The smallest nontrivial snarks of oddness 4

The oddness of a cubic graph is the smallest number of odd circuits in a 2-factor of the graph. This invariant is widely considered to be one of the most important measures of uncolourability of cubic graphs and as such has been repeatedly reoccurring in numerous investigations of problems and conjectures surrounding snarks (connected cubic graphs admitting no proper 3-edge-colouring). In [Ars Math. Contemp. 16 (2019), 277-298] we have proved that the smallest number of vertices of a snark with cyclic connectivity 4 and oddness 4 is 44. We now show that there are exactly 31 such snarks, all of them having girth 5. These snarks are built up from subgraphs of the Petersen graph and a small number of additional vertices. Depending on their structure they fall into six classes, each class giving rise to an infinite family of snarks with oddness at least 4 with increasing order. We explain the reasons why these snarks have oddness 4 and prove that the 31 snarks form the complete set of snarks with cyclic connectivity 4 and oddness 4 on 44 vertices. The proof is a combination of a purely theoretical approach with extensive computations performed by a computer.Comment: 38 pages; submitted for publicatio

Measurements of edge uncolourability in cubic graphs

Philosophiae Doctor - PhDThe history of the pursuit of uncolourable cubic graphs dates back more than a century. This pursuit has evolved from the slow discovery of individual uncolourable cubic graphs such as the famous Petersen graph and the Blanusa snarks, to discovering in nite classes of uncolourable cubic graphs such as the Louphekine and Goldberg snarks, to investigating parameters which measure the uncolourability of cubic graphs. These parameters include resistance, oddness and weak oddness, ow resistance, among others. In this thesis, we consider current ideas and problems regarding the uncolourability of cubic graphs, centering around these parameters. We introduce new ideas regarding the structural complexity of these graphs in question. In particular, we consider their 3-critical subgraphs, speci cally in relation to resistance. We further introduce new parameters which measure the uncolourability of cubic graphs, speci cally relating to their 3-critical subgraphs and various types of cubic graph reductions. This is also done with a view to identifying further problems of interest. This thesis also presents solutions and partial solutions to long-standing open conjectures relating in particular to oddness, weak oddness and resistance

Applications of dynamical systems with symmetry

This thesis examines the application of symmetric dynamical systems theory to two areas in applied mathematics: weakly coupled oscillators with symmetry, and bifurcations in flame front equations. After a general introduction in the first chapter, chapter 2 develops a theoretical framework for the study of identical oscillators with arbitrary symmetry group under an assumption of weak coupling. It focusses on networks with 'all to all' Sn coupling. The structure imposed by the symmetry on the phase space for weakly coupled oscillators with Sn, Zn or Dn symmetries is discussed, and the interaction of internal symmetries and network symmetries is shown to cause decoupling under certain conditions. Chapter 3 discusses what this implies for generic dynamical behaviour of coupled oscillator systems, and concentrates on application to small numbers of oscillators (three or four). We find strong restrictions on bifurcations, and structurally stable heteroclinic cycles. Following this, chapter 4 reports on experimental results from electronic oscillator systems and relates it to results in chapter 3. In a forced oscillator system, breakdown of regular motion is observed to occur through break up of tori followed by a symmetric bifurcation of chaotic attractors to fully symmetric chaos. Chapter 5 discusses reduction of a system of identical coupled oscillators to phase equations in a weakly coupled limit, considering them as weakly dissipative Hamiltonian oscillators with very weakly coupling. This provides a derivation of example phase equations discussed in chapter 2. Applications are shown for two van der Pol-Duffing oscillators in the case of a twin-well potential. Finally, we turn our attention to the Kuramoto-Sivashinsky equation. Chapter 6 starts by discussing flame front equations in general, and non-linear models in particular. The Kuramoto-Sivashinsky equation on a rectangular domain with simple boundary conditions is found to be an example of a large class of systems whose linear behaviour gives rise to arbitrarily high order mode interactions. Chapter 7 presents computation of some of these mode interactions using competerised Liapunov-Schmidt reduction onto the kernel of the linearisation, and investigates the bifurcation diagrams in two parameters

Strategies for teaching engineering mathematics

This thesis is an account of experiments into the teaching of mathematics to engineering undergraduates which have been conducted over twenty years against a background of changing intake ability, varying output requirements and increasing restrictions on the formal contact time available. The aim has been to improve the efficiency of the teaching-learning process. The main areas of experimentation have been the integration in the syllabus of numerical and analytical methods, the incorporation of case studies into the curriculum and the use of micro-based software to enhance the teaching process. Special attention is paid to courses in Mathematical Engineering and their position in the spectrum of engineering disciplines. A core curriculum in mathematics for undergraduate engineers is proposed and details are provided of its implementation. The roles of case studies and micro-based software are highlighted. The provision of a mathematics learning resource centre is considered a necessary feature of the implementation of the proposed course. Finally, suggestions for further research are made

Interplay of Spin-Orbit Interaction and Two-Dimensional Superconductivity in Al/InAs Heterostructures

In this work superconducting properties of Al/InAs heterostructures with strong spin-orbit coupling are investigated. The structure consists of 5.5 nm thick Al films epitaxially grown on an InGaAs/InAs/InGaAs 2DEG, which provides Rashba-type spin-orbit coupling. The films are characterized by standard DC transport measurements. In the first part a RLC resonator technique operating in the low MHz regime for measuring small inductances of the order of some nH is presented. It is then used to measure the inductive response of long meanders etched into the material as a function of external parameters such as magnetic field, temperature or bias current. Small out-of-plane magnetic fields induce vortices in the system and lead to a strong inductive signal, which can be attributed to oscillatory motion of pinned vortices. In superconductors without spin-orbit coupling the vortex pinning strength decreases with either in- or out-of-plane field due to pair-breaking effects, that reduce the superfluid stiffness. In the case of the Al/InAs heterostructure, unexpectedly an additional in-plane magnetic field leads to an enhancement of the vortex pinning. The pinning enhancement strongly depends on the direction of the in-plane magnetic field with respect to the current direction. A theoretical model capturing the basic phenomenological findings is presented. In the second part anisotropies of the DC transport properties are studied as a function of magnetic field. In straight wires etched in the Al/InAs, critical current measurements as well as second harmonic resistance measurements are performed. For in-plane magnetic fields perpendicular to the current direction, a pronounced second harmonic resistance signal is measured in the fluctuation and BKT regime of the heterostructure, which is due to the interplay of Rashba spin-orbit coupling and in-plane magnetic field. Unexpectedly also out-of-plane fields lead to a significant second harmonic resistance. At low temperatures, magnetic fields orthogonal to the current direction lead to a polarity-dependent critical current. In both critical current and second harmonic resistance measurements, complex symmetries with respect to combined in- and out-of-plane magnetic fields are found

Computational aspects of voting: a literature survey

Preference aggregation is a topic of study in different fields such as philosophy, mathematics, economics and political science. Recently, computational aspects of preference aggregation have gained especial attention and “computational politics” has emerged as a marked line of research in computer science with a clear concentration on voting protocols. The field of voting systems, rooted in social choice theory, has expanded notably in both depth and breadth in the last few decades. A significant amount of this growth comes from studies concerning the computational aspects of voting systems. This thesis comprehensively reviews the work on voting systems (from a computing perspective) by listing, classifying and comparing the results obtained by different researchers in the field. This survey covers a wide range of new and historical results yet provides a profound commentary on related work as individual studies and in relation to other related work and to the field in general. The deliverables serve as an overview where students and novice researchers in the field can start and also as a depository that can be referred to when searching for specific results. A comprehensive literature survey of the computational aspects of voting is a task that has not been undertaken yet and is initially realized here. Part of this research was dedicated to creating a web-depository that contains material and references related to the topic based on the survey. The purpose was to create a dynamic version of the survey that can be updated with latest findings and as an online practical reference

Çok parçacıklı sistemlerin geometri optimizasyonu için parelel kod geliştirilmesi: nano-sistemlerde hidrojen depolama ve nano ölçekli mıknatıs yapılarının incelenmesi uygulamaları

TÜBİTAK TBAG Proje01.07.200

Topological k · p Hamiltonians and their applications to uniaxially strained Mercury telluride

Topological insulators (TIs) are a new state of quantum matter that has fundamentally challenged our knowledge of insulator and metals. They are insulators in the bulk, but metallic on the edge. A TI is characterized by a so-called topological invariant. This characteristic integer number is associated to every mapping between two topological spaces and can be defined for an electronic system on the lattice. Due to the bulk-edge correspondence a non-trivial value leads to topologically protected edge states. To get insight into the electronic characteristics of these edge/surface states, however, an effective continuum theory is needed. Continuum models are analytical and are also able to model transport. In this thesis we will address the suitability of continuum low-energy theories to describe the topological characteristics of TIs. The models which are topologically well-defined are called topological k.p Hamiltonians. After introducing a necessary background in chapter 1 and 2, we will discuss in the methodological chapter 3 the strategies that have to be taken into account to allow for studying topological surface states. In chapter 4 we will study two different model classes associated to a spherical basis manifold. Both have an integer topological invariant, but one shows a marginal bulk-edge correspondence. In chapter 5 we will study a different continuum theory where the basis manifold corresponds to a hemisphere. We then apply all these ideas to a time-reversal invariant TI -- uniaxially strained Mercury Telluride (HgTe). We determine the spin textures of the topological surface states of strained HgTe using their close relations with the mirror Chern numbers of the system and the orbital composition of the surface states. We show that at the side surfaces with $C_{2v}$ point group symmetry an increase in the strain magnitude triggers a topological phase transition where the winding number of the surface state spin texture is flipped while the four topological invariants characterizing the bulk band structure are unchanged. In the last chapter we will give a summary

Measuring system qualification for LHC arc quadrupole magnets

Currently the LHC Project at CERN has reached the construction phase. The superconducting magnets of this new accelerator work at superfluid helium temperature. The "arc quadrupoles" (360 pieces), which focus the beam have to be measured at a temperature of 1.9 Kelvin with outstanding precision: The measurement aims to reach a reproducibility of 1.5 · 10^-4 for the field integral, 2 ppm for the harmonic content of the main field and 0.15mm for the position of the axis. A specially developed scanner allows the simultaneous measurement of the field axis and quality. This thesis demonstrates that the system as it stands fulfils the high requirements with respect to the magnetic measurement and the magnetic axis and thus provides the desired unique versatile equipment. The assessment was performed based on experimental results, direct calibration and using a new simulation tool. The main defects treated are mechanical torsion and vibration of moving parts, electrical noise and power supply ripple