Assouad dimension of self-affine carpets

We calculate the Assouad dimension of the self-affine carpets of Bedford and McMullen, and of Lalley and Gatzouras. We also calculate the conformal Assouad dimension of those carpets that are not self-similar.Comment: 10 pages, 3 figure

Quasi-hyperbolic planes in relatively hyperbolic groups

We show that any group that is hyperbolic relative to virtually nilpotent subgroups, and does not admit peripheral splittings, contains a quasi-isometrically embedded copy of the hyperbolic plane. In natural situations, the specific embeddings we find remain quasi-isometric embeddings when composed with the inclusion map from the Cayley graph to the coned-off graph, as well as when composed with the quotient map to "almost every" peripheral (Dehn) filling. We apply our theorem to study the same question for fundamental groups of 3-manifolds. The key idea is to study quantitative geometric properties of the boundaries of relatively hyperbolic groups, such as linear connectedness. In particular, we prove a new existence result for quasi-arcs that avoid obstacles.Comment: v1: 32 pages, 4 figures. v2: 38 pages, 4 figures. v3: 44 pages, 4 figures. An application (Theorem 1.2) is weakened as there was an error in its proof in section 7, all other changes minor, improved expositio

Conformal dimension via subcomplexes for small cancellation and random groups

We find new bounds on the conformal dimension of small cancellation groups. These are used to show that a random few relator group has conformal dimension 2+o(1) asymptotically almost surely (a.a.s.). In fact, if the number of relators grows like l^K in the length l of the relators, then a.a.s. such a random group has conformal dimension 2+K+o(1). In Gromov's density model, a random group at density d<1/8 a.a.s. has conformal dimension $\asymp dl / |\log d|$. The upper bound for C'(1/8) groups has two main ingredients: $\ell_p$-cohomology (following Bourdon-Kleiner), and walls in the Cayley complex (building on Wise and Ollivier-Wise). To find lower bounds we refine the methods of [Mackay, 2012] to create larger `round trees' in the Cayley complex of such groups. As a corollary, in the density model at d<1/8, the density d is determined, up to a power, by the conformal dimension of the boundary and the Euler characteristic of the group.Comment: v1: 42 pages, 21 figures; v2: 44 pages, 20 figures. Improved exposition, final versio

Poorly connected groups

We investigate groups whose Cayley graphs have poor\-ly connected subgraphs. We prove that a finitely generated group has bounded separation in the sense of Benjamini--Schramm--Tim\'ar if and only if it is virtually free. We then prove a gap theorem for connectivity of finitely presented groups, and prove that there is no comparable theorem for all finitely generated groups. Finally, we formulate a connectivity version of the conjecture that every group of type $F$ with no Baumslag-Solitar subgroup is hyperbolic, and prove it for groups with at most quadratic Dehn function.Comment: 14 pages. Changes to v2: Proof of the Theorem 1.2 shortened, Theorem 1.4 added completing the no-gap result outlined in v

Microstructure-property relationships in directionally solidified single crystal nickel-base superalloys

Some of the microstructural features which influence the creep properties of directionally solidified and single crystal nickel-base superalloys are discussed. Gamma precipitate size and morphology, gamma-gamma lattice mismatch, phase instability, alloy composition, and processing variations are among the factors considered. Recent experimental results are reviewed and related to the operative deformation mechanisms and to the corresponding mechanical properties. Special emphasis is placed on the creep behavior of single crystal superalloys at high temperatures, where directional gamma coarsening is prominent, and at lower temperatures, where gamma coarsening rates are significantly reduced. It can be seen that very subtle changes in microstructural features can have profound effects on the subsequent properties of these materials

The formation of mixed germanium–cobalt carbonyl clusters: an electrospray mass spectrometric study, and the structure of a high-nuclearity [Ge₂Co₁₀(CO)₂₄]²⁻ anion

The reaction of [µ₄-Ge{Co₂(CO)₇}₂] with [Co(CO)₄]⁻ has been monitored by electrospray mass spectrometry to detect the cluster anions generated. Conditions giving known mixed Ge–Co carbonyl clusters were established, and a new high nuclearity cluster anion, [Ge₂Co₁₀(CO)₂₄]²⁻ was detected. Conditions for its formation were optimised and it was subsequently isolated as its [Et₄N]⁺ salt and characterised by single-crystal X-ray crystallography. The Ge₂Co₁₀ cluster core has a novel geometry with the two germanium atoms in semi-encapsulated positions, forming seven formal Ge–Co bonds. There are also eighteen formal Co–Co bonds. Corresponding reactions of [µ₄-Si{Co₂(CO)₇}₂] with [Co(CO)₄]⁻ were also investigated

Conserved Charges and Supersymmetry in Principal Chiral Models

We report on investigations of local (and non-local) charges in bosonic and supersymmetric principal chiral models in 1+1 dimensions. In the bosonic PCM there is a classically conserved local charge for each symmetric invariant tensor of the underlying group. These all commute with the non-local Yangian charges. The algebra of the local charges amongst themselves is rather more subtle. We give a universal formula for infinite sets of mutually commuting local charges with spins equal to the exponents of the underlying classical algebra modulo its Coxeter number. Many of these results extend to the supersymmetric PCM, but with local conserved charges associated with antisymmetric invariants in the Lie algebra. We comment briefly on the quantum conservation of local charges in both the bosonic and super PCMs.Comment: 18 pages, LaTeX. Revised and up-dated version based on conference talks by JME and NJ

Towards gravitationally assisted negative refraction of light by vacuum

Propagation of electromagnetic plane waves in some directions in gravitationally affected vacuum over limited ranges of spacetime can be such that the phase velocity vector casts a negative projection on the time-averaged Poynting vector. This conclusion suggests, inter alia, gravitationally assisted negative refraction by vacuum.Comment: 6 page