An asymptotic existence result on compressed sensing matrices

For any rational number $h$ and all sufficiently large $n$ we give a deterministic construction for an $n\times \lfloor hn\rfloor$ compressed sensing matrix with $(\ell_1,t)$-recoverability where $t=O(\sqrt{n})$. Our method uses pairwise balanced designs and complex Hadamard matrices in the construction of $\epsilon$-equiangular frames, which we introduce as a generalisation of equiangular tight frames. The method is general and produces good compressed sensing matrices from any appropriately chosen pairwise balanced design. The $(\ell_1,t)$-recoverability performance is specified as a simple function of the parameters of the design. To obtain our asymptotic existence result we prove new results on the existence of pairwise balanced designs in which the numbers of blocks of each size are specified.Comment: 15 pages, no figures. Minor improvements and updates in February 201

Inequivalence of difference sets: on a remark of Baumert

An often cited statement of Baumert in his book Cyclic difference sets asserts that four well known families of cyclic (4t - 1,2t - 1,t - 1) difference sets are inequivalent, apart from a small number of exceptions with t ≤ 8. We are not aware of a proof of this statement in the literature. Three of the families discussed by Baumert have analogous constructions in non-cyclic groups. We extend his inequivalence statement to a general inequivalence result, for which we provide a complete and self-contained proof. We preface our proof with a survey of the four families of difference sets, since there seems to be some confusion in the literature between the cyclic and non-cyclic cases

Compressed sensing with combinatorial designs: theory and simulations

In 'An asymptotic result on compressed sensing matrices', a new construction for compressed sensing matrices using combinatorial design theory was introduced. In this paper, we use deterministic and probabilistic methods to analyse the performance of matrices obtained from this construction. We provide new theoretical results and detailed simulations. These simulations indicate that the construction is competitive with Gaussian random matrices, and that recovery is tolerant to noise. A new recovery algorithm tailored to the construction is also given.Comment: 18 pages, 3 figure

Gait re-training: a technology-based intervention for reducing impact loading in running

Purpose: The primary aim of this thesis was to examine if a compliant running technique reduces impact accelerations, what the associated kinematics and kinetics are, and what method should be employed in the teaching of runners to adopt this technique. A secondary aim was to determine the effect of compliant running kinematics on energy expenditure. Methods: Study 1 examined the use of a verbally directed compliant technique. Study 2 examined the success of an accelerometer-based biofeedback system using various accelerometer locations (tibia, sacrum and treadmill). Study 3 then compared the use of a tibial located accelerometer-based biofeedback system to verbal feedback. Study 4 examined a 4-week treadmill based biofeedback intervention with regard to how it altered kinematics, loading, and energy expenditure. Results: Treadmill accelerometer-based biofeedback appears to display an ability to reduce both tibial (-26%) and sacral (-17%) accelerations acutely, with reductions increasing further when the intervention period is extended to 4-weeks (tibia: -40%; sacrum: -42%). These reductions were associated with reduced vGRFs and joint moments. In comparison, an acute and a 3-week bout of verbal feedback produced lower reductions in impact accelerations (tibia: -8%, sacrum: - 22%; tibia: -10%, sacrum: -41%;); while segment based biofeedback produced large reductions but more localised to their source of feedback: sacral biofeedback (tibia: -1%; sacrum: -27%), and tibial biofeedback (tibia: -39%; sacrum: -4%). This reduced loading was achieved by increased cushioning at impact and decreased vertical oscillation of the COM. These kinematic changes demonstrated no effect on energy expenditure in study 4, but an increase in energy expenditure in study 1; possible due to the larger degree of knee and hip flexion in study 1. Conclusion: A 4-week treadmill accelerometer-based biofeedback intervention appears to reduce loading to a greater extent than verbal feedback, or biofeedback from the tibia or sacrum. This appeared to not influence energy expenditure

Good sequencings of partial Steiner systems

A partial (n, k, t)λ-system is a pair(X, B) where X is an n-set of vertices and B is a collection of k-subsets of X called blocks such that each t-set of vertices is a subset of at most λ blocks. A sequencing of such a system is a labelling of its vertices with distinct elements of {0,..., n − 1}. A sequencing is -block avoiding or, more briefly, -good if no block is contained in a set of vertices with consecutive labels. Here we give a short proof that, for fixed k, t and λ, any partial (n, k, t)λ-system has an -good sequencing for some = �(n1/t ) as n becomes large. This improves on results of Blackburn and Etzion, and of Stinson and Veitch. Our result is perhaps of most interest in the case k = t +1 where results of Kostochka, Mubayi and Verstraëte show that the value of cannot be increased beyond �((n log n)1/t ). A special case of our result shows that every partial Steiner triple system (partial (n, 3, 2)1- system) has an -good sequencing for each positive integer 0.0908 n1/2