On Sound Reflection in Superfluid

We consider reflection of the first and the second sound waves by a rigid flat wall in superfluid. Nontrivial dependence of the reflection coefficients on the angle of incidence is obtained. Sound conversion is predicted at slanted incidence.Comment: 9 pages, 4 figure

Error bounds for the large-argument asymptotic expansions of the Hankel and Bessel functions

In this paper, we reconsider the large-argument asymptotic expansions of the Hankel, Bessel and modified Bessel functions and their derivatives. New integral representations for the remainder terms of these asymptotic expansions are found and used to obtain sharp and realistic error bounds. We also give re-expansions for these remainder terms and provide their error estimates. A detailed discussion on the sharpness of our error bounds and their relation to other results in the literature is given. The techniques used in this paper should also generalize to asymptotic expansions which arise from an application of the method of steepest descents.Comment: 32 pages, 2 figures, accepted for publication in Acta Applicandae Mathematica

Basic Methods for Computing Special Functions

This paper gives an overview of methods for the numerical evaluation of special functions, that is, the functions that arise in many problems from mathematical physics, engineering, probability theory, and other applied sciences. We consider in detail a selection of basic methods which are frequently used in the numerical evaluation of special functions: converging and asymptotic series, including Chebyshev expansions, linear recurrence relations, and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. We give examples of recent software for special functions where these methods are used. We mention a list of new publications on computational aspects of special functions available on our website

Sea-level changes, river capture and the evolution of populations of the Eastern Cape and fiery redfins (Pseudobarbus afer and Pseudobarbus phlegethon, Cyprinidae) across multiple river systems in South Africa

AIM: The phylogeography of the two closely related species Pseudobarbus afer and Pseudobarbus phlegethon was investigated to assess the association of evolutionary processes, inferred from mitochondrial DNA (mtDNA) sequence variation, with hypothetical palaeoriver systems and other climatic and landscape changes. Location One western and several southern river systems in South Africa. METHODS: We sampled known populations and confirmed known distribution gaps. This was followed by an assessment of mtDNA control region sequence variation for 31 localities across 17 river systems across the range of the species complex. A map of possible offshore drainage patterns during the last major regression event was constructed based on bathymetry and geological studies. RESULTS: The genetic distinction of four major lineages of P. afer strongly correspond with proposed palaeoriver systems. However, a western ‘Forest’ lineage, is widespread across two such proposed systems and is closely related to P. phlegethon on the west coast of South Africa. Both the ‘Krom’ and ‘St Francis’ lineages were identified in the single palaeoriver system proposed for St Francis Bay. A fourth ‘Mandela’ lineage is restricted to the one or two palaeoriver systems proposed for Nelson Mandela Bay. Four minor lineages were identified within the Forest lineage and two within the Mandela lineage. Main conclusions The close relationship between P. phlegethon and the Forest lineage of P. afer can only be explained by a series of river captures. We suggest the Gourits River system as a historical link that could account for this relationship. On the south coast, lower sea levels than at present allowed confluence between currently isolated river systems, offering opportunities for dispersal among these populations. At present, isolation between different river systems rather than dispersal appears to have a dominant influence on mtDNA diversity