On Hadamard Product of Hypercomplex Numbers

Certain product rules take various forms in the set of hypercomplex numbers. In this paper, we introduce a new multiplication form of the hypercomplex numbers that will be called «the Hadamard product», inspired by the analogous product in the real matrix space, and investigate some algebraic properties of that, including the norm of inequality. In particular, we extend our new definition and its applications to the complex matrix theory.

Product rules and classification of nonassociative Moufang loops of order 81

In this paper, we produce the product rules of nonassociative Moufang loops of order 81 by using an analytical approach. We then explore all possible presentations on a suitable set of generators, thereby obtaining a total of five nonisomorphic cases. The result is in agreement with the classification by Nagy and Vojtěchovský using the GAP package LOOPS .

Littlewood–Richardson coefficients for Grothendieck polynomials from integrability

We study the Littlewood–Richardson coefficients of double Grothendieck polynomials indexed by Grassmannian permutations. Geometrically, these are the structure constants of the equivariant K-theory ring of Grassmannians. Representing the double Grothendieck polynomials as partition functions of an integrable vertex model, we use its Yang–Baxter equation to derive a series of product rules for the former polynomials and their duals. The Littlewood–Richardson coefficients that arise can all be expressed in terms of puzzles without gashes, which generalize previous puzzles obtained by Knutson–Tao and Vakil.

Microorganisms in Biostimulants

In March 2016, the EU Commission presented a proposal for new regulations on fertilising material. The regulation includes product rules for a wide range of organic and inorganic products. Microbial biostimulants is one of the categories of products that are included. Biostimulants, in the draft EU regulation, are defined as fertilising materials that affect nutrient processes independently of the product's own nutrient content and with the purpose of improving nutrient utilisation, tolerance for abiotic stress or quality of the crop. Positive list in which species of these bacterial genera are listed: Azotobacter spp, Rhizobium spp., Azospirillum spp and Mycorrhizal fungi are a part of the regulation. Since the import and use of these organisms are the responsibility of both the Norwegian Food Safety Authority and the Norwegian Environment Agency, they asked VKM to submit a joint report on effects on health (humans, plants and animals), biodiversity and dispersal, quality of agricultural land and on soil environment. Conclusions: Health risks: Based upon our literature review, we have found no indication of any specific diseases in plants, animals or humans induced by the discussed microorganisms. A few reported cases of human disease are caused through wound infections or injections in immunocompromised patients. These represent a situation where any microorganism may induce infections and is not specific for the agents discussed in this report. In summary, the risk of any disease caused by the discussed microorganisms is considered negligible. Environmental risks: In soil the biodiversity, competition, adaptation and functional redundancy of microorganisms are extremely high. This means that introduced microorganisms have a very small chance for establishing, and even less so for affecting biodiversity and soil functioning. Introduction of nitrogen fixing species or fungi that can transport P to plants (mycorrhiza) will lead to an increase in the primary production. However, even a large increased activity for these processes will not outcompete naturally occurring symbiotic N-fixation or growth of inherently non-mycorrhizal plant species. Thus, the risks associated with introduced non-pathogenic microorganisms are very low.

Some Applications of Fractional Velocities

Fractional velocity is defined as the limit of the difference quotient of the increments of a function and its argument raised to a fractional power. The fractional velocity can be suitable for characterizing singular behavior of derivatives of Hölderian functions and non differentiable functions. Relations to integer-order derivatives and other integral-based definitions are discussed.It is demonstrated that for Hölder functions under certain conditions the product rules deviates from the Leibniz rule. This deviation is expressed by another quantity, fractional co-variation.