Symmetry in Applied Mathematics

Applied mathematics and symmetry work together as a powerful tool for problem reduction and solving. We are communicating applications in probability theory and statistics (A Test Detecting the Outliers for Continuous Distributions Based on the Cumulative Distribution Function of the Data Being Tested, The Asymmetric Alpha-Power Skew-t Distribution), fractals - geometry and alike (Khovanov Homology of Three-Strand Braid Links, Volume Preserving Maps Between p-Balls, Generation of Julia and Mandelbrot Sets via Fixed Points), supersymmetry - physics, nanostructures -chemistry, taxonomy - biology and alike (A Continuous Coordinate System for the Plane by Triangular Symmetry, One-Dimensional Optimal System for 2D Rotating Ideal Gas, Minimal Energy Configurations of Finite Molecular Arrays, Noether-Like Operators and First Integrals for Generalized Systems of Lane-Emden Equations), algorithms, programs and software analysis (Algorithm for Neutrosophic Soft Sets in Stochastic Multi-Criteria Group Decision Making Based on Prospect Theory, On a Reduced Cost Higher Order Traub-Steffensen-Like Method for Nonlinear Systems, On a Class of Optimal Fourth Order Multiple Root Solvers without Using Derivatives) to specific subjects (Facility Location Problem Approach for Distributed Drones, Parametric Jensen-Shannon Statistical Complexity and Its Applications on Full-Scale Compartment Fire Data). Diverse topics are thus combined to map out the mathematical core of practical problems

Enabling 5G Technologies

The increasing demand for connectivity and broadband wireless access is leading to the fifth generation (5G) of cellular networks. The overall scope of 5G is greater in client width and diversity than in previous generations, requiring substantial changes to network topologies and air interfaces. This divergence from existing network designs is prompting a massive growth in research, with the U.S. government alone investing $400 million in advanced wireless technologies. 5G is projected to enable the connectivity of 20 billion devices by 2020, and dominate such areas as vehicular networking and the Internet of Things. However, many challenges exist to enable large scale deployment and general adoption of the cellular industries. In this dissertation, we propose three new additions to the literature to further the progression 5G development. These additions approach 5G from top down and bottom up perspectives considering interference modeling and physical layer prototyping. Heterogeneous deployments are considered from a purely analytical perspective, modeling co-channel interference between and among both macrocell and femtocell tiers. We further enhance these models with parameterized directional antennas and integrate them into a novel mixed point process study of the network. At the air interface, we examine Software-Defined Radio (SDR) development of physical link level simulations. First, we introduce a new algorithm acceleration framework for MATLAB, enabling real-time and concurrent applications. Extensible beyond SDR alone, this dataflow framework can provide application speedup for stream-based or data dependent processing. Furthermore, using SDRs we develop a localization testbed for dense deployments of 5G smallcells. Providing real-time tracking of targets using foundational direction of arrival estimation techniques, including a new OFDM based correlation implementation

Phase-field modeling of multi-domain evolution in ferromagnetic shape memory alloys and of polycrystalline thin film growth

The phase-field method is a powerful tool in computer-aided materials science as it allows for the analysis of the time-spatial evolution of microstructures on the mesoscale. A multi-phase-field model is adopted to run numerical simulations in two different areas of scientific interest: Polycrystalline thin films growth and the ferromagnetic shape memory effect. FFT-techniques, norm conservative integration and RVE-methods are necessary to make the coupled problems numerically feasible

Activation processes in biology

Many processes in physics and biology can be understand through the framework of escape from a metastable state, including (but not limited to) the rates of chemical reactions, the unfolding of proteins, the nucleation of bubbles, and the condensation of gases. To understand the kinetics of these processes, we have to be able to calculate the rate of escape. In this thesis, I solve several of such of escape problems, each addressing a specific physical or biological system. I first show how the forced unfolding of heteropolymers could be a process with non-exponential kinetics, developing ideas about the importance of unfolding pathways in determining kinetics of unfolding. Then, I consider forced unfolding when a molecule is attached to a yielding (viscoelastic) substrate, and a constant force is applied. I show that the rates of unfolding depend on both the elastic and viscous response of the substrate. This problem is related to the biological process of mechanosensing, when the unfolding `sensor' protein exposes catalytic residues and generates a chemical signal to the cell. Related to this is the analysis of population-dynamics study of cells adhesion on substrates, which allows me to extract key characteristics and parameters of the adhesome complex. Then, I apply the ideas of escape from a metastable state to ask about the rates of a ligand at the end of a tethered polymer binding to a surface receptor, using a mean field approach to reduce the problem to one dimension. I show that there is a trade-off between the entropic cost of reaching to a receptor vs the volumetric cost of expanding the tether length. I then show that for a Gaussian chain with multiple ligands along its length, there exists a finite, non-zero optimal number of ligands to minimise the time taken for the end of the chain to bind to the surface. Finally, I consider the problem of microswimmers in an obstacle lattice, calculating their transport properties, and showing how we can use lattices to examine the underlying stochastic dynamics.I was funded through an EPSRC studentship: EP/M508007/1

The effects of geometry and dynamics on biological pattern formation

This project examines the influence of geometry and dynamics on pattern formation in biological development . Since the work of Turing (1952) it has been known that patterns can form spontaneously given certain relatively simple conditions. The Turing mechanism involved a symmetry-breaking bifurcation from a stable spatially homogeneous state. However the development of patterns in developing organisms does not take place from such simple conditions, biological development causes pattern formation to occur within geometric structures which are complex and the environment is very noisy. This thesis examines the effects of such complexity and noise on pattern formation. The biological situations modelled in this thesis relate to the development of the mammalian cortex. The cortex is a very thin sheet, and there is evolutionary and developmental pressure to utilise cortical space to the maximum. This promotes the formation of spatial superstructures encompassing regions serving different functions. Also cortical development produces two types of pattern, one in the actual physical structure, this is common to much biological pattern formation, but also in terms of patterns of neural response which can be viewed as a feature mapping and is specific to cortical function. We examine the first type of pattern formation within the barrel field of the rat cortex, a geometric superstructure that has the properties of a Voronoi tessellation and apply a dynamical constraint from the observation that the patterns are sparse. We show that these constraints produce a distribution of patterns closer to what is observed than predictions derived from studies in a single domain of perfect circular shape. We also discover a novel effect of geometric alignment of patterns in neighbouring domains, without any physical communication between them, in a wide class of tessellations. This effect is confirmed by analysis of actual images of the subbarrel patterns in the developing rat cortex. The effect of geometry and dynamics of the second type of pattern formation is investigated in the patterns of orientation preference of neuronal response in the visual cortex of certain mammals. Where the domains are sufficiently small so that topological defects (pinwheels) cannot form the behaviour is similar to the reaction-diffusion equations. However, when there are many defects in the region alignment at the boundaries disappears

Phase-field modeling of multi-domain evolution in ferromagnetic shape memory alloys and of polycrystalline thin film growth

The phase-field method is a powerful tool for the analysis of microstructure evolution. A multi-phase-field model is developed to run computer simulations in different areas of materials science: The anisotropic growth of MFI zeolite-like crystallites on a smooth substrate into a hydrothermal solution constituting thin films, and the analysis of the behavior of magnetic shape memory alloys, a class of active smart martensitic materials that are used as components in actuators and dampers

Applied Mathematics to Mechanisms and Machines

This book brings together all 16 articles published in the Special Issue "Applied Mathematics to Mechanisms and Machines" of the MDPI Mathematics journal, in the section “Engineering Mathematics”. The subject matter covered by these works is varied, but they all have mechanisms as the object of study and mathematics as the basis of the methodology used. In fact, the synthesis, design and optimization of mechanisms, robotics, automotives, maintenance 4.0, machine vibrations, control, biomechanics and medical devices are among the topics covered in this book. This volume may be of interest to all who work in the field of mechanism and machine science and we hope that it will contribute to the development of both mechanical engineering and applied mathematics