Symbolic homotopy construction

The classical Theorem of Bézout yields an upper bound for the number of finite solutions to a given polynomial system, but is very often too large to be useful for the construction of a start system, for the solution of a polynomial system by means of homotopy continuation. The BKK bound gives a much lower upper bound for the number of solutions, but unfortunately, constructing a start system based on this bound seems as hard as solving the original given polynomial system. This paper presents a way for computing an upper bound together with the construction of a start system. The first computation is performed symbolically. Due to this symbolic computation, the constructed start system can be solved numerically more efficiently. The paper generalizes current approaches for homotopy construction towards the BKK bound.     

Computing dynamic output feedback laws

The pole placement problem asks to find laws to feed the output of a plant governed by a linear system of differential equations back to the input of the plant so that the resulting closed-loop system has a desired set of eigenvalues. Converting this problem into a question of enumerative geometry, efficient numerical homotopy algorithms to solve this problem for general multiple-input-multiple-output systems have been proposed recently. Despite the wider application range of dynamic feedback laws, the realization of the output of the numerical homotopies as a machine to control the plant in the time domain has not been addressed before. In this note, we present symbolic-numeric algorithms to turn the solution to the question of enumerative geometry into a useful control feedback machine. We report on numerical experiments with our publicly available software PHCpack and illustrate its application on various control problems from the literature.     

Characterization of functional residues in the interfacial recognition domain of lecithin cholesterol acyltransferase (LCAT)

Lecithin cholesterol acyltransferase (LCAT) is an interfacialenzyme active on both high-density (HDL) and low-density lipoproteins(LDL). Threading alignments of LCAT with lipases suggest thatresidues 50–74 form an interfacial recognition site andthis hypothesis was tested by site-directed mutagenesis. The(56–68) deletion mutant had no activity on any substrate.Substitution of W61 with F, Y, L or G suggested that an aromaticresidue is required for full enzymatic activity. The activityof the W61F and W61Y mutants was retained on HDL but decreasedon LDL, possibly owing to impaired accessibility to the LDLlipid substrate. The decreased activity of the single R52A andK53A mutants on HDL and LDL and the severer effect of the doublemutation suggested that these conserved residues contributeto the folding of the LCAT lid. The membrane-destabilizing propertiesof the LCAT 56–68 helical segment were demonstrated usingthe corresponding synthetic peptide. An M65N–N66M substitutiondecreased both the fusogenic properties of the peptide and theactivity of the mutant enzyme on all substrates. These resultssuggest that the putative interfacial recognition domain ofLCAT plays an important role in regulating the interaction ofthe enzyme with its organized lipoprotein substrates.     

Intra- and inter-annual variation of Cd, Zn, Mn and Cu in foliage of poplars on contaminated soil

The uptake of trace metals in the leaves of fast-growing woody species is a crucial factor in ecological risk assessment and in the evaluation of phytoextraction potentials. In this study, we present a long-term data series of foliar Cd, Zn, Mn and Cu concentrations in poplar (Populus trichocarpa x P. deltoides). Leaves were collected every three weeks from 2001 until 2007 on three sites, (i) a new plantation on an alluvial soil polluted by river sediments, (ii) a new plantation on an unpolluted soil and (iii) a 10-year old plantation on a polluted dredged sediment soil. In addition, tree rings were measured on the alluvial soil in order to better assess growth over the past seven years. Foliar concentrations of Cd, Zn and Mn decreased considerably with time in the new plantation on polluted soil. Concentrations of Zn and Mn decreased in the new plantation on unpolluted soil as well. The older plantation on polluted soil did not show changes in foliar concentrations for Cd, Zn or Mn. Foliar Cu concentrations slightly increased for all sites. Within one growing season, foliar concentrations of Cd, Zn, Cu and Mn increased towards the end of the season. The tree ring data of the poplars on the alluvial soil indicated a strong decrease in growth due to declining tree condition from 2005 onwards, the same year that foliar Cd and Zn concentrations markedly decreased. Lower transpiration rates probably induced a lower uptake of dissolved trace metals. It is concluded that stand health and growth rate have a strong impact on the variation of foliar trace metal concentrations over time.     

Tropical algebraic geometry in Maple: A preprocessing algorithm for finding common factors for multivariate polynomials with approximate coefficients

Finding a common factor of two multivariate polynomials with approximate coefficients is a problem in symbolic-numeric computing. Taking a tropical view of this problem leads to efficient preprocessing techniques, applying polyhedral methods to the exact exponents and numerical techniques to the approximate coefficients. With Maple we will illustrate our use of tropical algebraic geometry.