James Cruickshank

Software for calculating blood lactate endurance markers

Abstract Blood lactate markers are used as summary measures of the underlying model of an athletes blood lactate response to increasing work rate. Exercise physiologists use these endurance markers, typically corresponding to a work rate in the region of high curvature in the lactate curve, to predict and compare endurance ability. A short theoretical background of the commonly used markers is given and algorithms provided for their calculation. To date, no free software exists that allows the sports scientist to calculate these markers. In this paper, software is introduced for precisely this purpose that will calculate a variety of lactate markers for an individual athlete, an athlete at different instants (e.g. across a season), and simultaneously for a squad.

The generic rigidity of triangulated spheres with blocks and holes

A simple graph G=(V,E) is 3-rigid if its generic bar-joint frameworks in R3 are infinitesimally rigid. Block and hole graphs are derived from triangulated spheres by the removal of edges and the addition of minimally rigid subgraphs, known as blocks, in some of the resulting holes. Combinatorial characterisations of minimal

UNITARY GROUPS OVER LOCAL RINGS

3

Twisted homotopy theory and the geometric equivariant 1-stem

-rigidity are obtained for these graphs in the case of a single block and finitely many holes or a single hole and finitely many blocks. These results confirm a conjecture of Whiteley from 1988 and special cases of a stronger conjecture of Finbow-Singh and Whiteley from 2013.

Unitary groups and ramified extensions

Structural properties of unitary groups over local, not necessarily commutative, rings are developed, with applications to the computation of the orders of these groups (when finite) and to the degrees of the irreducible constituents of the Weil representation of a unitary group associated to a ramified extension of finite local rings.

Positive polynomials on Riesz spaces

Abstract We develop methods for computing the equivariant homotopy set [ M ,S V ] G , where M is a manifold on which the group G acts freely, and V is a real linear representation of G. Our approach is based on the idea that an equivariant invariant of M should correspond to a twisted invariant of the orbit space M/G. We use this method to make certain explicit calculations in the case dim M = dim V+ dim G+1 .

On Spaces of Infinitesimal Motions and Three Dimensional Henneberg Extensions

Abstract We classify all non-degenerate skew-hermitian forms defined over certain local rings, not necessarily commutative, and study some of the fundamental properties of the associated unitary groups, including their orders when the ring in question is finite.

Targeting Influential Nodes for Recovery in Bootstrap Percolation on Hyperbolic Networks

We prove some properties of positive polynomial mappings between Riesz spaces, using finite difference calculus. We establish the polynomial analogue of the classical result that positive, additive mappings are linear. And we prove a polynomial version of the Kantorovich extension theorem.

Rearrangement Inequalities and the Alternahedron

We investigate certain spaces of infinitesimal motions arising naturally in the rigidity theory of bar and joint frameworks. We prove some structure theorems for these spaces and, as a consequence, are able to deduce some special cases of a long standing conjecture of Graver, Tay and Whiteley concerning Henneberg extensions and generically rigid graphs.

Series parallel linkages

The influence of our peers is a powerful reinforcement for our social behaviour, evidenced in voter behaviour and trend adoption. Bootstrap percolation is a simple method for modelling this process. In this work we look at bootstrap percolation on hyperbolic random geometric graphs, which have been used to model the Internet graph, and introduce a form of bootstrap percolation with recovery, showing that random targeting of nodes for recovery will delay adoption, but this effect is enhanced when nodes of high degree are selectively targeted.