Robustness and perturbations of minimal bases II: The case with given row degrees

This paper studies generic and perturbation properties inside the linear space of $m\times (m+n)$ polynomial matrices whose rows have degrees bounded by a given list $d_1, \ldots, d_m$ of natural numbers, which in the particular case $d_1 = \cdots = d_m = d$ is just the set of $m\times (m+n)$ polynomial matrices with degree at most $d$. Thus, the results in this paper extend to a much more general setting the results recently obtained in [Van Dooren & Dopico, Linear Algebra Appl. (2017), http://dx.doi.org/10.1016/j.laa.2017.05.011] only for polynomial matrices with degree at most $d$. Surprisingly, most of the properties proved in [Van Dooren & Dopico, Linear Algebra Appl. (2017)], as well as their proofs, remain to a large extent unchanged in this general setting of row degrees bounded by a list that can be arbitrarily inhomogeneous provided the well-known Sylvester matrices of polynomial matrices are replaced by the new trimmed Sylvester matrices introduced in this paper. The following results are presented, among many others, in this work: (1) generically the polynomial matrices in the considered set are minimal bases with their row degrees exactly equal to $d_1, \ldots , d_m$, and with right minimal indices differing at most by one and having a sum equal to $\sum_{i=1}^{m} d_i$, and (2), under perturbations, these generic minimal bases are robust and their dual minimal bases can be chosen to vary smoothly.Comment: arXiv admin note: text overlap with arXiv:1612.0379

Evaluating the ≤10:1 wholegrain criterion in identifying nutrient quality and health implications of UK breads and breakfast cereals

This article has been published in a revised form in Public Health Nutrition DOI: https://doi.org/10.1017/S1368980017003718. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © 2017 The Authors. Under embargo until 26 June 2018.Objective: To evaluate the nutrient quality of breads and breakfast cereals identified using the wholegrain definition of ≤10:1 carbohydrate:fibre ratio. Design: Following a cross-sectional study design, nutritional information was systematically gathered from food labels of breads and breakfast cereals that met the ≤10:1 carbohydrate:fibre criterion. The median nutrient content was compared with the UK Food Standards Agency nutrient profiling standards and the association between carbohydrate:fibre ratio and other nutrients were analysed. Subgroup analyses were undertaken for products with and without fruit, nuts and/or seeds. Setting: Products from four major supermarket stores in the UK. Subjects: 162 breads and 266 breakfast cereals. Results: Breads which met the ≤10:1 criterion typically contained medium fat, low saturated fat, low sugar and medium sodium. Breakfast cereals typically contained medium fat, low saturated fat, high sugar and low sodium. In both groups, as the carbohydrate:fibre ratio decreased, fat content increased (bread: p=0.029, r=-0.171; breakfast cereal: p=0.033, r=-0.131) and, in breakfast cereals, as the ratio increased, sugar content increased (p<0.0005, r=0.381). Breakfast cereals with fruit, nuts and/or seeds contained, per 100 g, more energy (p=0.002), fat, saturated fat and sugar (all p<0.0005) while seeded breads had more energy, fat and saturated fat (all p<0.0005). Conclusions: Overall, breads and breakfast cereals meeting the ≤10:1 criterion have good nutritional quality, suggesting that the criterion could be useful in public health and/or food labelling. The utility of applying the 10:1 criterion to products containing fruit, nuts and/or seeds is less clear and requires further research.Peer reviewedFinal Accepted Versio

Narrow scope for resolution-limit-free community detection

Detecting communities in large networks has drawn much attention over the years. While modularity remains one of the more popular methods of community detection, the so-called resolution limit remains a significant drawback. To overcome this issue, it was recently suggested that instead of comparing the network to a random null model, as is done in modularity, it should be compared to a constant factor. However, it is unclear what is meant exactly by "resolution-limit-free", that is, not suffering from the resolution limit. Furthermore, the question remains what other methods could be classified as resolution-limit-free. In this paper we suggest a rigorous definition and derive some basic properties of resolution-limit-free methods. More importantly, we are able to prove exactly which class of community detection methods are resolution-limit-free. Furthermore, we analyze which methods are not resolution-limit-free, suggesting there is only a limited scope for resolution-limit-free community detection methods. Finally, we provide such a natural formulation, and show it performs superbly