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We consider a system of two identical Morris-Lecar neurons coupled via electrical coupling. We focus our study on the effects that the coupling strength, γ , and the coupling time delay, τ , cause on the dynamics of the system. For small γ we use the phase model reduction technique to analyze the system behavior. We determine the stable states of the system with respect to γ and τ using the appropriate phase models, and we estimate the regions of validity of the phase models in the γ , τ plane using both analytical and numerical analysis. Next we examine asymptotic of the arbitrary conductance-based neuronal model for γ → +∞ and γ → −∞. The theory of nearly linear systems developed in [30] is extended in the special case of matrices with non-positive eigenvalues. The asymptotic analysis for γ > 0 shows that with appropriate choice of γ the voltages of the neurons can be made arbitrarily close in finite time and will remain that close for all subsequent time, while the asymptotic analysis for γ < 0 suggests the method of estimation of the boundary between “weak” and “strong” coupling.

We consider a system of two identical Morris-Lecar neurons coupled via electrical coupling. We focus our study on the effects that the coupling strength, γ , and the coupling time delay, τ , cause on the dynamics of the system. For small γ we use the phase model reduction technique to analyze the system behavior. We determine the stable states of the system with respect to γ and τ using the appropriate phase models, and we estimate the regions of validity of the phase models in the γ , τ plane using both analytical and numerical analysis. Next we examine asymptotic of the arbitrary conductance-based neuronal model for γ → +∞ and γ → −∞. The theory of nearly linear systems developed in [30] is extended in the special case of matrices with non-positive eigenvalues. The asymptotic analysis for γ > 0 shows that with appropriate choice of γ the voltages of the neurons can be made arbitrarily close in finite time and will remain that close for all subsequent time, while the asymptotic analysis for γ < 0 suggests the method of estimation of the boundary between “weak” and “strong” coupling.

Common image compression standards are usually based on frequency transform such as Discrete Cosine Transform or Wavelets. We present a different approach for loss-less image compression, it is based on combinatorial transform. The main transform is Burrows Wheeler Transform (BWT) which tends to reorder symbols according to their following context. It becomes a promising compression approach based on contextmodelling. BWT was initially applied for text compression software such as BZIP2 ; nevertheless it has been recently applied to the image compression field. Compression scheme based on Burrows Wheeler Transform is usually lossless ; therefore we imple-ment this algorithm in medical imaging in order to reconstruct every bit. Many vari-ants of the three stages which form the original BWT-based compression scheme can be found in the literature. We propose an analysis of the more recent methods and the impact of their association. Then, we present several compression schemes based on this transform which significantly improve the current standards such as JPEG2000and JPEG-LS. In the final part, we present some open problems which are also further research directions

The aim of the thesis is to study the AdS/CFT correspondence and the AdS2/SYK connection as a very special example of the duality. While the first part of the thesis contains a review of AdS/CFT correspondence in arbitrary dimensions, the later parts focus on an interesting and speculative connection between the gravitational physics in two dimensional nearly AdS2 spacetime and one dimensional SYK model. More specifically, the connection is realized in terms of certain features of the SYK model in strong coupling limit, which resembles those of nearly AdS2 Jackiw-Teitelboim theory.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第22231号 / 理博第4545号 / 新制||理||1653(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 池田 保, 教授 雪江 明彦, 教授 並河 良典 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

The increasing demand for reclamation of periodically waterlogged nearly level agricultural lands in humid tropical areas and the hazard of soil deterioration and soil moisture balance disturbances by current land forming methods suggests the need for investigations of new surface drainage design procedures. This report presents a rainfall-runoff model for simulating hydrographs from ungaged agricultural plots. The model is based on routing procedures and utilizes common soil and hydrologic data. Tests made with several small agricultural water-sheds indicate that the model could be a useful tool in simulating surface drainage design. Input data for the model consists of (a) rainfall data, (b) infiltration, (c) watershed characteristics, and (d) soil parameters. The model could be used with any computer or desk calculator. Since the model was developed for surface drainage purposes, its use is limited to wet conditions in homogeneous and rectangular plots.

We give an Ansatz to construct pairs of locally homogeneous nearly Kaehler manifolds that are isospectral for the Dirac and the Hodge Laplace operator in dimensions higher than six and investigate the existence of generic isospectral pairs in dimension six.

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Krylov subspace iterative methods provide an effective tool for reducing the solution of large linear systems to a size for which a direct solver may be applied. However, the problems of limited storage and speed are still a concern. Therefore, in this dissertation work, we present iterative Krylov subspace algorithms for non-Hermitian systems which do have fixed memory requirements and have favorable convergence characteristics. This dissertation describes three projects. The first project concerns short-term recurrence Krylov subspace methods for nearly-Hermitian linear systems. In 2008, Beckermann and Reichel introduced a short-term recurrence progressive GMRES algorithm for nearly-Hermitian linear systems. However, we have found this method to be unstable. We document the instabilities and introduce a different fixed-memory algorithm to treat nearly-Hermitian problems. We present numerical experiments demonstrating that the performance of this algorithm is competitive. The other two projects involve extending a strategy called Krylov subspace recycling, introduced by Parks and colleagues in 2005. This method requires more overhead than other subspace augmentation methods but offers the ability to recycle subspace information between cycles for a single linear system and recycle information between related linear systems. In the first project, we extend subspace recycling to the block Krylov subspace setting. A block Krylov subspace is a generalization of Krylov subspace where a single starting vector is replaced with a block of linearly independent starting vectors. We then apply our method to a sequence of matrices arising in a Newton iteration applied to fluid density functional theory and present some numerical experiments. In the second project, we extend the methods of subspace recycling to a family of linear systems differing only by multiples of the identity. These problems arise in the theory of quantum chromodynamics, a theory of the behavior of subatomic particles. We wish to build on the class of Krylov methods which allow the simultaneous solution of all shifted linear systems while generating only one subspace. However, the mechanics of subspace recycling complicates this situation and interferes with our ability to simultaneously solve all systems using these techniques. Therefore, we introduce an algorithm which avoids this complication and present some numerical experiments demonstrating its effectiveness. / Mathematics