京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21540号 / 理博第4447号 / 新制||理||1639(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 玉川 安騎男, 教授 向井 茂, 教授 望月 新一 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21542号 / 理博第4449号 / 新制||理||1639(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 熊谷 隆, 准教授 福島 竜輝, 教授 牧野 和久 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

The subject of extra dimensions has experienced a renewed interest in recent years. Arkani-Hamed, Dimopoulos, and Dvali (ADD) have pointed out that it is possiblethat there exist extra dimensions that are as large as micrometer sized, if theStandard Model fields are restricted to a four-dimensional brane. In the ADD model, only the graviton is able to probe the extra dimensions. The main motivation for this model is that it could solve the hierarchy problem between the electroweak scale and the Planck scale by lowering the Planck scale to 1 TeV. However, in the ADD model, the radius of the extra dimensions is large, giving a new hierarchy problem between this radius and the electroweak scale. In addition, there are significant constraints on the model coming from astrophysics. To improve on the ADD model, it is possible to consider a model with curved extra dimensions. An interesting scenario is provided by extra dimensions of hyperbolic geometry. In this case, it is possible to bring the Planck scale down to 1 TeV without the need of a large radius. Also, the constraints that are important for the ADD model can be completely avoided. The most efficient probe of large extra dimensions is particle physics. In particular, it is possible to study their phenomenology in high-energy particle accelerators such as the Large Hadron Collider, which will be completed in 2008. The phenomenology of the ADD model has been extensively studied. In this thesis, we consider the phenomenology of a model where the internal space is a hyperbolic disc. We obtain the Kaluza-Klein spectrum approximately and study the Kaluza-Klein modes. The results are cross sections for production of a graviton together with a photon or a hadronic jet, which are the most important reactions for LHC physics.

<p>Understanding of thermal transport in small scales gains more importance with increasing demand in microelectronics and advancing fabrication technologies. In addition, scarce in energy sources adds more pressure with increasing expectations on research in energy conversion devices and renewable energies. In parallel to these, new phenomena observable only in small scales are discovered with the research, bringing more opportunities for engineers to solve real-world problems by applying the discoveries and more questions to answer. Thermal radiation as a thermal transport phenomenon is the epicenter of this research. Recent developments such as near-field radiative heat transfer exceeding blackbody radiation or control of radiative cooling via biasing grows the attraction on thermal radiation because these examples challenge our long-lasting understanding of nature. Exploring nature further in the small scale may help us meet the expectations mentioned above.</p> <p> </p> <p>In this thesis work, first, we carry out analyses on radiative heat transfer of natural hyperbolic material, calcite, and compare to that of a polar material SiC. Our study reveals that the high- modes within the hyperbolic bands are responsible for the substantial enhancement in near field radiation. Comparison of calcite with SiC illustrates the significance of the high- modes in calcite vs. surface polariton modes in SiC in their contributions to near-field radiation enhancement, for temperature differences ranging from 1 K to 400 K. We also noticed that the contributions of high- modes in calcite to near-field radiation is comparable to that of surface polaritons in SiC. The results of these analyses will be helpful in the search of hyperbolic materials that can enhance near field radiative transfer.</p> <p> </p> <p>Second, we demonstrate an experimental technique to measure near-field radiative heat transfer between two parallel plates at gap distances ranging from a few nanometers to far-field. A differential measurement circuit based on resistive thermometry to measure the defined temperatures are explained. To predict the defined temperatures, a computational method is utilized. We also detail an alignment technique that consists of a coarse and fine alignment in the relevant gap regions. This technique presents a method with high precision for gap measurement, dynamic gap control, and reliable sensitivity for extreme near-field measurements. Finally, we report experimental results that shows 18,000 times enhancement in radiative heat transfer between two parallel plates.</p> <p> </p> <p>Third, we analyze near-field radiative transfer due to hyperbolic phonon polaritons, driven by temperature gradient inside the bulk materials. We develop a mesoscale many-body scattering approach to account for the role of hyperbolic phonon polaritons in radiative transfer in the bulk and across a vacuum gap. Our study points out the equivalency between the bulk-generated mode and the surface mode in the absence of a temperature gradient in the material, and hence provide a unified framework for near-field radiative transfer by hyperbolic phonon polaritons. The results also elucidate contributions of the bulk-generated mode and the bulk temperature profile in the enhanced near-field radiative transfer.</p> <p> </p> <p>Forth, we study radiative heat transfer in hyperbolic material, hyperbolic boron nitride (hBN), and show a major contribution to energy transport arising from phonon polaritons supported in Reststrahlen bands. This contribution increases spectral radiative transfer by six orders of magnitude inside Reststrahlen bands compared to that outside Reststrahlen bands. The equivalent radiative thermal conductivity increases with temperature increase, and the radiative thermal conductivity can be of the same order of the phonon thermal conductivity. Experimental measurements are discussed. We showed the radiative contribution can account for as much as 27 % of the total thermal transport at 600 K. Hence, in hBN the radiative thermal transport can be comparable to thermal conduction by phonons. We also demonstrate contribution of polaritons to thermal transport in MoO₃. To calculate radiative heat transfer in three principal coordinates separately, we modify and apply the derived many-body model. Our analysis shows that radiative thermal conductivity in both in- and out-of-plane directions increases with temperature and contribution to energy transport by polaritons exceeds that by phonons.</p> <p> </p> Fifth, we build an experimental setup to examine near-field properties of materials using an external thermal source. The nanospectroscopy setup combines near-field microscopy technique, near-field scanning optical microscopy (NSOM), and Fourier-transform infrared (FTIR) spectroscopy. We further explain challenges in building a nanospectroscopy setup using a weak thermal source and coupling two techniques. This method enables us to investigate spectral thermal radiation and local dielectric properties in nanoscale.

This dissertation addresses the flow control problem in hyperbolic heat transfer mechanisms. It raises in concentrated distributed solar collectors to enhance their production efficiency under the unpredictable variations of the solar energy and the external disturbances. These factors which are either locally measured (the solar irradiance) or inaccessible for measurement (the collectors’ cleanliness) affect the source term of the distributed model and represent a major difficulty for the control design. Moreover, the temperature in the collector can only be measured at the boundaries. In this dissertation, we propose new adaptive control approaches to provide the adequate level of heat while coping with the unpredictable varying disturbances. First, we design model based control strategies for a better efficiency, in terms of accuracy and response time, with a relatively reduced complexity. Second, we enhance the controllers with on-line adaptation laws to continuously update the efficient value of the external conditions. In this study, we approach the control problem using both, the infinite dimensional model (late lumping) and a finite dimensional approximate representation (early lumping). For the early lumping approach, we introduce a new reduced order bilinear approximate model for system analysis and control design. This approximate state representation is then used to derive a nonlinear state feedback resorting to Lyapunov stability theory. To compensate for the external disturbances and the approximation uncertainties, an adaptive controller is developed based on a phenomenological representation of the system dynamics. For the late lumping approach, we propose two PDE based controllers by stabilization of the reference tracking error distributed profile. The control laws are explicitly defined as functions of the available measurement. The first one is obtained using a direct approach for error stabilization while the second one is derived through a nonlinear mapping. Furthermore, we endow the nonlinear controllers with an adaptation law to cope with the unpredictable unmeasured disturbances. The proposed adaptation law is based on a Proportional plus Integral correction feedback. We show that the control objectives with the required performance can be achieved following both approaches, but yet are conditioned with the physical limitations of the system.

Les équivalences sous-linéairement bilipschitziennes ont été introduites par Yves Cornulier afin de décrire les cônes asymptotiques des groupes de Lie. Elles généralisent les quasiisométries. Cette thèse construit des invariants pour l'équivalence sous-linéairement bilipschitzienne entre groupes et espaces hyperboliques au sens de Gromov, en utilisant l'analyse au bord de Gromov. Une classe d'application généralisant les homéomorphismes quasisymétriques, et une dimension conforme associée, sont introduites. Les espaces riemannien de type non-compact et de rang un, ainsi que certains espaces homogènes de courbure strictement négative, sont classifiés à équivalence sous-linéairement bilipschitzienne près. / Sublinearly biLipschitz equivalences have been introduced by Yves Cornulier as a means of describing the asymptotic cones of Lie groups; they include and generalize quasiisometries. This thesis provides invariants for sublinearly biLipschitz equivalence between Gromov-hyperbolic groups and spaces using analysis on the Gromov boundary. A class of applications generalizing quasisymmetric mappings, and a corresponding conformal dimension, are introduced as tools. Riemannian symmetric spaces of noncompact type as well as a subclass of homogeneous negatively curved Riemannian manifolds are classified up to sublinearly biLipschitz equivalence.

abstract: The main part of this work establishes existence, uniqueness and regularity properties of measure-valued solutions of a nonlinear hyperbolic conservation law with non-local velocities. Major challenges stem from in- and out-fluxes containing nonzero pure-point parts which cause discontinuities of the velocities. This part is preceded, and motivated, by an extended study which proves that an associated optimal control problem has no optimal $L^1$-solutions that are supported on short time intervals. The hyperbolic conservation law considered here is a well-established model for a highly re-entrant semiconductor manufacturing system. Prior work established well-posedness for $L^1$-controls and states, and existence of optimal solutions for $L^2$-controls, states, and control objectives. The results on measure-valued solutions presented here reduce to the existing literature in the case of initial state and in-flux being absolutely continuous measures. The surprising well-posedness (in the face of measures containing nonzero pure-point part and discontinuous velocities) is directly related to characteristic features of the model that capture the highly re-entrant nature of the semiconductor manufacturing system. More specifically, the optimal control problem is to minimize an $L^1$-functional that measures the mismatch between actual and desired accumulated out-flux. The focus is on the transition between equilibria with eventually zero backlog. In the case of a step up to a larger equilibrium, the in-flux not only needs to increase to match the higher desired out-flux, but also needs to increase the mass in the factory and to make up for the backlog caused by an inverse response of the system. The optimality results obtained confirm the heuristic inference that the optimal solution should be an impulsive in-flux, but this is no longer in the space of $L^1$-controls. The need for impulsive controls motivates the change of the setting from $L^1$-controls and states to controls and states that are Borel measures. The key strategy is to temporarily abandon the Eulerian point of view and first construct Lagrangian solutions. The final section proposes a notion of weak measure-valued solutions and proves existence and uniqueness of such. In the case of the in-flux containing nonzero pure-point part, the weak solution cannot depend continuously on the time with respect to any norm. However, using semi-norms that are related to the flat norm, a weaker form of continuity of solutions with respect to time is proven. It is conjectured that also a similar weak continuous dependence on initial data holds with respect to a variant of the flat norm. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2019

The diploma thesis deals with preparing the production process of the winning prize, which contains a selection in the form of a hyperbolic paraboloid in its upper part. The work is divided into four main chapters. First of all, the introduction part of the diploma thesis presents the possibilities of programming in the Sinumerik control system. Next, the mathematical requirements of the selected surface - hyperbolic paraboloid - are explained. The practical part consists of creating an NC program where all the theoretical knowledge from the introduction is applied. Finally, the diploma thesis ends with an economic evaluation of the selected element.

We model hyperbolic toral automorphisms by two types of Smale spaces; shifts of finite type and substitution tilings spaces. Smale spaces are dynamical systems with local hyperbolic product structure. In 1970, Bowen showed that an irreducible Smale space is a factor of a shift of finite type by showing that it has Markov partitions. Putnam extended Bowen's theorem by showing that every irreducible Smale space has a factor map that can be split into a s-bijective and u-bijective map; thereby better modelling a Smale space on its characterizing expanding and contracting spaces separately. In this thesis, we define two new constructions of Markov partitions for hyperbolic toral automorphisms inspired by the work of Adler, Weiss, and Praggastis. With one of the constructions, we investigate when a factor map from a shift of finite type to a hyperbolic toral automorphism can be written as a composition of a s-bijective and u-bijective map and we show that if such a splitting exists then the Markov partition must satisfy a Border Continuity condition. The second construction can be thought of as an explicit example of Putnam's theorem for the case of hyperbolic toral automorphisms whose defining matrix is in dimension 2 and has positive entries. We define a full splitting for all such hyperbolic toral automorphisms with one exception; the Arnold Cat map. / Graduate

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20158号 / 理博第4243号 / 新制||理||1610(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 望月 新一, 教授 岡本 久, 教授 玉川 安騎男 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM