Fault-tolerance in metric dimension of boron nanotubes lattices

The concept of resolving set and metric basis has been very successful because of multi-purpose applications both in computer and mathematical sciences. A system in which failure of any single unit, another chain of units not containing the faulty unit can replace the originally used chain is called a fault-tolerant self-stable system. Recent research studies reveal that the problem of finding metric dimension is NP-hard for general graphs and the problem of computing the exact values of fault-tolerant metric dimension seems to be even harder although some bounds can be computed rather easily. In this article, we compute closed formulas for the fault-tolerant metric dimension of lattices of two types of boron nanotubes, namely triangular and alpha boron. These lattices are formed by cutting the tubes vertically. We conclude that both tubes have constant fault tolerance metric dimension 4

Felt_space infrastructure: Hyper vigilant spatiality to valence the visceral dimension

Felt_space infrastructure: Hypervigilant spatiality to valence the visceral dimension. This thesis evolves perception as a hypothesis to reframe architectural praxis negotiated through agent-situation interaction. The research questions the geometric principles of architectural ordination to originate the ‘felt_space infrastructure’, a relational system of measurement concerned with the role of perception in mediating sensory space and the cognised environment. The methodological model for this research fuses perception and environmental stimuli, into a consistent generative process that penetrates the inner essence of space, to reveal the visceral parameter. These concepts are applied to develop a ‘coefficient of affordance’ typology, ‘hypervigilant’ tool set, and ‘cognitive_tope’ design methodology. Thus, by extending the architectural platform to consider perception as a design parameter, the thesis interprets the ‘inference schema’ as an instructional model to coordinate the acquisition of spatial reality through tensional and counter-tensional feedback dynamics. Three site-responsive case studies are used to advance the thesis. The first case study is descriptive and develops a typology of situated cognition to extend the ‘granularity’ of perceptual sensitisation (i.e. a fine-grained means of perceiving space). The second project is relational and questions how mapping can coordinate perceptual, cognitive and associative attention, as a ‘multi-webbed vector field’ comprised of attractors and deformations within a viewer-centred gravitational space. The third case study is causal, and demonstrates how a transactional-biased schema can generate, amplify and attenuate perceptual misalignment, thus triggering a visceral niche. The significance of the research is that it progresses generative perception as an additional variable for spatial practice, and promotes transactional methodologies to gain enhanced modes of spatial acuity to extend the repertoire of architectural practice

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

Decomposition of the unitary representation of SU(1,1) on the unit disk into irreducible components

In this thesis, we decompose the representation of SU(1,1) on the unit disk into ir reducible components. We start with the decomposition over the maximal compact subgroup K, we identify the modules of eigenfunctions which are square integrable with respect to the quasi invariant measure on the unit disk. These modules rep resent the discrete series representations. Then, we use the induction in stages method to find the principal series representation. The matrix coefficient with the principal series and a K-invariant vector turns to be an important function which is called a spherical function. There is a nice function (Harish Chandra’s function) controlling the decay of the spherical function at infinity. Finally, we use a new approach to find the inversion formula which is equivalent to decomposition into irreducible representations using the geometry of cycles with dual numbers and the covariant transform

Unconventional computing platforms and nature-inspired methods for solving hard optimisation problems

The search for novel hardware beyond the traditional von Neumann architecture has given rise to a modern area of unconventional computing requiring the efforts of mathematicians, physicists and engineers. Many analogue physical systems, including networks of nonlinear oscillators, lasers, condensates, and superconducting qubits, are proposed and realised to address challenging computational problems from various areas of social and physical sciences and technology. Understanding the underlying physical process by which the system finds the solutions to such problems often leads to new optimisation algorithms. This thesis focuses on studying gain-dissipative systems and nature-inspired algorithms that form a hybrid architecture that may soon rival classical hardware. Chapter 1 lays the necessary foundation and explains various interdisciplinary terms that are used throughout the dissertation. In particular, connections between the optimisation problems and spin Hamiltonians are established, their computational complexity classes are explained, and the most prominent physical platforms for spin Hamiltonian implementation are reviewed. Chapter 2 demonstrates a large variety of behaviours encapsulated in networks of polariton condensates, which are a vivid example of a gain-dissipative system we use throughout the thesis. We explain how the variations of experimentally tunable parameters allow the networks of polariton condensates to represent different oscillator models. We derive analytic expressions for the interactions between two spatially separated polariton condensates and show various synchronisation regimes for periodic chains of condensates. An odd number of condensates at the vertices of a regular polygon leads to a spontaneous formation of a giant multiply-quantised vortex at the centre of a polygon. Numerical simulations of all studied configurations of polariton condensates are performed with a mean-field approach with some theoretically proposed physical phenomena supported by the relevant experiments. Chapter 3 examines the potential of polariton graphs to find the low-energy minima of the spin Hamiltonians. By associating a spin with a condensate phase, the minima of the XY model are achieved for simple configurations of spatially-interacting polariton condensates. We argue that such implementation of gain-dissipative simulators limits their applicability to the classes of easily solvable problems since the parameters of a particular Hamiltonian depend on the node occupancies that are not known a priori. To overcome this difficulty, we propose to adjust pumping intensities and coupling strengths dynamically. We further theoretically suggest how the discrete Ising and $n$-state planar Potts models with or without external fields can be simulated using gain-dissipative platforms. The underlying operational principle originates from a combination of resonant and non-resonant pumping. Spatial anisotropy of pump and dissipation profiles enables an effective control of the sign and intensity of the coupling strength between any two neighbouring sites, which we demonstrate with a two dimensional square lattice of polariton condensates. For an accurate minimisation of discrete and continuous spin Hamiltonians, we propose a fully controllable polaritonic XY-Ising machine based on a network of geometrically isolated polariton condensates. In Chapter 4, we look at classical computing rivals and study nature-inspired methods for optimising spin Hamiltonians. Based on the operational principles of gain-dissipative machines, we develop a novel class of gain-dissipative algorithms for the optimisation of discrete and continuous problems and show its performance in comparison with traditional optimisation techniques. Besides looking at traditional heuristic methods for Ising minimisation, such as the Hopfield-Tank neural networks and parallel tempering, we consider a recent physics-inspired algorithm, namely chaotic amplitude control, and exact commercial solver, Gurobi. For a proper evaluation of physical simulators, we further discuss the importance of detecting easy instances of hard combinatorial optimisation problems. The Ising model for certain interaction matrices, that are commonly used for evaluating the performance of unconventional computing machines and assumed to be exponentially hard, is shown to be solvable in polynomial time including the Mobius ladder graphs and Mattis spin glasses. In Chapter 5 we discuss possible future applications of unconventional computing platforms including emulation of search algorithms such as PageRank, realisation of a proof-of-work protocol for blockchain technology, and reservoir computing

Symmetry in Graph Theory

This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view