Primitive Words, Free Factors and Measure Preservation

Let F_k be the free group on k generators. A word w \in F_k is called primitive if it belongs to some basis of F_k. We investigate two criteria for primitivity, and consider more generally, subgroups of F_k which are free factors. The first criterion is graph-theoretic and uses Stallings core graphs: given subgroups of finite rank H \le J \le F_k we present a simple procedure to determine whether H is a free factor of J. This yields, in particular, a procedure to determine whether a given element in F_k is primitive. Again let w \in F_k and consider the word map w:G x G x ... x G \to G (from the direct product of k copies of G to G), where G is an arbitrary finite group. We call w measure preserving if given uniform measure on G x G x ... x G, w induces uniform measure on G (for every finite G). This is the second criterion we investigate: it is not hard to see that primitivity implies measure preservation and it was conjectured that the two properties are equivalent. Our combinatorial approach to primitivity allows us to make progress on this problem and in particular prove the conjecture for k=2. It was asked whether the primitive elements of F_k form a closed set in the profinite topology of free groups. Our results provide a positive answer for F_2.Comment: This is a unified version of two manuscripts: "On Primitive words I: A New Algorithm", and "On Primitive Words II: Measure Preservation". 42 pages, 14 figures. Some parts of the paper reorganized towards publication in the Israel J. of Mat

Gifting personal interpretations in galleries

The designers of mobile guides for museums and galleries face three major challenges: fostering rich interpretation, delivering deep personalization, and enabling a coherent social visit. We propose an approach to tackling all three simultaneously by inviting visitors to design an interpretation that is specifically tailored for a friend or loved one that they then experience together. We describe a trial of this approach at a contemporary art gallery, revealing how visitors designed personal and sometimes provocative experiences for people they knew well. We reveal how pairs of visitors negotiated these experiences together, showing how our approach could deliver intense experiences for both, but also required them to manage social risk. By interpreting our findings through the lens of ‘gift giving’ we shed new light on ongoing explorations of interpretation, personalization and social visiting within HCI

Micromagnetic simulations of interacting dipoles on a fcc lattice: Application to nanoparticle assemblies

Micromagnetic simulations are used to examine the effects of cubic and axial anisotropy, magnetostatic interactions and temperature on M-H loops for a collection of magnetic dipoles on fcc and sc lattices. We employ a simple model of interacting dipoles that represent single-domain particles in an attempt to explain recent experimental data on ordered arrays of magnetoferritin nanoparticles that demonstrate the crucial role of interactions between particles in a fcc lattice. Significant agreement between the simulation and experimental results is achieved, and the impact of intra-particle degrees of freedom and surface effects on thermal fluctuations are investigated.Comment: 10 pages, 9 figure

The Beagle 2 optical microscope

Introduction to the Beagle2 optical microscope

Dynamical Spin Response Functions for Heisenberg Ladders

We present the results of a numerical study of the 2 by L spin 1/2 Heisenberg ladder. Ground state energies and the singlet-triplet energy gaps for L = (4-14) and equal rung and leg interaction strengths were obtained in a Lanczos calculation and checked against earlier calculations by Barnes et al. (even L up to 12). A related moments technique is then employed to evaluate the dynamical spin response for L=12 and a range of rung to leg interaction strength ratios (0 - 5). We comment on two issues, the need for reorthogonalization and the rate of convergence, that affect the numerical utility of the moments treatment of response functions.Comment: Revtex, 3 figure

Representation theory of some infinite-dimensional algebras arising in continuously controlled algebra and topology

In this paper we determine the representation type of some algebras of infinite matrices continuously controlled at infinity by a compact metrizable space. We explicitly classify their finitely presented modules in the finite and tame cases. The algebra of row-column-finite (or locally finite) matrices over an arbitrary field is one of the algebras considered in this paper, its representation type is shown to be finite.Comment: 33 page

Classification of unit-vector fields in convex polyhedra with tangent boundary conditions

A unit-vector field n on a convex three-dimensional polyhedron P is tangent if, on the faces of P, n is tangent to the faces. A homotopy classification of tangent unit-vector fields continuous away from the vertices of P is given. The classification is determined by certain invariants, namely edge orientations (values of n on the edges of P), kink numbers (relative winding numbers of n between edges on the faces of P), and wrapping numbers (relative degrees of n on surfaces separating the vertices of P), which are subject to certain sum rules. Another invariant, the trapped area, is expressed in terms of these. One motivation for this study comes from liquid crystal physics; tangent unit-vector fields describe the orientation of liquid crystals in certain polyhedral cells.Comment: 21 pages, 2 figure

The whole and its parts : why and how to disentangle plant communities and synusiae in vegetation classification

Most plant communities consist of different structural and ecological subsets, ranging from cryptogams to different tree layers. The completeness and approach with which these subsets are sampled have implications for vegetation classification. Non‐vascular plants are often omitted or sometimes treated separately, referring to their assemblages as “synusiae” (e.g. epiphytes on bark, saxicolous species on rocks). The distinction of complete plant communities (phytocoenoses or holocoenoses) from their parts (synusiae or merocoenoses) is crucial to avoid logical problems and inconsistencies of the resulting classification systems. We here describe theoretical differences between the phytocoenosis as a whole and its parts, and outline consequences of this distinction for practise and terminology in vegetation classification. To implement a clearer separation, we call for modifications of the International Code of Phytosociological Nomenclature and the EuroVegChecklist. We believe that these steps will make vegetation classification systems better applicable and raise the recognition of the importance of non‐vascular plants in the vegetation as well as their interplay with vascular plants