The model of methodical system and learning objectives of the foundations of mathematical informatics for students of technical universities

У статті наведені етапи розробки моделі методичної системи навчання спецкурсу «Основи математичної інформатики» студентів технічних університетів. Конкретизовано цільовий компонент методичної системи.Abstract. Introduction. Development of methodical system for training course “Foundations of mathematical informatics” plays a key role in forming the students’ of technical universities competencies in mathematical informatics. So it is very important to analyze of the components of methodical system, identify the weaknesses and problems that can significantly impair its quality and which can not be overcome without its further development. Purpose. Develop a model of methodical system of training course “Foundations of mathematical informatics” for students of technical universities and specify its target component. Methods. Using cloud technologies at learning the foundations of mathematical informatics requires the construction technology training, which results in the selection of appropriate cloud-oriented forms of organization and teaching methods. On the other hand, the theory, methods and tools for cloud significantly affect the primary content of learning and its goals. Thus, the cloud technology theory, methods and tools are the basis for constructing methodical system of training course “Foundations of mathematical science”. Results. Development of methodical system of training course “Foundations of mathematical informatics” plays a leading role in forming the students’ of technical universities competencies in mathematical informatics due to its fundamental impact and technology improvement. Originality. Theoretically grounded and constructed the model of methodical system of training course “Foundations of mathematical informatics” and learning goals. Conclusion. The model of methodical system of training course “Foundations of mathematical informatics” includes content, objectives and learning technology. The last one contains forms of organization, methods and teaching tools, including leading cloud technologies. The main purpose of training course “Foundations of mathematical informatics” is developing of students competencies of technical universities in mathematical informatics

Divergence Measures

Data science, information theory, probability theory, statistical learning and other related disciplines greatly benefit from non-negative measures of dissimilarity between pairs of probability measures. These are known as divergence measures, and exploring their mathematical foundations and diverse applications is of significant interest. The present Special Issue, entitled “Divergence Measures: Mathematical Foundations and Applications in Information-Theoretic and Statistical Problems”, includes eight original contributions, and it is focused on the study of the mathematical properties and applications of classical and generalized divergence measures from an information-theoretic perspective. It mainly deals with two key generalizations of the relative entropy: namely, the R_ényi divergence and the important class of f -divergences. It is our hope that the readers will find interest in this Special Issue, which will stimulate further research in the study of the mathematical foundations and applications of divergence measures

Learning Theory and Approximation

The main goal of this workshop – the third one of this type at the MFO – has been to blend mathematical results from statistical learning theory and approximation theory to strengthen both disciplines and use synergistic effects to work on current research questions. Learning theory aims at modeling unknown function relations and data structures from samples in an automatic manner. Approximation theory is naturally used for the advancement and closely connected to the further development of learning theory, in particular for the exploration of new useful algorithms, and for the theoretical understanding of existing methods. Conversely, the study of learning theory also gives rise to interesting theoretical problems for approximation theory such as the approximation and sparse representation of functions or the construction of rich kernel reproducing Hilbert spaces on general metric spaces. This workshop has concentrated on the following recent topics: Pitchfork bifurcation of dynamical systems arising from mathematical foundations of cell development; regularized kernel based learning in the Big Data situation; deep learning; convergence rates of learning and online learning algorithms; numerical refinement algorithms to learning; statistical robustness of regularized kernel based learning

Towards Collaborative Conceptual Exploration

In domains with high knowledge distribution a natural objective is to create principle foundations for collaborative interactive learning environments. We present a first mathematical characterization of a collaborative learning group, a consortium, based on closure systems of attribute sets and the well-known attribute exploration algorithm from formal concept analysis. To this end, we introduce (weak) local experts for subdomains of a given knowledge domain. These entities are able to refute and potentially accept a given (implicational) query for some closure system that is a restriction of the whole domain. On this we build up a consortial expert and show first insights about the ability of such an expert to answer queries. Furthermore, we depict techniques on how to cope with falsely accepted implications and on combining counterexamples. Using notions from combinatorial design theory we further expand those insights as far as providing first results on the decidability problem if a given consortium is able to explore some target domain. Applications in conceptual knowledge acquisition as well as in collaborative interactive ontology learning are at hand.Comment: 15 pages, 2 figure

Introduction to Univalent Foundations of Mathematics with Agda

We introduce Voevodsky's univalent foundations and univalent mathematics, and explain how to develop them with the computer system Agda, which is based on Martin-L\"of type theory. Agda allows us to write mathematical definitions, constructions, theorems and proofs, for example in number theory, analysis, group theory, topology, category theory or programming language theory, checking them for logical and mathematical correctness. Agda is a constructive mathematical system by default, which amounts to saying that it can also be considered as a programming language for manipulating mathematical objects. But we can assume the axiom of choice or the principle of excluded middle for pieces of mathematics that require them, at the cost of losing the implicit programming-language character of the system. For a fully constructive development of univalent mathematics in Agda, we would need to use its new cubical flavour, and we hope these notes provide a base for researchers interested in learning cubical type theory and cubical Agda as the next step. Compared to most expositions of the subject, we work with explicit universe levels.Comment: 200 pages, extended version of Midlands Graduate School course (2019), includes Agda-verified mathematics. Sources available at github (as explained in the pdf file), but not in LaTe

Introducing Anthropological Foundations of Economic Behavior, Organization, and Control

One of the rude awakenings for economists from the current recession is an emerging understanding that economics has gone wild with its highbrow mathematical models that bear little resemblance to reality. The failure of economics to predict and solve the current global recession, has restored the “dismal science” title to the profession. It stems from the cultish fascination with Adam Smith’s Invisible Hand, centered around the belief in the efficient market hypothesis. This fascination exposes analytical holes in the methodology that economics has followed lately, as economics itself has moved too far away from its key social foundations. This situation is incomprehensible and needs redressing. I compiled a book - Selected Readings on the Anthropological Bases of Economic Behavior, Organization, and Control - to help economists and others get back in touch with their genealogy. This current paper is a watered down version of the introduction to Selected Readings. It concludes that real life is neither as simple nor economic as economic theory sometimes suggests. In fact, and for much of human history, non-economic factors and forces have driven economic activities. Selected Readings provides an opportunity for reinforced focus on economics principles. Alone and detached from its social foundations economics has a future only as fiction. As serious non-fiction, economics cannot successfully divorce itself from its elemental foundations in the social sciences. To advance its theory in the past economics needed to borrow from mathematics and physics. Such learning must continue. However, to remain policy-relevant economics cannot wish away the very social bases upon which it is founded. Without social foundations one might as well kiss economics goodbye.economic behavior, economic organization, economic control, Frank Knight, W. Arthur Lewis, Joan Robinson, Janos Kornai, J. Herskovits, B. Malinowski, anthropological bases