京都大学 / 新制・課程博士 / 博士(医学) / 甲第23367号 / 医博第4736号 / 新制||医||1051(附属図書館) / 京都大学大学院医学研究科医学専攻 / (主査)教授 森田 智視, 教授 川上 浩司, 教授 佐藤 俊哉 / 学位規則第4条第1項該当 / Doctor of Medical Science / Kyoto University / DFAM

The present thesis intents on analysing the truncated matricial Hamburger power moment problem in the general (degenerate and non-degenerate) case. Initiated due to manifold lines of research, by this time, outnumbering results and thoughts have been established that are concerned with specific subproblems within this field. The resulting presence of such a diversity as well as an extensively considered topic si- multaneously involves advantageous as well as obstructive aspects: on the one hand, we adopt the favourable possibility to capitalise on essential available results that proved beneficial within subsequent research. Nevertheless, on the other hand, we are obliged to illustrate major preparatory work in order to illucidate the comprehension of the attaching examination. Moreover, treating the matricial cases of the respective problems requires meticulous technical demands, in particular, in view of the chosen explicit approach to solving the considered tasks. Consequently, the first part of this thesis is dedicated to furnishing the necessary basis arranging the prime results of this research paper. Compul- sary notation as well as objects are introduced and thoroughly explained. Furthermore, the required techniques in order to achieve the desired results are characterised and ex- haustively discussed. Concerning the respective findings, we are afforded the opportunity to seise presentations and results that are, by this time, elaborately studied. Being equipped with mandatory cognisance, the thematically bipartite second and pivo- tal part objectives to describe all the possible values of all the solution functions of the truncated matricial Hamburger power moment problem M P [R; (s j ) 2n j=0 , ≤]. Aming this, we realise a first paramount achievement epitomising one of the two parts of the main results: Capturing an established representation of the solution set R 0,q [Π + ; (s j ) 2n j=0 , ≤] of the assigned matricial Hamburger moment problem via operating a specific algorithm of Schur-type, we expand these findings. We formulate a parameterisation of the set R 0,q [Π + ; (s j ) 2n j=0 , ≤] which is compatible with establishing respective equivalence classes within a certain subset of Nevanlinna pairs and utilise specific systems of orthogonal polynomials in order to entrench novel representations. In conclusion, we receive a para- meterisation that is valid within the entire upper open complex half-plane Π + . The second of the two prime parts changes focus to analysing all possible values of the functions belonging to R 0,q [Π + ; (s j ) 2n j=0 , ≤] in an arbitrary point w ∈ Π + . We gain two decisive conclusions: We identify these respective values to exhaust particular matrix balls 2n K[(s j ) 2n j=0 , w] := {F (w) | F ∈ R 0,q [Π + ; (s j ) j=0 , ≤]} the parameters of which are feasable to being described by specific rational matrix-valued functions and, in this course, enhance formerly established analyses. Moreover, we compile an alternative representation of the semi-radii constructing the respective matrix balls which manifests supportive in further consideration. We seise the achieved parameterisation of the set K[(s j ) 2n j=0 , w] and examine the behaviour of the respective sequences of left and right semi-radii. We recognise that these sequences of semi-radii associated with the respective matrix balls in the general case admit a particular monotonic behaviour. Consequently, with increasing number of given data, the resulting matrix balls are identified as being nested. Moreover, a proper description of the limit case of an infinite number of prescribed moments is facilitated.:1. Brief Historic Embedding and Introduction 2. Part I: Initialising Compulsary Cognisance Arranging Principal Achievements 2.1. Notation and Preliminaries 2.2. Particular Classes of Holomorphic Matrix-Valued Functions 2.3. Nevanlinna Pairs 2.4. Block Hankel Matrices 2.5. A Schur-Type Algorithm for Sequences of Complex p × q Matrices 2.6. Specific Matrix Polynomials 3. Part II: Momentous Results and Exposition – Improved Parameterisations of the Set R 0,q [Π + ; (s j ) 2n j=0 , ≤] 3.1. An Essential Step to a Parameterisation of the Solution Set R 0,q [Π + ; (s j ) 2n j=0 , ≤] 3.2. Parameterisation of the Solution Set R 0,q [Π + ; (s j ) 2n j=0 3.3. Particular Matrix Polynomials 3.4. Description of the Solution Set of the Truncated Matricial Hamburger Moment Problem by a Certain System of Orthogonal Matrix Polynomials 4. Part III: Prime Results and Exposition – Novel Description Balls 4.1. Particular Rational Matrix-Valued Functions 4.2. Description of the Values of the Solutions 4.3. Monotony of the Semi-Radii and Limit Balls of the Weyl Matrix 5. Summary of Principal Achievements and Prospects A. Matrix Theory B. Integration Theory of Non-Negative Hermitian Measures

The thesis studies truncated moment problems and related reconstruction techniques. It transfers the main aspects of Prony's method from finitely-supported measures to the classes of signed or non-negative measures supported on algebraic varieties of any dimension. The Zariski closure of the support of these measures is shown to be determined by finitely many moments and can be computed from the kernel of moment matrices. Moreover, several reconstruction algorithms are developed which are based on the computation of generalized eigenvalues and allow to recover the components of mixtures of such measures.

In this thesis we analyze the phase of the heavy symmetric top and the tippe top. These tops are two examples of physical systems for which the usefulness of integrals of motion and invariant manifolds, in phase space picture analysis, can be illustrated In the case of the heavy symmetric top, simplified proofs of stability of the vertical rotation have been perpetuated by successive textbooks during the last century. In these proofs correct perturbations of integrals of motion are missing. This may seem harmless since the deduced threshold value for stability is correct. However, perturbations of first integrals are essential in rigorous proofs of stability of motions for both tops. The tippe top is a toy that has the form of a truncated sphere equipped with a little peg. When spun fast on the spherical bottom its center of mass rises above its geometrical center and after a few seconds the top is spinning vertically on the peg. We study the tippe top through a sequence of embedded invariant manifolds to unveil the structure of the top's phase space. The last manifold, consisting of the asymptotic trajectories, is analyzed completely. We prove that trajectories in this manifold attract solutions in contact with the plane of support at all times and we give a complete description of their stability/instability properties for all admissible choices of model parameters and of the initial conditions. / <p>Report code: LiU-TEK-LIC-2004:35.</p>

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Proper Orthogonal Decomposition (POD), combined with the Method of Snapshots and Galerkin projection, is a popular method for the model order reduction of nonlinear PDEs. The POD requires the left singular vectors from the singular value decomposition (SVD) of an n-by-m "snapshot matrix" S, each column of which represents the computed state of the system at a given time. However, the direct computation of this decomposition can be computationally expensive, particularly for snapshot matrices that are too large to fit in memory. Hierarchical Approximate POD (HAPOD) (Himpe 2016) is a recent method for the approximate truncated SVD that requires only a single pass over S, is easily parallelizable, and can be computationally cheaper than direct SVD, all while guaranteeing the requested accuracy for the resulting basis. This method processes the columns of S in blocks based on a predefined rooted tree of processors, concatenating the outputs from each stage to form the inputs for the next. However, depending on the selected parameter values and the properties of S, the performance of HAPOD may be no better than that of direct SVD. In this work, we numerically explore the parameter values and snapshot matrix properties for which HAPOD is computationally advantageous over the full SVD and compare its performance to that of a parallelized incremental SVD method (Brand 2002, Brand 2003, and Arrighi2015). In particular, in addition to the two major processor tree structures detailed in the initial publication of HAPOD (Himpe2016), we explore the viability of a new structure designed with an MPI implementation in mind. / Master of Science / Singular Value Decomposition (SVD) provides a way to represent numeric data that breaks the data up into its most important components, as well as measuring how significant each part is. This decomposition is widely used to assist in finding patterns in data and making decisions accordingly, or to obtain simple, yet accurate, representations of complex physical processes. Examples of useful data to decompose include the velocity of water flowing past an obstacle in a river, a large collection of images, or user ratings for a large number of movies. However, computing the SVD directly can be computationally expensive, and usually requires repeated access to the entire dataset. As these data sets can be very large, up to hundreds of gigabytes or even several terabytes, storing all of the data in memory at once may be infeasible. Thus, repeated access to the entire dataset requires that the files be read repeatedly from the hard disk, which can make the required computations exceptionally slow. Fortunately, for many applications, only the most important parts of the data are needed, and the rest can be discarded. As a result, several methods have surfaced that can pick out the most important parts of the data while accessing the original data only once, piece by piece, and can be much faster than computing the SVD directly. In addition, the recent bottleneck in individual computer processor speeds has motivated a need for methods that can efficiently run on a large number of processors in parallel. Hierarchical Approximate POD (HAPOD) [1] is a recently-developed method that can efficiently pick out the most important parts of the data while only accessing the original data once, and which is very easy to run in parallel. However, depending on a user-defined algorithm parameter (weight), HAPOD may return more information than is needed to satisfy the requested accuracy, which determines how much data can be discarded. It turns out that the input weights that result in less extra data also result in slower computations and the eventual need for more data to be stored in memory at once. This thesis explores how to choose this input weight to best balance the amount of extra information used with the speed of the method, and also explores how the properties of the data, such as the size of the data or the distribution of levels of significance of each part, impact the effectiveness of HAPOD.

By means of a Monte Carlo study, this paper investigates the relative performance of Listwise Deletion, the EM-algorithm and the default algorithm in the MICE-package for R (PMM) in estimating regression coefficients under a left truncated covariate with missing values. The intention is to investigate whether the three frequently used missing data techniques are robust against left truncation when missing values are MCAR or MAR. The results suggest that no technique is superior overall in all combinations of factors studied. The EM-algorithm is unaffected by left truncation under MCAR but negatively affected by strong left truncation under MAR. Compared to the default MICE-algorithm, the performance of EM is more stable across distributions and combinations of sample size and missing rate. The default MICE-algorithm is improved by left truncation but is sensitive to missingness pattern and missing rate. Compared to Listwise Deletion, the EM-algorithm is less robust against left truncation when missing values are MAR. However, the decline in performance of the EM-algorithm is not large enough for the algorithm to be completely outperformed by Listwise Deletion, especially not when the missing rate is moderate. Listwise Deletion might be robust against left truncation but is inefficient.

We study the application of Bose-Einstein condensates (BECs) to simulations of phenomena across a number of disciplines in physics, using theoretical and computational methods. ¶ Collapsing condensates as created by E. Donley et al. [Nature 415, 39 (2002)] exhibit potentially useful parallels to an inflationary universe. To enable the exploitation of this analogy, we check if current quantum field theories describe collapsing condensates quantitatively, by targeting the discrepancy between experimental and theoretical values for the time to collapse. To this end, we couple the lowest order quantum field correlation functions to the condensate wavefunction, and solve the resulting Hartree-Fock-Bogoliubov equations numerically. Complementarily, we perform stochastic truncated Wigner simulations of the collapse. Both methods also allow us to study finite temperature effects. ¶ We find with neither method that quantum corrections lead to a faster collapse than is predicted by Gross-Pitaevskii theory. We conclude that the discrepancy between the experimental and theoretical values of the collapse time cannot be explained by Gaussian quantum fluctuations or finite temperature effects. Further studies are thus required before the full analogue cosmology potential of collapsing condensates can be utilised. ¶ As the next project, we find experimental parameter regimes in which stable three-dimensional Skyrmions can exist in a condensate. We show that their stability in a harmonic trap depends critically on scattering lengths, atom numbers, trap rotation and trap anisotropy. In particular, for the Rb87 |F=1,m_f=-1>, |F=2,m_f=1> hyperfine states, stability is sensitive to the scattering lengths at the 2% level. We find stable Skyrmions with slightly more than 2*10^6 atoms, which can be stabilised against drifting out of the trap by laser pinning. ¶ As a stepping stone towards Skyrmions, we propose a method for the stabilisation of a stack of parallel vortex rings in a Bose-Einstein condensate. The method makes use of a ``hollow'' laser beam containing an optical vortex, which realises an optical tunnel for the condensate. Using realistic experimental parameters, we demonstrate numerically that our method can stabilise up to 9 vortex rings. ¶ Finally, we focus on analogue gravity, further exploiting the analogy between flowing condensates and general relativistic curved space time. We compare several realistic setups, investigating their suitability for the observation of analogue Hawking radiation. We link our proposal of stable ring flows to analogue gravity, by studying supersonic flows in the optical tunnel. We show that long-living immobile condensate solitons generated in the tunnel exhibit sonic horizons, and discuss whether these could be employed to study extreme cases in analogue gravity. ¶ Beyond these, our survey indicates that for conventional analogue Hawking radiation, simple outflow from a condensate reservoir, in effectively one dimension, has the best properties. We show with three dimensional simulations that stable sonic horizons exist under realistic conditions. However, we highlight that three-body losses impose limitations on the achievable analogue Hawking temperatures. These limitations vary between the atomic species and favour light atoms. ¶ Our results indicate that Bose-Einstein condensates will soon be useful for interdisciplinary studies by analogy, but also show that the experiments will be challenging.