Digitisation, representation, and formalisation - Digital libraries of mathematics

One of the main tasks of the mathematical knowledge management community must surely be to enhance access to mathematics on digital systems. In this paper we present a spectrum of approaches to solving the various problems inherent in this task, arguing that a variety of approaches is both necessary and useful. The main ideas presented are about the differences between digitised mathematics, digitally represented mathematics and formalised mathematics. Each has its part to play in managing mathematical information in a connected world. Digitised material is that which is embodied in a computer file, accessible and displayable locally or globally. Represented material is digital material in which there is some structure (usually syntactic in nature) which maps to the mathematics contained in the digitised information. Formalised material is that in which both the syntax and semantics of the represented material, is automatically accessible. Given the range of mathematical information to which access is desired, and the limited resources available for managing that information, we must ensure that these resources are applied to digitise, form representations of or formalise, existing and new mathematical information in such a way as to extract the most benefit from the least expenditure of resources. We also analyse some of the various social and legal issues which surround the practical tasks.

Monte Carlo phase diagram for diblock copolymer melts

A partial phase diagram is constructed for diblock copolymer melts using lattice-based Monte Carlo simulations. This is done by locating the order-disorder transition (ODT) with the aid of a recently proposed order parameter and identifying the ordered phase over a wide range of copolymer compositions (0.2 <= f <= 0.8). Consistent with experiments, the disordered phase is found to exhibit direct first-order transitions to each of the ordered morphologies. This includes the spontaneous formation of a perforated-lamellar phase, which presumably forms in place of the gyroid morphology due to finite-size and/or nonequilibrium effects. Also included in our study is a detailed examination of disordered cylinder-forming (f=0.3) diblock copolymers, revealing a substantial degree of pretransitional chain stretching and short-range order that set in well before the ODT, as observed previously in analogous studies on lamellar-forming (f=0.5) molecules. (c) 2006 American Institute of Physics.

A wavenumber independent boundary element method for an acoustic scattering problem

In this paper we consider the impedance boundary value problem for the Helmholtz equation in a half-plane with piecewise constant boundary data, a problem which models, for example, outdoor sound propagation over inhomogeneous. at terrain. To achieve good approximation at high frequencies with a relatively low number of degrees of freedom, we propose a novel Galerkin boundary element method, using a graded mesh with smaller elements adjacent to discontinuities in impedance and a special set of basis functions so that, on each element, the approximation space contains polynomials ( of degree.) multiplied by traces of plane waves on the boundary. We prove stability and convergence and show that the error in computing the total acoustic field is O( N-(v+1) log(1/2) N), where the number of degrees of freedom is proportional to N logN. This error estimate is independent of the wavenumber, and thus the number of degrees of freedom required to achieve a prescribed level of accuracy does not increase as the wavenumber tends to infinity.

A high-wavenumber boundary-element method for an acoustic scattering problem

In this paper we show stability and convergence for a novel Galerkin boundary element method approach to the impedance boundary value problem for the Helmholtz equation in a half-plane with piecewise constant boundary data. This problem models, for example, outdoor sound propagation over inhomogeneous flat terrain. To achieve a good approximation with a relatively low number of degrees of freedom we employ a graded mesh with smaller elements adjacent to discontinuities in impedance, and a special set of basis functions for the Galerkin method so that, on each element, the approximation space consists of polynomials (of degree $\nu$) multiplied by traces of plane waves on the boundary. In the case where the impedance is constant outside an interval $[a,b]$, which only requires the discretization of $[a,b]$, we show theoretically and experimentally that the $L_2$ error in computing the acoustic field on $[a,b]$ is ${\cal O}(\log^{\nu+3/2}|k(b-a)| M^{-(\nu+1)})$, where $M$ is the number of degrees of freedom and $k$ is the wavenumber. This indicates that the proposed method is especially commendable for large intervals or a high wavenumber. In a final section we sketch how the same methodology extends to more general scattering problems.

Feeding rumen-inert fats differing in their degree of saturation decreases intake and increases plasma concentrations of gut peptides in lactating dairy cows

Our objective was to determine the effect of feeding rumen-inert fats differing in their degree of saturation on dry matter intake (DMI), milk production, and plasma concentrations of insulin, glucagon-like peptide 1 (7-36) amide (GLP-1), glucose-dependent insulinotropic polypeptide (GIP), and cholecystokinin (CCK) in lactating dairy cows. Four midlactation, primiparous Holstein cows were used in a 4 x 4 Latin square experiment with 2-wk periods. Cows were fed a control mixed ration ad libitum, and treatments were the dietary addition (3.5% of ration dry matter) of 3 rumen-inert fats as sources of mostly saturated fatty acids (SFA), monounsaturated fatty acids (MUFA), or polyunsaturated fatty acids (PUFA). Daily DMI, milk yield, and composition were measured on the last 4 d of each period. Jugular vein blood was collected every 30 min over a 7-h period on d 12 and 14 of each period for analysis of plasma concentrations of hormones, glucose, and nonesterified fatty acids. Feeding fat decreased DMI, and the decrease tended to be greater for MUFA and PUFA compared with SFA. Plasma concentration of GLP-1 increased when fat was fed and was greater for MUFA and PUFA. Feeding fat increased plasma glucose-dependent insulinotropic polypeptide and CCK concentrations and decreased plasma insulin concentration. Plasma CCK concentration was greater for MUFA and PUFA than for SFA and was greater for MUFA than PUFA. Decreases in DMI in cows fed fat were associated with increased plasma concentrations of GLP-1 and CCK and a decreased insulin concentration. The role of these peptides in regulating DMI in cattle fed fat requires further investigation.