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For real application and theoretical investigation of ordinary hypergraphs and non-ordinary hypergraphs, researchers need to establish standard rules and feasible operating methods. We propose a visualization tool for investigating…

Wedderburn's theorem on the structure of finite dimensional semisimple algebras is proved by using minimal prerequisites.

Sequences that are defined by multisums of hypergeometric terms with compact support occur frequently in enumeration problems of combinatorics, algebraic geometry and perturbative quantum field theory. The standard recipe to study the…

Orbit-finite models of computation generalise the standard models of computation, to allow computation over infinite objects that are finite up to symmetries on atoms, denoted by $\mathbb{A}$. Set theory with atoms is used to reason about…

In the course of classifying the homogeneous permutations, Cameron introduced the viewpoint of permutations as structures in a language of two linear orders, and this structural viewpoint is taken up here. The majority of this thesis is…

In this paper, we prove a theorem that gives a simple criterion for generating commuting pairs of generalized almost complex structures on spaces that are the product of two generalized almost contact metric spaces. We examine the…

We use semi--classical and perturbation methods to establish the quantum theory of the Neumann model, and explain the features observed in previous numerical computations.

Let $X$ be a toric $\mbox{hyperK\"ahler}$ manifold. The purpose of this note is to describe the topological $K$-ring $K^*(X)$ of $X$. We give a presentation for the topological $K$-ring in terms of generators and relations similar to the…

We use automated theorem provers to significantly shorten a formal development in higher order set theory. The development includes many standard theorems such as the fundamental theorem of arithmetic and irrationality of square root of…

In some recent theories including Quantum SuperString theory we encounter duality - it arises due to a non commutative geometry which in effect adds an extra term to the Heiserberg Uncertainity Principle. The result is that the micro world…

We construct two combinatorially equivalent line arrangements in the complex projective plane such that the fundamental groups of their complements are not isomorphic. The proof uses a new invariant of the fundamental group of the…

We provide a framework to classify hyperbolic monopoles with continuous symmetries and find a Structure Theorem, greatly simplifying the construction of all those with spherically symmetry. In doing so, we reduce the problem of finding…

Subatomic systems were recently introduced to identify the structural principles underpinning the normalization of proofs. "Subatomic" means that we can reformulate logical systems in accordance with two principles. Their atomic formulas…

We compare three notions of genericity of separable metric structures. Our analysis provides a general model theoretic technique of showing that structures are generic in descriptive set theoretic (topological) sense and in measure…

In this project, we will develop the foundations of quantum mechanics using the methods of supersymmetry. We will discuss the use of the superpotential to derive the supersymmetric partner of a potential in one dimension, and explore…

We survey some of the main conceptual developments in the study of PT-symmetric and pseudo-Hermitian Hamiltonian operators that have taken place during the past ten years or so. We offer a precise mathematical description of a quantum…

In this paper we use the strength of the constraint method in combination with a generalized Borsuk-Ulam type theorem and a cohomological intersection lemma to show how one can obtain many new topological transversal theorems of Tverberg…

We prove that if two additive functions (from a certain class) take large values with roughly the same probability then they must be identical. This is a consequence of a structure theorem making clear the inter-relation between the…

We generalize the asymptotic shape theorem in first passage percolation on $\mathbb{Z}^d$ to cover the case of general semimetrics. We prove a structure theorem for equivariant semimetrics on topological groups and an extended version of…

In this paper, we prove a structure theorem for the infinite union of $n$-adic doubling measures via techniques which involve far numbers. Our approach extends the results of Wu in 1998, and as a by product, we also prove a classification…