This thesis is devoted to show various applications of fixed point theorems on dif- ferential equations. In the beginning we use a notion of topological degree to derive several fixed points theorems, primarily Brouwer, Schauder and Kakutani-Ky Fan the- orem. Then we apply them on a wide range of relatively simple problems from ordinary and partial differential equations (ode and pde). Finally, we take a look on a few more complex problems. First is an existence of a solution to the model of mechanical os- cillator with non-monotone dependence of both displacement and velocity. Second is a solution to so called Gause predator-prey model with a refuge. The last one is cer- tain partial differential equation with a constraint which determines maximal monotone graph. 1

A convertible bond (CB) is a financial derivative, a so called hybrid security. It is an issued contract from a company or a government, which is paid for up-front. The contract yields a known amount at the specified maturity date, unless the holder chooses to convert it into an amount of the underlying asset. This kind of financial products can have complex features affecting the contract price and the optimal exercising situation. The partial differential equation (PDE) approach used for pricing financial derivatives makes it possible to describe convertible bonds with a physical model, a reversed diffusion described by a parabolic PDE. One can sometimes find both analytical and numerical solutions for this type of PDEs and interpret the solutions from a financial point of view, as they suggest predictable behaviour of the contract price.

There exist many numerical methods for numerical solutions of the systems of stochastic differential equations. We choose the method of deterministic quadrature formulae proposed by Müller–Gronbach, and Yaroslavtseva in 2016. The idea is to apply a simplified version of the cubature in Wiener space. We explain the method and check how good it works in the simplest case of the classical Black–Scholes model.

In this paper, we develop a model for predation movements of a subsocial spider species, Anelosimus studiosus. We expand on a previous model to include multiple spider interaction on the web as well as a latency period during predation. We then use the model to test different spatial configurations to determine the optimal spacing of spiders within a colony for successful capture during predation. The model simulations indicate that spiders uniformly spacing out along the edge of the web results in the most successful predation strategy. This is similar to the behavior observed by Ross (2013) in which it was determined to be statistically significant that during certain times of the day, spiders were positioned along the edge more than expected under complete spatial randomness.

We consider two distinct techniques for estimating random parameters in random differential equation (RDE) models. In one approach, the solution to a RDE is represented by a collection of solution trajectories in the form of sample deterministic equations. In a second approach we employ pointwise equivalent stochastic differential equation (SDE) representations for certain RDEs. Each of the approaches is tested using deterministic model comparison techniques for a logistic growth model which is viewed as a special case of a more general Bernoulli growth model. We demonstrate efficacy of the preferred method with experimental data using algae growth model comparisons.

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第19121号 / 情博第567号 / 新制||情||100(附属図書館) / 32072 / 京都大学大学院情報学研究科複雑系科学専攻 / (主査)教授 木上 淳, 教授 磯 祐介, 教授 西村 直志 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

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The probit function is the inverse of the cumulative distribution function associated with the standard normal distribution. It is of great utility in statistical modelling. The Carleman embedding technique has been shown to be effective in solving first order and, less efficiently, second order nonlinear differential equations. In this thesis, we show that solutions to the second order nonlinear differential equation for the probit function can be approximated efficiently using a variant of the Carleman embedding technique.

The Vector Quantization (VQ) model proposes a powerful solution for data clustering. Its design indicates a specific combination of concepts from machine learning and dynamical systems theory to classify input data into distinct groups. The model evolves over time to better match the distribution of the input data. This adaptive feature is a strength of the model, as it allows the cluster centers to shift according to the input patterns, effectively quantizing the data distribution. It is a gradient dynamical system, using the energy function V as its Lyapunov function, and thus possesses properties of convergence and stability. These characteristics make the VQ model a promising tool for complex data analysis tasks, including those encountered in machine learning, data mining, and pattern recognition.In this study, we have applied the dynamic model to the "Breast Cancer Wisconsin Diagnostic" dataset, a comprehensive collection of features derived from digitized images of fine needle aspirate (FNA) of breast masses. This dataset, comprising various diagnostic measurements related to breast cancer, poses a unique challenge for clustering due to its high dimensionality and the critical nature of its application in medical diagnostics. By employing the model, we aim to demonstrate its efficacy in handling complex, multidimensional data, especially in the realm of medical pattern recognition and data mining. This integration not only highlights the model's versatility in different domains but also showcases its potential in contributing significantly to medical diagnostics, particularly in breast cancer identification and classification.

No / In this paper we discuss a novel method of mathematically modelling facial action units for accurate representation of human facial expressions in 3- dimensions. Our method utilizes the approach of Facial Action Coding System (FACS). It is based on a boundary-value approach, which utilizes a solution to a fourth order elliptic Partial Differential Equation (PDE) subject to a suitable set of boundary conditions. Here the PDE surface generation method for human facial expressions is utilized in order to generate a wide variety of facial expressions in an efficient and realistic way. For this purpose, we identify a set of boundary curves corresponding to the key features of the face which in turn define a given facial expression in 3-dimensions. The action units (AUs) relating to the FACS are then efficiently represented in terms of Fourier coefficients relating to the boundary curves which enables us to store both the face and the facial expressions in an efficient way.