Current projects “Scaling in stochastic models – Young Researchers’ Career Development Project – Training of Doctoral Students”, (School of Applied Mathematics and Informatics, J. J. Strossmayer University of Osijek – Croatian Science Foundation) – Project coordinator: Danijel Grahovac Summary: The PhD student will primarily study the limiting behavior of superpositions of different classes of stationary processes and their properties. Special focus will be on the so-called ambit processes. The advantages of the ambit process are the explicit definition of the process as a stochastic integral and the availability of the basic characteristics. The goal of the doctoral dissertation of the PhD student is to provide a deeper insight into the properties of some classes of ambit processes and especially the limiting behavior on large time scales. As part of his research, the PhD student will deal with processes such as superpositions of Ornstein-Uhlenbeck-type processes (supOU). One of the goals of the doctoral thesis is to investigate the limits of the supOU processes in the so-called triangular scheme. While in the classical scheme only self-similar processes can be obtained as limits, in the triangular scheme it is also possible to obtain processes with different scaling properties. Such processes often have interesting properties and provide transitional models between different self-similar processes, which will be intensively studied in the doctoral thesis. Programme: Young Researchers’ Career Development Project – Training of Doctoral Students (DOK-2023-10) Mentor’s name and surname: Danijel Grahovac PhD name and surname: Magdalena Mikić Project duration: 1 October 2024 – 30 September 2030 “Homogenization and Optimization in Mechanics of Solids”, – Young Researchers’ Career Development Project – Training of Doctoral Students (School of Applied Mathematics and Informatics, J. J. Strossmayer University of Osijek – Croatian Science Foundation) – Project coordinator: Krešimir Burazin Summary: The assistant will primarily focus on studying the homogenization of the Reissner-Mindlin plate equation. The goal of the assistant’s doctoral work is to fully develop the homogenization theory for this equation in the sense of operator theory (H-convergence). This includes establishing a compactness result for H-convergence, as well as verifying/proving properties of H-convergence (such as independence from boundary conditions, metrizability, energy convergence, etc.). Special attention will be given to applying H-convergence for modeling composite Reissner-Mindlin plates and calculating the corresponding Hashin-Shtrikman bounds on the energy of the resulting composite material. The research will also explore the characterization of G-closure, which is expected to remain partially unresolved, as is the case with other elasticity theory equations. However, in the low-contrast regime of mixed materials, we expect to obtain considerably more information on G-closure, paving the way for potential applications in the optimal design of elastic plates. Programme: Young Researchers’ Career Development Project – Training of Doctoral Students (DOK-2023-10) Mentor’s name and surname: Krešimir Burazin PhD name and surname: Ivan Vuković Project duration: 1 October 2024 – 30 September 2030 “Assessment of chronic exposure to food contaminants”, (School of Applied Mathematics and Informatics, J. J. Strossmayer University of Osijek – Croatian Agency for Agriculture and Food) – Project coordinator: Nenad Šuvak Summary: Within this project an application software solution will be developed to assess the exposure of the population of the Republic of Croatia to contaminants from different categories of food. The exposure assessment will be carried out according to different categories of respondents, for example by gender and by regions where the respondents live. In addition to the detailed results of the exposure assessment, which will be organized into tables, the software solution will contain various graphical displays that help illustrate and understand the assessment results: box and whisker plots, bar charts, histograms and graphs of empirical distribution functions of the analyzed sample characteristics. The programming code will be written in the open source programming language R (https://www.r-project.org/), and the interface itself will be created using its Shiny extension (https://www.rstudio.com/products/shiny /). Technically, the procedure will consist of uploading tables of data on the eating habits of a sample of residents of the Republic of Croatia in csv (comma setarated values) format into the created application. After that, tables and graphics will be automatically generated. Programme: Internal project of the School of Applied Mathematics and Informatics – Strengthening cooperation with the economy Project collaborators: School of Applied Mathematics and Informatics, J. J. Strossmayer University of Osijek: Nenad Šuvak (collaborator), Danijel Grahovac, Dominik Mihalčić Project duration: 18 December 2023 – 18 December 2024 “Scaling in stochastic models”, (School of Applied Mathematics and Informatics, J. J. Strossmayer University of Osijek – Croatian Science Foundation) – Project coordinator: Danijel Grahovac Link: Scaling in stochastic models Summary: Stochastic models are of great importance in describing numerous phenomena in various fields of science and life. An essential property of time-dependent models is their behavior under time and space scaling. The characteristics of many phenomena can be significantly changed by changing the scale. For example, as we move from the tiny world of particles to the astronomical scales, the laws of nature may behave drastically different. The need for models with specific scaling properties comes from numerous practical applications in medicine, biology, physics, finance, economics and other fields of science. The goal of the project is the construction and investigation of stochastic models with special scaling properties. The emphasis is placed on the study of properties of models in the limits of small or large scales. Among other results, a possible explanation will be provided for the phenomenon related to the scaling of empirical moments, known as Taylor’s law. By replacing the time scale of some standard processes, the new classes of stochastic processes will be constructed and analyzed. These new models will be applied in epidemiology and EEG signal analysis. Furthermore, model for the clasification of neurocognitive outcomes based on scaling parameters will be developed. Programme: Research projects – Croatian Science Foundation (IP-2022-10) Team members: Nenad Šuvak, (School of Applied Mathematics and Informatics , J. J. Strossmayer University of Osijek), Ivan Papić (School of Applied Mathematics and Informatics , J. J. Strossmayer University of Osijek), Nikolai Leonenko (Cardiff University), Alla Sikorski (Michigan State University) Project duration: 15 December 2023 – 14 December 2027 “Homogenization and Optimization in Mechanics of Solids”, (School of Applied Mathematics and Informatics, J. J. Strossmayer University of Osijek – Croatian Science Foundation) – Project coordinator: Krešimir Burazin Link: Homogenization and Optimization in Mechanics of Solids Summary: The main topics of this project are different problems in homogenization, dimension reduction and structural optimization with the special emphasis on the applications in mechanics of solids. The physical idea of homogenization, i.e., averaging a highly heterogeneous media to derive its effective properties has a rather long history. Mathematically, it emerged from difficulties faced with (numerical) analysis of governing partial differential equastions in cases when the material oscilates an a small scale or is significantly thinner in one or two directions. A passage to the limit (with respect to some material parameters,i.e., in appropriate topology) usually leads to models that are expected to be simpler, both from the analytical and numerical point of view. The nature of the oscilations of the material can be assumed to be ergodic, periodic or completely general and thus analysed by (stochastic) two-scale convergence method, compensated compactness approach (H-convergence, H-measures) or calculus of variations techniques (Gamma-convergence). Described procedure is well suited for modelling of composite materials, which are fine mixtures of two or more constituent materials. The special interest arises in high-contrast composites, where one obtains a metamaterial with a band gap structure of the spectrum. Optimizing design of devices made of such responsive material and structural material is of great interest. The research proposed by the project encompasses modelling of such materials, and constitutes of challenging and contemporary problems, including topics of stochastic homogenization of high-contrast media, homogenization and dimension reduction in elasto-plasticity, crystallization, quantitative homogenization for evolution for linear elastic thin structures, homogenization of Kirchhoff-Love and Reissner-Mindlin plate equastion, structural optimization of thin elastic plates and in 3D linearized elasticity, homogenization of Friedrich’s systems. Programme: Research projects – Croatian Science Foundation (IP-2022-10) Team members: Matteo Capoferri (Heriot-Watt University), Paolo Piovano (University of Vienna), Sergio Gutierrez (Pontificia Universidad Catolica de Chile), Marin Bužančić (University of Zagreb, Faculty of Chemical Engineering and Technology), Mikhail Cherdantsev (Cardiff University), Ivana Crnjac, (School of Applied Mathematics and Informatics , J. J. Strossmayer University of Osijek), Jelena Jankov (School of Applied Mathematics and Informatics , J. J. Strossmayer University of Osijek), Igor Velčić (University of Zagreb, Faculty of Electrical Engineering and Computing), Josip Žubrinić (University of Zagreb, Faculty of Electrical Engineering and Computing) Project duration: 23 December 2023 – 22 December 2027 “Development of an interactive virtual environment” (Department of Mathematics, J. J. Strossmayer University of Osijek) – Project coordinator: Luka Borozan Summary: The project aims to create a three-dimensional virtual environment containing a large number of objects with which the user can interact. Besides user interactions, each object can have its own behavior controlled by artificial intelligence. Moreover, over a local network or the internet, multiple users can access the same virtual environment simultaneously, allowing interaction between users. Ultimately, the project should quickly adapt to create virtual environments applicable across numerous fields. Programme: Internal project of the Department of Mathematics – Strengthening cooperation with the economy Project collaborators: Luka Borozan (coordinator), Bartol Borozan (collaborator), Antonio Janjić (collaborator), Leon Vilagoš (collaborator) Neven Kopecki (collaborator), Bruno Bugarić (collaborator) i Sven Viland (collaborator) Project duration: 1 May 2021 – 1 May 2025