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This paper describes the development of a method for the investigation and comparison of materials for use in stab resistant body armour. A number of polymer composite panels of different thicknesses and construction have been tested. A dynamic test which simulated the real threat has been used and the results compared to a simpler quasi-static test that might be used in initial materials selection. The materials tested were glass-epoxy, and glass-nylon composite panels of several thicknesses between 1.8 and 5.8mm. Additional tests were also performed on similar composites containing tungsten wires. An accelerated instrumented drop-tower was used to drive a knife through composite panels and record the force resisting penetration by the knife. The final penetration of the knife through the armour into a soft backing was also measured. For comparison,a similar geometry quasi-static test was carried out on the same specimens. It was found that energy absorbtion took the form of an initial resistance to perforation and then by a resistance to further penetration. This is thought to stem from resistance to cutting ofthe panel material and gripping of the knife blade. The energy required to produce a given penetration in dynamic tests was found to be in good agreement with the penetration achieved at similar energies under quasi-static conditions. For the materials tested there was no significant difference between the penetration resistance of single or two layer systems. The penetration achieved through a panel of a given material was approximately proportional to the inverse square of the panel's thickness. The relative performance of different armour materials was assessed by plotting the energy required to penetrate a fixed distance against the areal density of the panel.

We observe that the group of all lifts of elements of Thompson’s group T T to the real line is finitely presented and contains the additive group Q \mathbb {Q} of the rational numbers. This gives an explicit realization of the Higman embedding theorem for Q \mathbb {Q} , answering a Kourovka notebook question of Martin Bridson and Pierre de la Harpe.

We prove Thompson's group $F$ has quadratic conjugator length function. That is, for any two conjugate elements of $F$ of length $n$ or less, there exists an element of $F$ of length $O(n^2)$ that conjugates one to the other. Moreover, there exist conjugate pairs of elements of $F$ of length at most $n$ such that the shortest conjugator between them has length $\Omega(n^2)$. This latter statement holds for $T$ and $V$ as well., Comment: 17 pages, 13 figures. Fixed some minor typos

We prove that all Gromov hyperbolic groups embed into the asynchronous rational group defined by Grigorchuk, Nekrashevych and Sushchanski\u{i}. The proof involves assigning a system of binary addresses to points in the Gromov boundary of $G$, and proving that elements of $G$ act on these addresses by transducers. These addresses derive from a certain self-similar tree of subsets of $G$, whose boundary is naturally homeomorphic to the horofunction boundary of $G$., Comment: 73 pages, 17 figures

We prove that every finitely-generated right-angled Artin group can be embedded into some Brin-Thompson group $nV$. It follows that many other groups can be embedded into some $nV$ (e.g., any finite extension of any of Haglund and Wise's special groups), and that various decision problems involving subgroups of $nV$ are unsolvable., Comment: 7 pages, no figures

We prove that the asynchronous rational group defined by Grigorchuk, Nekrashevych, and Sushchanskii is simple and not finitely generated. Our proofs also apply to certain subgroups of the rational group, such as the group of all rational bilipschitz homeomorphisms., Comment: 12 pages, 3 figures

We prove that Claas Röver's Thompson-Grigorchuk simple group V G has type F∞. The proof involves constructing two complexes on which V G acts: a simplicial complex analogous to the Stein complex for V , and a polysimplicial complex analogous to the Farley complex for V . We then analyze the descending links of the polysimplicial complex, using a theorem of Belk and Forrest to prove increasing connectivity.

We present an efficient implementation of the solution to the conjugacy problem in Thompson's group F. This algorithm checks for conjugacy by constructing and comparing directed graphs called strand diagrams. We provide a description of our solution algorithm, including the data structure that represents strand diagrams and supports simplifications.