Quantum field theories are very good at describing the world around us but use complicated computations that cannot always be solved exactly. Introducing conformal symmetry to quantum field theory can reduce this complexity and allow for quite simple calculation in the best case. This report aims to describe the critical part of the Ising model in 2 dimensions using conformal field theory while assuming only some knowledge of quantum mechanics and complex analysis from the reader. This is done by using the book Conformal Field Theory as the source for information about conformal field theory. / Kvantfältteorier är mycket bra på att beskriva verkligheten runt om oss men de använder sig av avancerade beräkningar som inte alltid kan lösas exakt. Genom att ge systemet konform symmetri så kan dessa avancerade beräkningar förenklas och bli ganska enkla i de bästa fallen. Målet med denna rapport är att beskriva hur en modell som kallas för "Ising model" kan beskrivas i sitt kritiska tillstånd med hjälp utav konform fältteori. Läsaren antas kunna kvantmekanik samt komplex analys. Informationen om konform fältteori hämtas ifrån boken Conformal Field Theory

京都大学 / 新制・課程博士 / 博士(理学) / 甲第24391号 / 理博第4890号 / 新制||理||1699(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 梶野 直孝, 教授 並河 良典, 准教授 Croydon David Alexander / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

京都大学 / 新制・課程博士 / 博士(理学) / 甲第24403号 / 理博第4902号 / 新制||理||1700(附属図書館) / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)教授 高柳 匡, 教授 杉本 茂樹, 教授 橋本 幸士 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Mesh adaptation techniques are used to modify complex finite element meshes to reduce analysis time and improve accuracy. Modification of all-hexahedral meshes has proven difficult to the unique connectivity constraints they exhibit. This thesis presents an automated tool for local, conformal refinement of all-hexahedral meshes based on the insertion of multi-directional twist planes into the spatial twist continuum. The contributions of this thesis are (1) the ability to conformally refine all entities of an all-hexahedral element mesh, (2) the simplification of template insertion to multi-directional refinement. The refinement algorithm is divided into single hex sheet operations, where individual refinement steps are performed completely within a single hex sheet, and parallel sheet operation, where each refinement step occurs within two parallel hex sheets. Combining these two procedures facilitates the refinement of any mesh feature. Refinement is accomplished by replacing original mesh elements with one or more of six base templates selected by the number of nodes, flagged for refinement on the element. The refinement procedures are covered in detail with representative graphics and examples that illustrate the application of the techniques and the results of the refinement.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第23685号 / 理博第4775号 / 新制||理||1684(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 荒川 知幸, 教授 玉川 安騎男, 教授 並河 良典 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

This study focuses on the cycle-averaged mold temperature of additively manufactured injection molding tooling and how it is affected by conformal cooling channels. This was done by producing a benchmark mold out of Digital ABS produced by Stratasys, an acrylic based photopolymer, which was then used to produce injection molded parts until tool failure. Another, more cost-effective material, High Temp Resin produced by Formlabs, another acrylic based photopolymer, was also tested but yielded very little success. Then the mold design was altered by adding conformal cooling channels and again tested by producing injection molded parts while tracking the mold temperature. This experimentation was then compared to an injection molding cooling channel model in order to validate the model for use with additively manufactured tooling with conformal cooling channels for use in injection molding. The benchmark Digital ABS mold was able to produce 66 shots in the injection molding machine before complete mold failure. The Digital ABS mold had a cycle-averaged mold temperature of about 155°F. The High Temp Resin mold was able to produce 3 shots before complete mold failure. The High Temp Resin material is much more brittle, and the mold design did not take into account how brittle the material was. The Digital ABS mold with conformal cooling channels had a cycle-averaged mold temperature of 111°F. This is significantly lower than without cooling channels and has a high potential for improving tooling life. The cooling channel model predicted the cycle-averaged mold temperature to be 116°F. This proved to be a very good model and can be used as a design tool when choosing cooling channel geometry and position in additively manufactured tooling. This research shows the potential that conformal cooling channels have to help improve additively manufactured tooling life for injection molding. As shown in other research done, the ability to maintain the mold below 120°F significantly improves the life of additively manufactured tooling. The results of this study demonstrate the effectiveness of conformal cooling channels in controlling mold temperature. It should be researched further, but the use of conformal cooling channels has the potential to produce more production or prototype parts with additively manufactured tooling for injection molding.

The goal of this research is to expand the field of conformal predictions using Random Survival Forests. The standard Conformal Random Survival Forest can predict with a fixed certainty whether something will survive up until a certain time point. This research is the first to show that there is little practical use in the standard Conformal Random Survival Forest algorithm. It turns out that the confidence guarantees of the conformal prediction framework are violated if the Standard algorithm makes predictions for a user-controlled fixed time point. To solve this challenge, this thesis proposes two algorithms that specialize in conformal predictions for a fixed point in time: a Fixed Time algorithm and a Hybrid algorithm. Both algorithms transform the survival data that is used by the split evaluation metric in the Random Survival Forest algorithm. The algorithms are evaluated and compared along six different set prediction evaluation criteria. The prediction performance of the Hybrid algorithm outperforms the prediction performance of the Fixed Time algorithm in most cases. Furthermore, the Hybrid algorithm is more stable than the Fixed Time algorithm when the predicting job extends to various time points. The hybrid Conformal Random Survival Forest should thus be considered by anyone who wants to make conformal survival predictions at usercontrolled time points. / Målet med denna avhandling är att utöka området för konformitetsprediktion med hjälp av Random Survival Forests. Standardutförandet av Conformal Random Survival Forest kan förutsäga med en viss säkerhet om någonting kommer att överleva fram till en viss tidpunkt. Denna avhandling är den första som visar att det finns liten praktisk användning i standardutförandet av Conformal Random Survival Forest-algoritmen. Det visar sig att konfidensgarantierna för konformitetsprediktionsramverket bryts om standardalgoritmen gör förutsägelser för en användarstyrd fast tidpunkt. För att lösa denna utmaning, föreslår denna avhandling två algoritmer som specialiserar sig i konformitetsprediktion för en bestämd tidpunkt: en fast-tids algoritm och en hybridalgoritm. Båda algoritmerna omvandlar den överlevnadsdata som används av den delade utvärderingsmetoden i Random Survival Forest-algoritmen. Uppskattningsförmågan för hybridalgoritmen överträffar den för fast-tids algoritmen i de flesta fall. Dessutom är hybrid algoritmen stabilare än fast-tids algoritmen när det förutsägelsejobbet sträcker sig till olika tidpunkter. Hybridalgoritmen för Conformal Random Survival Forest bör därför föredras av den som vill göra konformitetsprediktion av överlevnad vid användarstyrda tidpunkter.

The central objective of this work is to present an analysis of the asymptotic conformal Killing vectors in asymptotically-flat space-times of general relativity. This problem has been examined by two different methods; in Chapter 5 the asymptotic expansion technique originated by Newman and Unti [31] leads to a solution for asymptotically-flat spacetimes which admit an asymptotically shear-free congruence of null geodesics, and in Chapter 6 the conformal rescaling technique of Penrose [54] is used both to support the findings of the previous chapter and to set out a procedure for solution in the general case. It is pointed out that Penrose's conformal technique is preferable to the use of asymptotic expansion methods, since it can be established in a rigorous manner without leading to the possible convergence difficulties associated with asymptotic expansions. Since the asymptotic conformal symmetry groups of asymptotically flat space-times Are generalisations of the conformal group of Minkowski space-time we devote Chapters 3 and 4 to a study of the flat space case so that the results of later chapters may receive an interpretation in terms of familiar concepts. These chapters fulfil a second, equally important, role in establishing local isomorphisms between the Minkowski-space conformal group, 90(2,4) and SU(2,2). The SO(2,4) representation has been used by Kastrup [61] to give a physical interpretation using space-time gauge transformations. This appears as part of the survey of interpretative work in Chapter 7. The SU(2,2) representation of the conformal group has assumed a theoretical prominence in recent years. through the work of Penrose [9-11] on twistors. In Chapter 4 we establish contact with twistor ideas by showing that points in Minkowski space-time correspond to certain complex skew-symmetric rank two tensors on the SU(2,2) carrier space. These objects are, in Penrose's terminology [91, simple skew-symmetric twistors of valence [J. A particularly interesting aspect of conformal objects in space-time is explored in Chapter 8, where we extend the work of Geroch [16] on multipole moments of the Laplace equation in 3-space to the consideration. of Q tý =0 in Minkowski space-time. This development hinges upon the fact that multipole moment fields are also conformal Killing tensors. In the final chapter some elementary applications of the results of Chapters 3 and 5 are made to cosmological models which have conformal flatness or asymptotic conformal flatness. In the first class here we have 'models of the Robertson-Walker type and in the second class we have the asymptotically-Friedmann universes considered by Hawking [73]. / University of Bradford Research Studenship

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