This paper aims at presenting the necessary tools to prove that a scheme of finite type over Z exhibits the same singularities as those which occur on a Grassmann variety. First, basic theory regarding the combinatorial objects matroids is presented. Some important examples for the remainder of the paper are given, which also serve to aid the reader in intuition and understanding of matroids. Basic algebraic geometry is presented, and the building blocks affine varieties, projective varieties and general varieties are introduced. These object are generalised in the following subsection as affine schemes and schemes, which are the central object of study in modern algebraic geometry. Important results from the theory of algebraic groups are shown in order to better understand the formulation and proof of the Gelfand–MacPherson theorem, which in turn is utilised, together with Mnëv’s universality theorem, to prove the main result of the paper.

The aim of this thesis is to examine the connection between the Riemannhypothesis and the distribution of prime numbers. We first derive theanalytic continuation of the zeta function and prove some of its propertiesusing a functional equation. Results from complex analysis such asJensen’s formula and Hadamard factorization are introduced to facilitatea deeper investigation of the zeros of the zeta function. Subsequently, therelation between these zeros and the asymptotic distribution of primesis rendered explicit: they determine the error term when the prime-counting function π(x) is approximated by the logarithmic integral li(x).We show that this absolute error is O(x exp(−c√log x) ) and that the Riemannhypothesis implies the significantly improved upper bound O(√x log x).

Building a large library of mathematical knowledge is a complex and labour-intensive task. By examining current libraries of mathematics, we see that the human effort put in building them is not entirely directed towards tasks that need human creativity. Instead, a non-trivial amount of work is spent on providing definitions that could have been mechanically derived. In this work, we propose a generative approach to library building, so definitions that can be automatically derived are computed by meta-programs. We focus our attention on libraries of algebraic structures, like monoids, groups, and rings. These structures are highly inter-related and their commonalities have been well-studied in universal algebra. We use theory presentation combinators to build a library of algebraic structures. Definitions from universal algebra and programming languages meta-theory are used to derive library definitions of constructions, like homomorphisms and term languages, from algebraic theory presentations. The result is an interpreter that, given 227 theory expressions, builds a library of over 5000 definitions. This library is, then, exported to Agda and Lean. / Dissertation / Doctor of Philosophy (PhD)

The well-known Eneström–Kakeya Theorem states that a polynomial with real, nonnegative, monotone increasing coefficients has all its complex zeros in the closed unit disk in the complex plane. In this paper, we extend this result by showing that all quaternionic zeros of such a polynomial lie in the unit sphere in the quaternions. We also extend related results from the complex to quaternionic setting.

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Many unique attributes of quantum cryptography arise from the no-cloning property of quantum information. We study this using two closely-related types of uncloneability game: no-cloning and monogamy-of-entanglement games. In a no-cloning game, a referee sends a quantum state encoding classical information to two cooperating players who split the state, then try simultaneously guessing the information, provided the key. In a monogamy-of-entanglement game, two cooperating players try to guess the referee's measurement result on a tripartite state the players prepared. In this work, we prove winning probability bounds on no-cloning games based on coset states, which have the interesting property that the players guess two different strings. We also show a rigidity property for the original monogamy-of-entanglement game, letting it be used as a test of separability. Finally, we apply these properties to construct a variety of novel cryptographic protocols for uncloneable encryption, quantum key distribution, bit commitment, and randomness expansion.

We study the Lyapunov exponents and their associated invariant subspaces for infinite dimensional random dynamical systems in a Banach space, which are generated by, for example, stochastic or random partial differential equations. We prove a multiplicative ergodic theorem. Then, we use this theorem to establish the stable and unstable manifold theorem for nonuniformly hyperbolic random invariant sets.

Let M_k^2 be the complete, simply connected, Riemannian 2-manifold of constant curvature k ± 0. Let E be a closed, simply connected subspace of M_k^2 with the property that every two points in E are connected by a rectifi able path in E. We show that E is CAT(k) under the induced path metric.

The Hellmann-Feynman theorem is presented together with certain allied theorems. The origin of the Hellmann-Feynman theorem in quantum physical chemistry is described. The theorem is stated with proof and with discussion of applicability and reliability. Some adaptations of the theorem to the study of the variation of zeros of special functions and orthogonal polynomials are surveyed. Possible extensions are discussed.

This thesis highlights a method of arriving at form that minimizes the need for high end technology and complex mathematical models, yet has structural principles of load flow at the highlighted methods core. Similar to how graphical statics assisted earlier architects and engineers to arrive at form by relating form and forces, this thesis suggests a method of form finding that relates the flow of stresses within solid masses to possible load-bearing façades. Looking to nature, where an abundance of efficient structural solutions can be found, this thesis focuses on a tree-like structural form called the dendriform. In doing so, this thesis explores the idea that through an understanding of typical load flow patterns and the removal of minimally stressed material of the solid body, dendriforms can be revealed that qualitatively exemplify load flow yet maintain an architectural aesthetic.