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

The Color Number of Cubic Graphs Having a Spanning Tree with a Bounded Number of Leaves

The color number c(G) of a cubic graph G is the minimum cardinality of a color class of a proper 4-edge-coloring of G. It is well-known that every cubic graph G satisfies c(G) = 0 if G has a Hamiltonian cycle, and c(G) ≤ 2 if G has a Hamiltonian path. In this paper, we extend these observations by obtaining a bound for the color number of cubic graphs having a spanning tree with a bounded number of leaves

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

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

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

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

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

Dynamic interrelationship between technology and architecture in tall buildings

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Architecture, 2005.Page 230 blank.Includes bibliographical references (p. 225-229).The interrelationship between the technology and architecture of tall buildings is investigated from the emergence of tall buildings in the late 19th century to the present. Through the historical research, a filtering concept is developed - original technology and remedial technology - through which one can clearly understand the interrelationship between the technological evolution and architectural esthetic and further stylistic transition of tall buildings. More desirable visions for the future can be constructed based on this concept. Contemporary design practice of tall buildings is reviewed, and design guidelines are provided for new design trends. Investigated in depth are the behavioral characteristics and design methodology for diagrid structures, which emerge as a new direction in the design of tall buildings with their powerful structural rationale and symbolic architectural expression. Moreover, new technologies for tall building structures and facades are developed for performance enhancement through design integration, and their architectural potentials are explored. Special emphasis is placed on the research on the structural dynamic motion control using double skin facades / distributed tuned mass dampers. Design integration among architecture-related disciplines is emphasized throughout the research process as a means to more effectively overcome or at least minimize contemporary technological limitations and to create architecture of higher quality.(cont.) While each study makes its own contribution theoretically and in a particular design situation, from a wider viewpoint, the contribution of this thesis is to create more constructive relationships of architecture-related disciplines to produce better architecture through synergistic effects.by Kyoung-Sun Moon.Ph.D

Ranteesta mitattavat tunteet tietokoneen ja ihmisen vuorovaikutuksessa

The role of emotion in human-computer interaction (HCI) has seen an increase in interest during the last decades. Technological advancements have made studying them much more viable for example because of the availability of affordable and accurate wrist-based sensors. However, this subfield of HCI still lacks theory and it has many unsolved engineering problems, especially considering naturalistic and automated emotion recognition. This thesis provides an overview of wrist- based emotion recognition in human-computer interaction by tying in the views and theoretical background of emotion from philosophy, psychology, neuroscience and economics. The thesis also includes an experimental set-up in naturalistic settings. The experiment uses an Empatica E4 device that can be worn on the wrist and which can be used to measure electrodermal activity (EDA) and heartrate variability (HRV). Both EDA and HRV are known biomarkers for various emotional reactions, such as emotional arousal or mental stress. The study explores the possibilities of EDA and HRV to measure emotional arousal and valence. Furthermore, the correlations between psychological surveys and emotional biosignal markers are explored. We used the Affect Intensity Measure (AIM) -survey, which measures the intensity of experienced and shown emotion, and Rational- Experiential Inventory (REI) -survey, which measures an individual preferred style of information processing. Five custom experiments and a data analysis method with custom analyzer code were designed for this thesis. Our findings suggest that EDA is a good marker for arousal, but that HRV is a problematic measure. Furthermore, we found evidence that there would be correlations between psychological traits and biosignals. However, there were limitations within our experiments. In conclusions, we provide suggestions for futher research and a new theoretical framework that could be used to understand emotions better in HCI.Kiinnostus tunteiden merkityksestä ihmisen ja tietokoneen vuorovaikutuksessa on kasvanut. Teknologian kehityksen myötä tunteisiin liittyviä biosignaaleja voidaan mitata hyvinkin huomaamattomasti esimerkiksi rannetietokoneilla. Alan teoria on kuitenkin vähäistä ja erityisesti naturalistiseen ja automatisoituun tunteiden tunnistamiseen liittyy monia ratkaisemattomia teknologisia ongelmia. Tämän diplomityön tarkoituksena on tarjota lukijalleen kattava teoreettinen näkemys monilta tieteen aloilta, jotka tutkivat tunteita. Työ yhdistää tunteisiin liittyvää teoriaa filosofiasta, psykologiasta, neurotieteestä sekä ihmistietokonevuorovaikutuksen tutkimuksesta rakentaakseen yhtenäisen teoreettisen viitekehyksen ongelman ymmärtämiseksi. Työhön kuuluu myös kokeellinen osuus, jossa mitataan tunteita oikeassa ympäristössä. Kokeessa käytetään Empatica E4-rannetietokonetta, jolla voidaan mitata ihon sähkönjohtavuutta (EDA) ja sydämen sykevälivaihtelua (HRV). Sekä EDA että HRV ovat molemmat tunnettuja biosignaaleja erilaisissa tunnetiloissa. Kokeen tarkoitus on tutkia EDA:n ja HRV:n kykyä mitata tunteellista virittäytyneisyyttä ja tunnearvoa. Tämän lisäksi koe tutkii erilaisten psykologisten kyselylomakkeiden korrelaatioita mitattujen biosignaalejen välillä. Kokeessa käytetään Affect Intensity Measure (AIM) -kyselykaavaketta, joka mittaa koettujen ja näytettyjen tunteiden vahvuutta, sekä Rational Experiential Inventory (REI) -kyselykaavaketta, joka mittaa yksilön suosimaa sisäisen tiedonkäsittelyn menetelmää. Koetta varten kehitettiin viisi koeasetelmaa ja metodi, jolla voitiin analysoida mitattua dataa. Tulokset vahvistavat käsityksen, että EDA on hyvä virittäytyneisyyden mittari, mutta HRV:n käytössä löydettiin vain ongelmia. Tuloksissa on myös todisteita psykologisten luonteenpiirteiden ja biosignaalien korrelaatiolle. Lopussa annamme suosituksia seuraaville tutkimuksille ja esittelemme kehittämämme uuden teoreettisen viitekehyksen, jolla tunteita voisi ymmärtää paremmin ihmisen ja tietokoneen vuorovaikutuksessa