Polynomial Roots and Calabi-Yau Geometries

The examination of roots of constrained polynomials dates back at least to Waring and to Littlewood. However, such delicate structures as fractals and holes have only recently been found. We study the space of roots to certain integer polynomials arising naturally in the context of Calabi-Yau spaces, notably Poincare and Newton polynomials, and observe various salient features and geometrical patterns.Comment: 22 pages, 13 Figure

An Etude on Recursion Relations and Triangulations

Following~\cite{Arkani-Hamed:2017thz}, we derive a recursion relation by applying a one-parameter deformation of kinematic variables for tree-level scattering amplitudes in bi-adjoint $\phi^3$ theory. The recursion relies on properties of the amplitude that can be made manifest in the underlying kinematic associahedron, and it provides triangulations for the latter. Furthermore, we solve the recursion relation and present all-multiplicity results for the amplitude: by reformulating the associahedron in terms of its vertices, it is given explicitly as a sum of "volume" of simplicies for any triangulation, which is an analogy of BCFW representation/triangulation of amplituhedron for ${\cal N}=4$ SYM.Comment: 26 pages, 3 figure

Marketing development strategies to attract domestic customers

The purpose of this research is to improve the sales of a product in the New Zealand domestic market. The product is an eco-friendly way to deal with the problem of insects, including flies and mosquitos. It is a traditional Chinese product which is well known by Chinese and widely accepted in China and other counties like Australia and the United States of America. It has a potential market in New Zealand. The method of this research is based on the Ansoff matrix, and use of quantitative data. Sixty people participated in the questionnaire. The result of the survey shows that most New Zealanders (78%) have trouble with insects and 91% of participants would like to try an eco-friendlier way to deal with this problem rather than use insect spray. Most of the participants care about the price and quality of the product. This research will provide valuable information regarding the habit of domestic customers, recommendations for increasing sales, such as adverts and focus on price and quality, and creating a CRM system

Elements with finite Coxeter part in an affine Weyl group

Let $W_a$ be an affine Weyl group and $\eta:W_a\longrightarrow W_0$ be the natural projection to the corresponding finite Weyl group. We say that $w\in W_a$ has finite Coxeter part if $\eta(w)$ is conjugate to a Coxeter element of $W_0$. The elements with finite Coxeter part is a union of conjugacy classes of $W_a$. We show that for each conjugacy class $\mathcal{O}$ of $W_a$ with finite Coxeter part there exits a unique maximal proper parabolic subgroup $W_J$ of $W_a$, such that the set of minimal length elements in $\mathcal{O}$ is exactly the set of Coxeter elements in $W_J$. Similar results hold for twisted conjugacy classes.Comment: 9 page