High frequency limit of the Transport Cross Section and boundedness of the Total Cross Section in scattering by an obstacle with impedance boundary conditions

The scalar scattering of the plane wave by a strictly convex obstacle with impedance boundary conditions is considered. The uniform boundedness of the Total Cross Section for all values of frequencies is proved. The high frequency limit of the Transport Cross Section is founded and presented as a classical functional of the variational theory

Geometry of random interactions

It is argued that spectral features of quantal systems with random interactions can be given a geometric interpretation. This conjecture is investigated in the context of two simple models: a system of randomly interacting d bosons and one of randomly interacting fermions in a j=7/2 shell. In both examples the probability for a given state to become the ground state is shown to be related to a geometric property of a polygon or polyhedron which is entirely determined by particle number, shell size, and symmetry character of the states. Extensions to more general situations are discussed

Optimal topological simplification of discrete functions on surfaces

We solve the problem of minimizing the number of critical points among all functions on a surface within a prescribed distance {\delta} from a given input function. The result is achieved by establishing a connection between discrete Morse theory and persistent homology. Our method completely removes homological noise with persistence less than 2{\delta}, constructively proving the tightness of a lower bound on the number of critical points given by the stability theorem of persistent homology in dimension two for any input function. We also show that an optimal solution can be computed in linear time after persistence pairs have been computed.Comment: 27 pages, 8 figure

A Closed Contour of Integration in Regge Calculus

The analytic structure of the Regge action on a cone in $d$ dimensions over a boundary of arbitrary topology is determined in simplicial minisuperspace. The minisuperspace is defined by the assignment of a single internal edge length to all 1-simplices emanating from the cone vertex, and a single boundary edge length to all 1-simplices lying on the boundary. The Regge action is analyzed in the space of complex edge lengths, and it is shown that there are three finite branch points in this complex plane. A closed contour of integration encircling the branch points is shown to yield a convergent real wave function. This closed contour can be deformed to a steepest descent contour for all sizes of the bounding universe. In general, the contour yields an oscillating wave function for universes of size greater than a critical value which depends on the topology of the bounding universe. For values less than the critical value the wave function exhibits exponential behaviour. It is shown that the critical value is positive for spherical topology in arbitrary dimensions. In three dimensions we compute the critical value for a boundary universe of arbitrary genus, while in four and five dimensions we study examples of product manifolds and connected sums.Comment: 16 pages, Latex, To appear in Gen. Rel. Gra

Prevalence and incidence of neuromuscular conditions in the UK between 2000 and 2019: A retrospective study using primary care data.

BACKGROUND: In the UK, large-scale electronic primary care datasets can provide up-to-date, accurate epidemiological information on rarer diseases, where specialist diagnoses from hospital discharges and clinic letters are generally well recorded and electronically searchable. Current estimates of the number of people living with neuromuscular disease (NMD) have largely been based on secondary care data sources and lacked direct denominators. OBJECTIVE: To estimate trends in the recording of neuromuscular disease in UK primary care between 2000-2019. METHODS: The Clinical Practice Research Datalink (CPRD) database was searched electronically to estimate incidence and prevalence rates (per 100,000) for a range of NMDs in each year. To compare trends over time, rates were age standardised to the most recent CPRD population (2019). RESULTS: Approximately 13 million patients were actively registered in each year. By 2019, 28,230 active patients had ever received a NMD diagnosis (223.6), which was higher among males (239.0) than females (208.3). The most common classifications were Guillain-Barre syndrome (40.1), myasthenia gravis (33.7), muscular dystrophy (29.5), Charcot-Marie-Tooth (29.5) and inflammatory myopathies (25.0). Since 2000, overall prevalence grew by 63%, with the largest increases seen at older ages (≥65-years). However, overall incidence remained constant, though myasthenia gravis incidence has risen steadily since 2008, while new cases of muscular dystrophy fell over the same period. CONCLUSIONS: Lifetime recording of many NMDs on primary care records exceed current estimates of people living with these conditions; these are important data for health service and care planning. Temporal trends suggest this number is steadily increasing, and while this may partially be due to better recording, it cannot be simply explained by new cases, as incidence remained constant. The increase in prevalence among older ages suggests increases in life expectancy among those living with NMDs may have occurred

Triangulations and Severi varieties

We consider the problem of constructing triangulations of projective planes over Hurwitz algebras with minimal numbers of vertices. We observe that the numbers of faces of each dimension must be equal to the dimensions of certain representations of the automorphism groups of the corresponding Severi varieties. We construct a complex involving these representations, which should be considered as a geometric version of the (putative) triangulations

World citation and collaboration networks: uncovering the role of geography in science

Modern information and communication technologies, especially the Internet, have diminished the role of spatial distances and territorial boundaries on the access and transmissibility of information. This has enabled scientists for closer collaboration and internationalization. Nevertheless, geography remains an important factor affecting the dynamics of science. Here we present a systematic analysis of citation and collaboration networks between cities and countries, by assigning papers to the geographic locations of their authors' affiliations. The citation flows as well as the collaboration strengths between cities decrease with the distance between them and follow gravity laws. In addition, the total research impact of a country grows linearly with the amount of national funding for research & development. However, the average impact reveals a peculiar threshold effect: the scientific output of a country may reach an impact larger than the world average only if the country invests more than about 100,000 USD per researcher annually.Comment: Published version. 9 pages, 5 figures + Appendix, The world citation and collaboration networks at both city and country level are available at http://becs.aalto.fi/~rajkp/datasets.htm

Combinatorial 3-manifolds with transitive cyclic symmetry

In this article we give combinatorial criteria to decide whether a transitive cyclic combinatorial d-manifold can be generalized to an infinite family of such complexes, together with an explicit construction in the case that such a family exists. In addition, we substantially extend the classification of combinatorial 3-manifolds with transitive cyclic symmetry up to 22 vertices. Finally, a combination of these results is used to describe new infinite families of transitive cyclic combinatorial manifolds and in particular a family of neighborly combinatorial lens spaces of infinitely many distinct topological types.Comment: 24 pages, 5 figures. Journal-ref: Discrete and Computational Geometry, 51(2):394-426, 201