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Ubiquitous computing systems and the digital economy
Published in James Juniper, The Economic Philosophy of the Internet of Things, 2018
Category theory provides ubiquitous computing systems with a variety of formal approaches including the coalgebraic representation of automatons and transition systems, Domain theory, and the Geometry of Interaction, along with specific mathematical substrates such as the elementary topos. For its part, categorical logic links inferential procedures based on resource-using or linear logics with functional programming and algebraic topology, while string diagrams can represent everything from graphical linear algebra to signal flow graphs and topological quantum field theory (with the latter using string diagrams to account for the functional relationships holding between quantum phenomena and the cobordisms deriving from general relativity theory).
Interface Superconductivity
Published in David A. Cardwell, David C. Larbalestier, I. Braginski Aleksander, Handbook of Superconductivity, 2023
A final comment on these considerations of the statistical mechanics of two-dimensional superconductors follows from the analysis of Ref. [35]. If one includes longitudinal and transverse fluctuations of the electromagnetic field, the corresponding quantum Ginzburg–Landau theory becomes a genuine gauge theory. This implies that the usual order parameter is not a gauge-invariant object and should not have physical meaning in the sense of an observable. In this regime, one can demonstrate that the proper description of a superconductor is in terms of a topological quantum field theory [35]. A superconductor, even a s-wave pairing state as it occurs in conventional materials, is in fact a state of topological order. This behavior is closely connected to the fact that Bogoliubov quasiparticles of charged superconductors are truly fractionalized degrees of freedom [36]. This topological order is in many ways a more fundamental topological state than what is often discussed in the context of topological superconductivity with Majorana edge states [11]. The topological quantum field theory allows to link the degeneracy of the many-body ground state to the genus of the manifold on which the superconductor exists [35] and offers a fundamental explanation for the known degeneracy of the spectrum of superconducting rings with external flux [37]. Despite these subtle phenomena it is reassuring that one can still employ the Ginzburg–Landau formalism to draw conclusions about observable quantities. Non-local, yet gauge-invariant observables of a superconductor can, under a specific gauge choice, be directly related to the results that follow the usual analysis of the Ginzburg–Landau theory, see Ref. [35]. Thus, if properly interpreted, a Ginzburg–Landau formalism of two-dimensional superconductors continues to be possible despite the exotic topological nature of the superconducting state.
The knowledge of knots: an interdisciplinary literature review
Published in Spatial Cognition & Computation, 2019
Paulo E. Santos, Pedro Cabalar, Roberto Casati
Topological knot theory has far-reaching applications in various fields. Detailed descriptions of these applications have been the subject of various books in the past (Buck & Flapan, 2009; Kauffman, 2013; Murasugi, 1996b). Here we present a few recent examples of key applications of knot theory in the natural sciences. The importance of knot theory in chemistry has become apparent recently as knot topologies have been synthesized in chemical compounds. The physical constraints imposed by molecular knotting have impacted on molecular properties such as chirality, ion binding, and catalytic activity (Fielden, Leigh & Woltering, 2017). In molecular biology, knot theory has been used to describe the loops and links present in DNA chains and to help understanding how cells execute DNA detangling efficiently (Moore, Vazquez & Lidman, 2019; Stolz et al., 2017). In theoretical physics, the orbit of a charged particle in spacetime can be modeled as a knot. Thus, the knot invariants are used in the field of topological quantum field theory to predict trajectories of particles in four spacetime dimensions (Kauffman, 2005; Witten, 2011).
On the mathematics of higher structures
Published in International Journal of General Systems, 2019
For state assignments there is a plethora of new possibilities, extending assignments in topological quantum field theory (TQFT). In such a level structure (hyperstructure) of states represents the local states associated with the lowest level bonds , and represents the global states associated with the top bonds .
On the philosophy of higher structures
Published in International Journal of General Systems, 2019
If a tensor product may be provided in we may have and sometimes when it makes sense, , like in topological quantum field theory: See also Baas (2013b).