Homotopy Quantum Field Theories and Related Ideas

In this short note we provide a review of some developments in the area of homotopy quantum field theories, loosely based on a talk given by the second author at the Xth Oporto Meeting on Geometry, Topology and Physics.Comment: 8 pages, 2 figures; correcte

The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials

Magnitude is an invariant of metric spaces with origins in category theory. Using potential theoretic methods, Barceló and Carbery gave an algorithm for calculating the magnitude of any odd dimensional ball in Euclidean space, and they proved that it was a rational function of the radius of the ball. In this paper an explicit formula is given for the magnitude of each odd dimensional ball in terms of a ratio of Hankel determinants of reverse Bessel polynomials. This is done by finding a distribution on the ball which solves the weight equations. Using Schröder paths and a continued fraction expansion for the generating function of the reverse Bessel polynomials, combinatorial formulae are given for the numerator and denominator of the magnitude of each odd dimensional ball. These formulae are then used to prove facts about the magnitude such as its asymptotic behaviour as the radius of the ball grows

Heuristic and computer calculations for the magnitude of metric spaces

The notion of the magnitude of a compact metric space was considered in arXiv:0908.1582 with Tom Leinster, where the magnitude was calculated for line segments, circles and Cantor sets. In this paper more evidence is presented for a conjectured relationship with a geometric measure theoretic valuation. Firstly, a heuristic is given for deriving this valuation by considering 'large' subspaces of Euclidean space and, secondly, numerical approximations to the magnitude are calculated for squares, disks, cubes, annuli, tori and Sierpinski gaskets. The valuation is seen to be very close to the magnitude for the convex spaces considered and is seen to be 'asymptotically' close for some other spaces

The Legendre-Fenchel transform from a category theoretic perspective

The Legendre-Fenchel transform is a classical piece of mathematics with many applications. In this paper we show how it arises in the context of category theory using categories enriched over the extended real numbers R¯¯¯¯:=[−∞,+∞]. A key ingredient is Pavlovic's 'nucleus of a profunctor' construction. The pairing between a vector space and its dual can be viewed as an R¯¯¯¯-profunctor; the construction of the nucleus of this profunctor is the construction of a lot of the theory of the Legendre-Fenchel transform. For a relation between sets viewed as a {true,false}-valued profunctor, the construction of the nucleus is the construction of the Galois connection associated to the relation. One insight given by this approach is that the relevant structure on the function spaces involved in the Legendre-Fenchel transform is something like a metric but is asymmetric and can take negative values. This 'R¯¯¯¯-structure' is a considerable refinement of the usual partial order on real-valued function space and it allows a natural interpretation of Toland-Singer duality and of the two tropical module structures on the set of convex functions

Categorifying the magnitude of a graph

The magnitude of a graph can be thought of as an integer power series associated to a graph; Leinster introduced it using his idea of magnitude of a metric space. Here we introduce a bigraded homology theory for graphs which has the magnitude as its graded Euler characteristic. This is a categorification of the magnitude in the same spirit as Khovanov homology is a categorification of the Jones polynomial. We show how properties of magnitude proved by Leinster categorify to properties such as a Kunneth Theorem and a Mayer-Vietoris Theorem. We prove that joins of graphs have their homology supported on the diagonal. Finally, we give various computer calculated examples

On the asymptotic magnitude of subsets of Euclidean space

Magnitude is a canonical invariant of finite metric spaces which has its origins in category theory; it is analogous to cardinality of finite sets. Here, by approximating certain compact subsets of Euclidean space with finite subsets, the magnitudes of line segments, circles and Cantor sets are defined and calculated. It is observed that asymptotically these satisfy the inclusion-exclusion principle, relating them to intrinsic volumes of polyconvex sets.Comment: 23 pages. Version 2: updated to reflect more recent work, in particular, the approximation method is now known to calculate (rather than merely define) the magnitude; also minor alterations such as references adde

Invasive meningococcal disease in patients with complement deficiencies: a case series (2008-2017).

BACKGROUND: To describe patients with inherited and acquired complement deficiency who developed invasive meningococcal disease (IMD) in England over the last decade. METHODS: Public Health England conducts enhanced surveillance of IMD in England. We retrospectively identified patients with complement deficiency who developed IMD in England during 2008-2017 and retrieved information on their clinical presentation, vaccination status, medication history, recurrence of infection and outcomes, as well as characteristics of the infecting meningococcal strain. RESULTS: A total of 16 patients with 20 IMD episodes were identified, including four with two episodes. Six patients had inherited complement deficiencies, two had immune-mediated conditions associated with complement deficiency (glomerulonephritis and vasculitis), and eight others were on Eculizumab therapy, five for paroxysmal nocturnal haemoglobinuria and three for atypical haemolytic uraemic syndrome. Cultures were available for 7 of 11 episodes among those with inherited complement deficiencies/immune-mediated conditions and the predominant capsular group was Y (7/11), followed by B (3/11) and non-groupable (1/11) strains. Among patients receiving Eculizumab therapy, 3 of the 9 episodes were due to group B (3/9), three others were NG but genotypically group B, and one case each of groups E, W and Y. CONCLUSIONS: In England, complement deficiency is rare among IMD cases and includes inherited disorders of the late complement pathway, immune-mediated disorders associated with low complement levels and patients on Eculizumab therapy. IMD due to capsular group Y predominates in patient with inherited complement deficiency, whilst those on Eculizumab therapy develop IMD due to more diverse capsular groups including non-encapsulated strains

Geographically widespread invasive meningococcal disease caused by a ciprofloxacin resistant non-groupable strain of the ST-175 clonal complex.

INTRODUCTION: Invasive meningococcal disease (IMD) caused by non-serogroupable (NG) strains mainly affects immunocompromised individuals. Reduced susceptibility to penicillin in meningococci is increasing in Europe but ciprofloxacin resistance remains rare. In 2019, three travel-related meningococcal disease cases caused by a ciprofloxacin-resistant NG strain were identified in England, leading Germany to report four additional IMD cases (2016 to 2019). We describe these and newly identified cases and characterise the strain responsible. METHODS: Cases were identified as part of national surveillance and by analysing available genomes using PubMLST tools. RESULTS: Of the cases identified in England in 2019, two geographically distinct cases developed conjunctivitis after returning from Mecca (Kingdom of Saudi Arabia) and a third linked case presented with IMD. Of the four cases from Germany, three occurred in asylum seekers - two familial and a further geographically distinct case. Further IMD cases were identified in Italy (n = 2; 2017-2018), Sweden (n = 1; 2016) and England (n = 1; 2015). A single ST-175 clonal complex (cc175) strain with genosubtype P1.22-11,15-25 was responsible. Decreased susceptibility to penicillin was widespread with three ciprofloxacin resistant subclusters. Constituent isolates were potentially covered by subcapsular vaccines. CONCLUSION: This disease associated NG cc175 strain exhibits resistance to antibiotics commonly used to prevent IMD but is potentially covered by subcapsular (meningococcal B) vaccines

Nutritional Ketosis Alters Fuel Preference and Thereby Endurance Performance in Athletes.

Ketosis, the metabolic response to energy crisis, is a mechanism to sustain life by altering oxidative fuel selection. Often overlooked for its metabolic potential, ketosis is poorly understood outside of starvation or diabetic crisis. Thus, we studied the biochemical advantages of ketosis in humans using a ketone ester-based form of nutrition without the unwanted milieu of endogenous ketone body production by caloric or carbohydrate restriction. In five separate studies of 39 high-performance athletes, we show how this unique metabolic state improves physical endurance by altering fuel competition for oxidative respiration. Ketosis decreased muscle glycolysis and plasma lactate concentrations, while providing an alternative substrate for oxidative phosphorylation. Ketosis increased intramuscular triacylglycerol oxidation during exercise, even in the presence of normal muscle glycogen, co-ingested carbohydrate and elevated insulin. These findings may hold clues to greater human potential and a better understanding of fuel metabolism in health and disease