Mathematics

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Research Topic Summary.

The aim of this research is to develop and investigate adaptive classification methods that are effective under heteroskedastic conditions. Heteroskedasticity, where data variance changes depending on the characteristics of an object, is a common issue in various fields, such as economics, medicine, and social sciences. Traditional classification methods may be ineffective when data has a non-homogeneous structure, making it essential to create new algorithms that can accurately classify heterogeneous data. This study will focus on creating a mathematical model to describe object distribution and developing procedures based on this model to enable precise data classification. Additionally, methods will be proposed for integrating additional information about object positions and context into classification algorithms to further improve accuracy. At the conclusion of the study, a comparison of the developed procedures with alternative methods will be conducted using real data to assess their practical applicability. The expected outcomes include the development of classification procedures with higher accuracy and efficiency, tailored for heteroskedastic data. These results will contribute to advancements across fields—from finance to biomedicine—ensuring more precise data analysis and forecasting.

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Research Topic Summary.

The objective of the research is to develop a common methodology for accurate and fast estimation of state-space model parameters and application of Bayesian statistics by combining particle filtering and Markov chain Monte Carlo methods. Tasks linked to expected results: 1. To analyse the state-of-the-art literature on state-space modelling and parameter estimation with particle filtration and MCMC methods, and to develop an initial estimation methodology; 2. To process experimental and real data and research on the application of Bayesian statistics; 3. To develop the particle MCMC algorithm and investigate other accuracy and speed factors; 4. To review and test acceleration alternatives for the MCMC algorithm, e.g. parallelisation; 5. To compare parameter updating with new and varying uncertainty data options; 6. To summarise the research results and disseminate the final achievements. For further information on the topic and relevant R&D work please contact the supervisor.

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Research Topic Summary.

The objective of the research is the creation of methodology and test calculations based on data science development and information integration for the most accurate diagnosis and risk prevalence prediction. Tasks: 1. Review and compare possibilities of Bayes methods and other information integration methods and models as well as the application of related software and data science methods intended for diagnostic and risk prediction. 2. Define information integration methods and relevant models accuracy metrics, criteria and their evaluation procedures in the context of data relevant to diagnostic and risk prediction. 3. Develop and demonstrate the information integration tools whose usage increases the accuracy and/or reduces the risk of incorrect decision-making. 4. Perform the pilot studies of methods and models intended for diagnostic and risk prevalence prediction, and create a methodology for the effective application of these means. For further information please contact the supervisor of the topic.

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Research Topic Summary.

Overparameterization and the related double-descent phenomena are important aspects of how very deep neural networks learn. Implicit biases, generalization properties, robustness, convergence are the different sides of a statistical learning problem related to overparameterized mathematical models and especially to neural networks. However, there are still many open problems: double descent in non-standard learning scenarios (e.g. unsupervised learning), how to balance good generalizability properties of overparameterized models with robustness, how using multiple data modalities affect the dynamics of double descent, etc. Additionally, practical implications of these learning regimes within specific fields like medical image analysis are also unclear. The aim of this research is to investigate using mathematical modelling techniques (for example, random matrix theory) various learning regimes of extremely overparameterized (i.e. far beyond interpolation threshold) neural networks and related models. An important part of this project is consideration of possible implications to the specific context of medical diagnostics mainly via classification and regression tasks. For example, understanding mathematical properties of very deep neural networks and how they learn will help in better understanding of how spurious correlations in medical images are learned and how to avoid it.

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Research Topic Summary.

Blockchain technologies are widely used in the modern world. Surely, almost every adult has heard of Bitcoin cryptocurrency. This technology is also useful for banks, where it can be implemented in the development of payment systems. Cryptography has a major role in this development. Using cryptographic algorithms blocks are linked into a chain. These algorithms must be cryptographically secure to withstand quantum cryptanalysis. However, algorithms used nowadays do not provide sufficient protection from such attacks. Moreover, quantum algorithms to be applied for the analysis of this technology are known. Hence it is time to implement the elements of non-commutative cryptography in blockchain technology. Relying on the previously published results of the research of our group we plan to propose a payment system based on quantum cryptanalysis-resistant blockchain technology.

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Research Topic Summary.

Multi-state models are used to describe complex processes with multiple possible states and transitions among them. They allow for detailed analysis of time-dependent events and help quantify the risks and probabilities associated with different outcomes. A convenient modeling assumption is that the multi-state stochastic process follows a Markovian property, which, however, does not valid for many real-world applications. As such, this research aims to enhance non-Markovian models by incorporating external impacts, such as spatial data, panel data, shocks, causal relationships, and missing data reconstruction, with the aim to improve the dynamics of stochastic system to be modelled. Key tasks include extending and formalizing existing models, estimating parameters, developing transition matrices and probabilities, simulating outcomes, and comparing with similar models. In general, the application fields cover finance and economics, healthcare, engineering and environmental studies.

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Research Topic Summary.

Thin films covered with nano-particles having antimicrobial properties are gaining a lot of attention in recent years due to the increased antibiotics resistance. Understanding bacterial behavior is crucial in various fields including microbiology, biotechnology, and healthcare. Predictive modeling of bacterial behavior can provide insights into microbial dynamics, antibiotic resistance, and disease pathogenesis. By leveraging mathematical modeling and neural networks, which excel in learning complex patterns from data, this research aims to advance our ability to predict and understand bacterial behavior with high accuracy and efficiency. The aim of this research is to develop and evaluate methodology for predictive modeling of bacterial behavior on thin films using mathematical modeling and neural networks.

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Research Topic Summary.

Statistical hypothesis testing is a crucial step in decision-making within modern data analysis. Various fields of data analysis, such as economics, medicine, and social sciences, require reliable statistical methods, as incorrect assumptions can lead to inaccurate results and wrong decisions. The relevance of this research arises from the need to develop adaptive criteria that can effectively test hypotheses such as homogeneity, compatibility, symmetry, and independence, in the presence of data heterogeneity. The aim of the dissertation is to develop statistical criteria based on N-metric theory that are sensitive to different types of data distributions and demonstrate high efficiency. The research will explore how N-metric theory can be applied to hypothesis testing, evaluating its effectiveness using Bahadur's asymptotic relative efficiency. It is expected that these criteria will provide greater accuracy and reliability in both practice and theory. Furthermore, the study will compare the proposed methods with classical criteria, using real data and Monte Carlo simulation techniques. The goal is to create new, adaptive, and powerful statistical hypothesis testing criteria that can be applied in practice. These methods will help ensure more accurate data analysis, allowing for well-informed decision-making across various fields.

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Research Topic Summary.

Relevance. Coupled Map Lattices (CML) are widely used and applied in the research of dynamical systems. Coupled Map Lattice of Matrices (CMLM) represents a new class of models and a promising research area introduced by the author of the proposed dissertation topic along with co-authors in 2018. Matrix iterative maps and their networks have been introduced and ongoing research is being presented in high-level scientific journals - indicating the importance of the research and its application possibilities. There is an increasing number of publications citing applied CMLM models and attracting growing attention from scientists willing to collaborate by combining their own research topics with CMLM studies. The scientific research problem is to examine CMLM tasks aiming to explore and apply nth order matrix CMLs that have never been explored before. The aim of the scientific research is to explore and apply networks of nxn matrix iterative maps. The abundance of scalar coupled map lattices and related studies examined in the literature only confirms the countless possible directions in which new extensions of matrix iterative models can be introduced. However, more specific research objectives can also be identified: trantient processes (e.g., traveling or spiral waves, solitons, chimeras) in coupled map lattices of matrices; theorethical aspects and application of divergence processes in coupled map lattices of matrices, as well as the finite-time stabilization of unstable solutions in CML of matrices. The proposed research aims to intensify breakthroughs in the respective areas guaranteeing CMLM studies.