On the Extension of Complex Numbers

This paper proposes an extension of the complex numbers, adding further imaginary units and preserving the idea of the product as a geometric construction. These `supercomplex numbers\u27, denoted S, are studied, and it is found that the algebra contains both new and old phenomena. It is established that equal-dimensional subspaces of S containing R are isomorphic under algebraic operations, whereby a symmetry within the space of imaginary units is illuminated. Certain equations are studied, and also a connection to special relativity is set up and explored. Finally, abstraction leads to the notion of a `generalised supercomplex algebra\u27; both the supercomplex numbers and the quaternions are found to be such algebras

A Mathematical Framework for Causally Structured Dilations and its Relation to Quantum Self-Testing

The motivation for this thesis was to recast quantum self-testing [MY98,MY04] in operational terms. The result is a category-theoretic framework for discussing the following general question: How do different implementations of the same input-output process compare to each other? In the proposed framework, an input-output process is modelled by a causally structured channel in some fixed theory, and its implementations are modelled by causally structured dilations formalising hidden side-computations. These dilations compare through a pre-order formalising relative strength of side-computations. Chapter 1 reviews a mathematical model for physical theories as semicartesian symmetric monoidal categories. Many concrete examples are discussed, in particular quantum and classical information theory. The key feature is that the model facilitates the notion of dilations. Chapter 2 is devoted to the study of dilations. It introduces a handful of simple yet potent axioms about dilations, one of which (resembling the Purification Postulate [CDP10]) entails a duality theorem encompassing a large number of classic no-go results for quantum theory. Chapter 3 considers metric structure on physical theories, introducing in particular a new metric for quantum channels, the purified diamond distance, which generalises the purified distance [TCR10,Tom12] and relates to the Bures distance [KSW08a]. Chapter 4 presents a category-theoretic formalism for causality in terms of '(constructible) causal channels' and 'contractions'. It simplifies aspects of the formalisms [CDP09,KU17] and relates to traces in monoidal categories [JSV96]. The formalism allows for the definition of 'causal dilations' and the establishment of a non-trivial theory of such dilations. Chapter 5 realises quantum self-testing from the perspective of chapter 4, thus pointing towards the first known operational foundation for self-testing.Comment: PhD thesis submitted to the University of Copenhagen (ISBN 978-87-7125-039-8). Advised by prof. Matthias Christandl, submitted 1st of December 2020, defended 11th of February 2021. Keywords: dilations, applied category theory, quantum foundations, causal structure, quantum self-testing. 242 pages, 1 figure. Comments are welcom

Dilations and information flow axioms in categorical probability

We study the positivity and causality axioms for Markov categories as properties of dilations and information flow in Markov categories, and in variations thereof for arbitrary semicartesian monoidal categories. These help us show that being a positive Markov category is merely an additional property of a symmetric monoidal category (rather than extra structure). We also characterize the positivity of representable Markov categories and prove that causality implies positivity, but not conversely. Finally, we note that positivity fails for quasi-Borel spaces and interpret this failure as a privacy property of probabilistic name generation.Comment: 42 page

Harmonic oscillator model of the insulin and IGF1 receptors' allosteric binding and activation

The insulin and insulin-like growth factor 1 receptors activate overlapping signalling pathways that are critical for growth, metabolism, survival and longevity. Their mechanism of ligand binding and activation displays complex allosteric properties, which no mathematical model has been able to account for. Modelling these receptors' binding and activation in terms of interactions between the molecular components is problematical due to many unknown biochemical and structural details. Moreover, substantial combinatorial complexity originating from multivalent ligand binding further complicates the problem. On the basis of the available structural and biochemical information, we develop a physically plausible model of the receptor binding and activation, which is based on the concept of a harmonic oscillator. Modelling a network of interactions among all possible receptor intermediaries arising in the context of the model (35, for the insulin receptor) accurately reproduces for the first time all the kinetic properties of the receptor, and provides unique and robust estimates of the kinetic parameters. The harmonic oscillator model may be adaptable for many other dimeric/dimerizing receptor tyrosine kinases, cytokine receptors and G-protein-coupled receptors where ligand crosslinking occurs

A Comparative Structural Bioinformatics Analysis of the Insulin Receptor Family Ectodomain Based on Phylogenetic Information

The insulin receptor (IR), the insulin-like growth factor 1 receptor (IGF1R) and the insulin receptor-related receptor (IRR) are covalently-linked homodimers made up of several structural domains. The molecular mechanism of ligand binding to the ectodomain of these receptors and the resulting activation of their tyrosine kinase domain is still not well understood. We have carried out an amino acid residue conservation analysis in order to reconstruct the phylogeny of the IR Family. We have confirmed the location of ligand binding site 1 of the IGF1R and IR. Importantly, we have also predicted the likely location of the insulin binding site 2 on the surface of the fibronectin type III domains of the IR. An evolutionary conserved surface on the second leucine-rich domain that may interact with the ligand could not be detected. We suggest a possible mechanical trigger of the activation of the IR that involves a slight ‘twist’ rotation of the last two fibronectin type III domains in order to face the likely location of insulin. Finally, a strong selective pressure was found amongst the IRR orthologous sequences, suggesting that this orphan receptor has a yet unknown physiological role which may be conserved from amphibians to mammals

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