leave its confines, and the smallest down of a bird in autumn awaits its power to assume form.-Chuang Tzu (tr. Lin Yutang)