Spatially partitioned embedded Runge-Kutta Methods

We study spatially partitioned embedded Runge–Kutta (SPERK) schemes for partial differential equations (PDEs), in which each of the component schemes is applied over a different part of the spatial domain. Such methods may be convenient for problems in which the smoothness of the solution or the magnitudes of the PDE coefficients vary strongly in space. We focus on embedded partitioned methods as they offer greater efficiency and avoid the order reduction that may occur in non-embedded schemes. We demonstrate that the lack of conservation in partitioned schemes can lead to non-physical effects and propose conservative additive schemes based on partitioning the fluxes rather than the ordinary differential equations. A variety of SPERK schemes are presented, including an embedded pair suitable for the time evolution of fifth-order weighted non-oscillatory (WENO) spatial discretizations. Numerical experiments are provided to support the theory

Effective order strong stability preserving Runge–Kutta methods

We apply the concept of effective order to strong stability preserving (SSP) explicit Runge–Kutta methods. Relative to classical Runge–Kutta methods, effective order methods are designed to satisfy a relaxed set of order conditions, but yield higher order accuracy when composed with special starting and stopping methods. The relaxed order conditions allow for greater freedom in the design of effective order methods. We show that this allows the construction of four-stage SSP methods with effective order four (such methods cannot have classical order four). However, we also prove that effective order five methods—like classical order five methods—require the use of non-positive weights and so cannot be SSP. By numerical optimization, we construct explicit SSP Runge–Kutta methods up to effective order four and establish the optimality of many of them. Numerical experiments demonstrate the validity of these methods in practice

Parafermions, parabosons and representations of so(\infty) and osp(1|\infty)

The goal of this paper is to give an explicit construction of the Fock spaces of the parafermion and the paraboson algebra, for an infinite set of generators. This is equivalent to constructing certain unitary irreducible lowest weight representations of the (infinite rank) Lie algebra so(\infty) and of the Lie superalgebra osp(1|\infty). A complete solution to the problem is presented, in which the Fock spaces have basis vectors labelled by certain infinite but stable Gelfand-Zetlin patterns, and the transformation of the basis is given explicitly. We also present expressions for the character of the Fock space representations

Hatching Strategies in Monogenean (Platyhelminth) Parasites that Facilitate Host Infection

In parasites, environmental cues may influence hatching of eggs and enhance the success of infections. The two major endoparasitic groups of parasitic platyhelminths, cestodes (tapeworms) and digeneans (flukes), typically have high fecundity, infect more than one host species, and transmit trophically. Monogeneans are parasitic flatworms that are among the most host specific of all parasites. Most are ectoparasites with relatively low fecundity and direct life cycles tied to water. They infect a single host species, usually a fish, although some are endoparasites of amphibians and aquatic chelonian reptiles. Monogenean eggs have strong shells and mostly release ciliated larvae, which, against all odds, must find, identify, and infect a suitable specific host. Some monogeneans increase their chances of finding a host by greatly extending the hatching period (possible bet-hedging). Others respond to cues for hatching such as shadows, chemicals, mechanical disturbance, and osmotic changes, most of which may be generated by the host. Hatching may be rhythmical, larvae emerging at times when the host is more vulnerable to invasion, and this may be combined with responses to other environmental cues. Different monogenean species that infect the same host species may adopt different strategies of hatching, indicating that tactics may be more complex than first thought. Control of egg assembly and egg-laying, possibly by host hormones, has permitted colonization of frogs and toads by polystomatid monogeneans. Some monogeneans further improve the chances of infection by attaching eggs to the host or by retaining eggs on, or in, the body of the parasite. The latter adaptation has led ultimately to viviparity in gyrodactylid monogeneans

Establishing the potential for using routine data on Incapacity Benefit to assess the local impact of policy initiatives

<i>Background</i>: Incapacity Benefit (IB) is the key contributory benefit for people who are incapable of work because of illness or disability. <i>Methods</i>: The aims were to establish the utility of routinely collected data for local evaluation and to provide a descriptive epidemiology of the IB population in Glasgow and Scotland for the period 2000–05 using data supplied by the Department for Work and Pensions. <i>Results</i>: Glasgow's IB population is large in absolute and relative terms but is now falling, mainly due to a decrease in on flow. Claimants, tend to be older, have a poor work history and suffer from mental health problems. The rate of decline has been greater in Glasgow than Scotland, although the rate of on flow is still higher. <i>Conclusions</i>: Department for Work and Pensions (DWP) data can be used locally to provide important insights into the dynamics of the IB population. However, to be truly useful, more work needs to be undertaken to combine the DWP data with other information

Bilinear identities on Schur symmetric functions

A series of bilinear identities on the Schur symmetric functions is obtained with the use of Pluecker relations.Comment: Accepted to Journal of Nonlinear Mathematical Physics. A reference to a connected result is adde

Some Properties of the Calogero-Sutherland Model with Reflections

We prove that the Calogero-Sutherland Model with reflections (the BC_N model) possesses a property of duality relating the eigenfunctions of two Hamiltonians with different coupling constants. We obtain a generating function for their polynomial eigenfunctions, the generalized Jacobi polynomials. The symmetry of the wave-functions for certain particular cases (associated to the root systems of the classical Lie groups B_N, C_N and D_N) is also discussed.Comment: 16 pages, harvmac.te

Interval structure of the Pieri formula for Grothendieck polynomials

We give a combinatorial interpretation of a Pieri formula for double Grothendieck polynomials in terms of an interval of the Bruhat order. Another description had been given by Lenart and Postnikov in terms of chain enumerations. We use Lascoux's interpretation of a product of Grothendieck polynomials as a product of two kinds of generators of the 0-Hecke algebra, or sorting operators. In this way we obtain a direct proof of the result of Lenart and Postnikov and then prove that the set of permutations occuring in the result is actually an interval of the Bruhat order.Comment: 27 page