A Special Homotopy Continuation Method For A Class of Polynomial Systems

A special homotopy continuation method, as a combination of the polyhedral homotopy and the linear product homotopy, is proposed for computing all the isolated solutions to a special class of polynomial systems. The root number bound of this method is between the total degree bound and the mixed volume bound and can be easily computed. The new algorithm has been implemented as a program called LPH using C++. Our experiments show its efficiency compared to the polyhedral or other homotopies on such systems. As an application, the algorithm can be used to find witness points on each connected component of a real variety

Which wets TiB2 inoculant particles: Al or Al3Ti?

TiB2 particles are proven effective nucleants of commercial purity aluminium, resulting in smaller grains and hence greater desired mechanical properties; however, there is uncertainty as to the mechanism by which it operates. Here we clarify what happens in the initial stages by computing the total Gibbs energy change associated with four possible nucleation mechanisms, each characterised by the termination of the TiB2(0001) substrate (Ti or B) and the solid that forms on it (Al or Al3Ti). The appropriate solid//solid interfacial energies are derived from Density Functional Theory (DFT) calculations, while the bulk energies are derived from thermodynamic data, supplemented with strain energies calculated from DFT. Solid//liquid interfacial energies are estimated using simple models with parameters based on the literature and DFT calculations. The results suggest that the Ti termination of TiB2 is more stable than the B termination in the melt, and that the direct formation of Al off a Ti-terminated TiB2 substrate is the most favourable mechanism for the nucleation of Al rather than the previously proposed formation of a Al3Ti interlayer. On the B termination of TiB2, Al formation is more stable for thick solid layers, but this is much more uncertain for thin solid layers where it is possible that Al3Ti formation is more stable

Classical and quantum calculations of the temperature dependence of the free energy of argon

The free energy is central to statistical mechanics and thermodynamics, and its accurate calculation via. computational modelling is important for a large number of applications, especially when its experimental value is hard to obtain. Several established and general methods for calculating the Helmholtz free energy across different length scales, including continuum, atomistic and quantum mechanical, are compared and analyzed. A computational approach is then proposed to calculate the temperature dependences of internal energy and absolute Helmholtz free energy for solid and liquid phases with the coupling of thermodynamic integration (TI) and harmonic approximation calculations from both classical molecular dynamics (MD) and density functional theory (DFT). We use the Lennard-Jones system as an example (i.e. argon) for the demonstration of the approach. It is observed that the free energy transits smoothly from being describable by the harmonic approximation to including anharmonic effects at a transition temperature around 0.56 Tm; below this temperature, the quantum behavior of atoms is important. At higher temperatures (T > 0.56 Tm), the TI and harmonic approximation results for the Helmholtz free energy functions become increasingly divergent with the increase of temperature. This work demonstrates that a multiscale approach employing TI, MD, and DFT can provide accurate calculations of the temperature dependence of absolute Helmholtz free energy for both solid and liquid phases

Emergent excitations in a geometrically frustrated magnet

Frustrated systems are ubiquitous and interesting because their behavior is difficult to predict. Magnetism offers extreme examples in the form of spin lattices where all interactions between spins cannot be simultaneously satisfied. Such geometrical frustration leads to macroscopic degeneracies, and offers the possibility of qualitatively new states of matter whose nature has yet to be fully understood. Here we have discovered how novel composite spin degrees of freedom can emerge from frustrated interactions in the cubic spinel ZnCr2O4. Upon cooling, groups of six spins self-organize into weakly interacting antiferromagnetic loops whose directors, defined as the unique direction along which the spins are aligned parallel or antiparallel, govern all low temperature dynamics. The experimental evidence comes from a measurement of the magnetic form factor by inelastic neutron scattering. While the data bears no resemblance to the atomic form factor for chromium, they are perfectly consistent with the form factor for hexagonal spin loop directors. The hexagon directors are to a first approximation decoupled from each other and hence their reorientations embody the long-sought local zero energy modes for the pyrochlore lattice.Comment: 10 pages, 4 figures upon reques

From Molecular Cores to Planet-forming Disks with SIRTF

The SIRTF mission and the Legacy programs will provide coherent data bases for extra-galactic and Galactic science that will rapidly become available to researchers through a public archive. The capabilities of SIRTF and the six legacy programs are described briefly. Then the cores to disks (c2d) program is described in more detail. The c2d program will use all three SIRTF instruments (IRAC, MIPS, and IRS) to observe sources from molecular cores to protoplanetary disks, with a wide range of cloud masses, stellar masses, and star-forming environments. The SIRTF data will stimulate many follow-up studies, both with SIRTF and with other instruments.Comment: 6 pages, from Fourth Cologne-Bonn-Zermatt-Symposium, The Dense Interstellar Matter in Galaxie

Computational algebraic methods in efficient estimation

A strong link between information geometry and algebraic statistics is made by investigating statistical manifolds which are algebraic varieties. In particular it it shown how first and second order efficient estimators can be constructed, such as bias corrected Maximum Likelihood and more general estimators, and for which the estimating equations are purely algebraic. In addition it is shown how Gr\"obner basis technology, which is at the heart of algebraic statistics, can be used to reduce the degrees of the terms in the estimating equations. This points the way to the feasible use, to find the estimators, of special methods for solving polynomial equations, such as homotopy continuation methods. Simple examples are given showing both equations and computations. *** The proof of Theorem 2 was corrected by the latest version. Some minor errors were also corrected.Comment: 21 pages, 5 figure

Farmer intentional pathways for net zero carbon: exploring the lock-in effects of forestry and renewables

Climate smart farming requires food production to sit alongside practices which sequester greenhouse gas emissions. Given the requirement to meet net zero emissions by the middle of the century, agricultural policies are now seeking to embed climate smart approaches within future support schemes. Path dependency, the influence of past choices on decision making, has been found to constrain future growth pathways. We apply this concept within a survey of 2494 farmers in Scotland to understand their intentions towards uptake of two prominent climate smart approaches, namely forestry expansion and on-farm renewable energy. We employ a bivariate probit model to estimate the single and joint dependences of these two activities within a farm decision making framework. Factors such as succession planning, the level of agricultural diversification and risk seeking perceptions were found to be positively related to influencing uptake. However, the strongest predictors for uptake were past expansion of these activities and, conversely, a limiting factor for those who did not intend to increase activities. This provides some evidence that path dependencies will limit large scale adoption to meet a net zero target. We argue for a dual approach to intervention which differentiates between past adopters and those who are reluctant to adopt. More targetted support for these two cohorts would address these high level policy ambitions.</p

Signal peptide peptidases and gamma-secretase: Cousins of the same protease family?

Signal peptide peptidase (SPIP) is an unusual aspartyl protease, which mediates clearance of signal peptides by proteolysis within the endoplasmic reticulum (ER). Like presenilins, which provide the proteolytically active subunit of the,gamma-secretase complex, SPP contains a conserved GxGD motif in its C-terminal domain which is critical for its activity. While SPIP is known to be an aspartyl protease of the GxGD type, several presenilin homologues/SPP-like proteins (PSHs/ SPPL) of unknown function have been identified by database searches. In contrast to SPP and SPPL3, which are both restricted to the endoplasmic reticulum, SPPL2b is targeted through the secretory pathway to endosomes/lysosomes. As suggested by the differential subcellular localization of SPPL2b and SPPL3 distinct phenotypes were found upon antisense gripNA-mediated knockdown in zebrafish. spp and sppl3 knockdowns in zebrafish result in cell death within the central nervous system, whereas reduction of sppl2b expression causes erythrocyte accumulation in an enlarged caudal vein. Moreover, expression of D/A mutants of the putative C-terminal active sites of spp, sppl2, and spp13 produced phenocopies of the respective knockdown phenotypes. These data suggest that all investigated PSHs/SPPLs are members of the novel family of GxGD aspartyl proteases. More recently, it was shown that SPPL2b utilizes multiple intramembrane cleavages to liberate the TNF(x intracellular domain into the cytosol and to release the C-terminal counterpart into the lumen. These findings suggest common principles of intramembrane proteolysis by GxGD type aspartyl proteases. In this article,we will review the similarities of SPPs and gamma-secretase based on recent findings by us and others