Resolutions of subsets of finite sets of points in projective space

Abstract

Eisenbud et al. proved a number of results regarding Gröbner bases and initial ideals of those ideals J in the free associative algebra K ?X ₁,…, X _n ? which contain the commutator ideal. We prove similar results for ideals which contains the anti-commutator ideal (the defining ideal of the exterior algebra). We define one weak notion of generic initial ideals in K ?X ₁,…, X _n ?, and show that generic initial ideals of ideals containing the anti-commutator ideal, or the commutator ideal, are finitely generated.     

Half Space and Quarter Space Dirichlet Problems for the Partial Differential Equation

A customary, heuristic, method, by which the Poisson integral formula for the Dirichlet problem, for the half space, for Laplace's equation is obtained, involves Green's function, and Kelvin's method of images. Although this heuristic method leads one to guess the correct result, this Poisson formula still has to be verified directly, independently of the method by which it was arrived at, in order to be absolutely certain that a solution of the Dirichlet problem for the half space, for Laplace's equation, has been actually obtained. A similar heuristic method, as seems to be generally known, could be followed in solving the Dirichlet problem, for the half space, for the equation where is a real constant. However, in Part 1, a different, labor-saving, method is used to study Dirichlet problems for the equation. This method is essentially based on what Hadamard called the method of descent. Indeed, it is shown that he who has solved the half space Dirichlet problem for Laplace's equation has already solved the half space Dirichlet problem for the equation In Part 2, the solution formula for the quarter space Dirichlet problem for Laplace's equation is obtained from the Poisson integral formula for the half space Dirichlet problem for Laplace's equation. A representation theorem for harmonic functions in the quarter space is deduced. The method of descent is used, in Part 3, to obtain the solution formula for the quarter space Dirichlet problem for the equation by means of the solution formula for the quarter space Dirichlet problem for Laplace's equation. So that, indeed, it is also shown that he who has solved the quarter space Dirichlet problem for Laplace's equation has already solved the quarter space Dirichlet problem for the " equation" For the sake of completeness and clarity, and for the convenience of the reader, the appendix, at the end of Part 3, contains a detailed proof that the Poisson integral formula solves the half space Dirichlet problem for Laplace's equation. The Bibliography for Parts 1,2, 3 is to be found at the end of Part 1.     

Development and characterisation of an immunoaffinity chromatographic column for the on-line determination of the pesticide triclopyr

An immunoaffinity column for the selective extraction and concentration of the herbicide triclopyr from water samples and quantification on line by HPLC has been developed. The immunoaffinity device was prepared by immobilising triclopyr antibodies to hydrazide derivatized azlactona beds. Efficient desorption of bound triclopyr was achieved with 70% ethanol/water solution. The column was evaluated regarding selectivity, recovery, capacity, saturation volume and reusability. Obtained results show that immunoaffinity chromatography (IAC) can be used for quantitative extraction, concentration and determination of triclopyr from water samples.     

General theory of domain decomposition: Indirect methods

According to a general theory of domain decomposition methods (DDM), recently proposed by Herrera, DDM may be classified into two broad categories: direct and indirect (or Trefftz‐Herrera methods). This article is devoted to formulate systematically indirect methods and apply them to differential equations in several dimensions. They have interest since they subsume some of the best‐known formulations of domain decomposition methods, such as those based on the application of Steklov‐Poincaré operators. Trefftz‐Herrera approach is based on a special kind of Green's formulas applicable to discontinuous functions, and one of their essential features is the use of weighting functions which yield information, about the sought solution, at the internal boundary of the domain decomposition exclusively. A special class of Sobolev spaces is introduced in which boundary value problems with prescribed jumps at the internal boundary are formulated. Green's formulas applicable in such Sobolev spaces, which contain discontinuous functions, are established and from them the general framework for indirect methods is derived. Guidelines for the construction of the special kind of test functions are then supplied and, as an illustration, the method is applied to elliptic problems in several dimensions. A nonstandard method of collocation is derived in this manner, which possesses significant advantages over more standard procedures. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 296–322, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10008     

Alkalimetalle

Ohne Zusammenfassung     

The Molecular Weaponry Produced by the Bacterium Hafnia alvei in Foods

Hafnia alvei is receiving increasing attention from both a medical and veterinary point of view, but the diversity of molecules it produces has made the interest in this bacterium extend to the field of probiotics, the microbiota, and above all, to its presence and action on consumer foods. The production of Acyl Homoserine Lactones (AHLs), a type of quorum-sensing (QS) signaling molecule, is the most often-studied chemical signaling molecule in Gram-negative bacteria. H. alvei can use this communication mechanism to promote the expression of certain enzymatic activities in fermented foods, where this bacterium is frequently present. H. alvei also produces a series of molecules involved in the modification of the organoleptic properties of different products, especially cheeses, where it shares space with other microorganisms. Although some strains of this species are implicated in infections in humans, many produce antibacterial compounds, such as bacteriocins, that inhibit the growth of true pathogens, so the characterization of these molecules could be very interesting from the point of view of clinical medicine and the food industry. Lastly, in some cases, H. alvei is responsible for the production of biogenic amines or other compounds of special interest in food health. In this article, we will review the most interesting molecules that produce the H. alvei strains and will discuss some of their properties, both from the point of view of their biological activity on other microorganisms and the properties of different food matrices in which this bacterium usually thrives.