Quiver Gauge Theories: Finitude and Trichotomoty

D-brane probes, Hanany-Witten setups and geometrical engineering stand as a trichotomy of standard techniques of constructing gauge theories from string theory. Meanwhile, asymptotic freedom, conformality and IR freedom pose as a trichotomy of the beta-function behaviour in quantum field theories. Parallel thereto is a trichotomy in set theory of finite, tame and wild representation types. At the intersection of the above lies the theory of quivers. We briefly review some of the terminology standard to the physics and to the mathematics. Then, we utilise certain results from graph theory and axiomatic representation theory of path algebras to address physical issues such as the implication of graph additivity to finiteness of gauge theories, the impossibility of constructing completely IR free string orbifold theories and the unclassifiability of N < 2 Yang-Mills theories in four dimensions

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