Teachers commonly consider the number of teaching hours in a particular subject to be insufficient for achieving sustainable results in the acquisition of knowledge and skills. It is assumed that increasing the number of hours should lead to an improvement in the quality of education. Thus, the question of the optimal amount of teaching time remains open. This study focuses on this methodological problem by analyzing the opinions of teachers from schools in the capital who teach two types of students simultaneously – those included in extended mathematics training and those who are taught only within the compulsory program. A comparative approach is used to examine the added value of extended training as perceived by teachers, as well as its impact on the quality of teaching and learning outcomes. The results contribute to a deeper understanding of the relationship between the quantitative parameters of teaching time and the qualitative dimensions of the educational process. In the European Schools attached to the institutions of the European Union, additional hours are organized in different ways. This study also examines these differences and seeks to answer the question of how to use additional teaching time most effectively.

In this paper, we propose a new system that occurs in modelling “the battle of the sexes” in evolutionary biology (Hofbauer & Sigmund 1988). The existence of a heteroclinic cycle and a continuous family of periodic orbits of the system is established; then the dynamical characteristics of a time-periodic perturbation of the system are investigated. By using the well-known Melnikov’s method, a sufficient condition is obtained for the perturbed system to have a transverse hetero-clinic cycle and hence to possess chaotic behaviour in the sense of Smale. One possible application that Melnikov functions may find in the modelling and synthesis of radiating antenna patterns is considered. We demonstrate some modules for investigating the dynamics of the proposed model. This will be included as an integral part of a planned much more general Web-based application for scientific computing. The proposed new extended model contains many free parameters (the coefficients ai, i = 1, 2, . . . ,N), which makes it attractive for use in the fields of biological applications, chemistry, sociology, lifetime analysis, reaction kinetics, biostatistics, population dynamic, medical research, games theory etc. Finally, a special case of subharmonic solutions is discussed.

The present study is dedicated to describing a sample approach for integrating digital tools powered by artificial intelligence (AI) and project-based method in mathematics education. Numerous studies have demonstrated the significant potential of AI to revolutionize mathematics teaching and learning, while also highlighting the importance of addressing the associated challenges to ensure its effective and responsible implementation in the educational process. The project-based method enables students to collaborate, engage in discussion, think critically, and express their creativity. This paper presents a possible application of AI-based digital tools and the project-based method in teaching the topic of Trigonometry from the 11th grade mathematics curriculum. The results from a diagnostic study conducted with students from the Foreign Language High School “Antoine de Saint-Exupery” in Plovdiv are also analyzed.

In the modern Bulgarian school of the 21st century, the issue of using innovative strategies for teaching students from the initial stage of primary education is increasingly being addressed. This paper examines the problem of improving students' mathematical competencies through project-based learning activities. By implementing such activities, students apply the acquired mathematical knowledge in practice. A model of project-based learning in mathematics in the 3rd grade is proposed. The model is based on 5 problem-oriented tasks, which are implemented in teams in two experimental classes with a total of 38 students. The tasks in the project and an analysis of the most common difficulties encountered by students are presented.

A variant for interactive presentation of content with the inclusion of augmented reality using dynamic software GeoGebra is described. A concretization is made with the Pythagorean Tree fractal. The goal is to support the understanding and memorization of the relevant content by providing conditions for research and experience, as well as for gaining experience in using augmented reality as a technology and motivating teachers to implement it in school. Models are described with which the growth of the fractal is observed through animation, as well as changes depending on several parameters. Basic ideas for creating the relevant compositions with dynamic software GeoGebra are described. The goal is to facilitate the use of this resource when working in STEAM centers. An expert assessment of the proposed resources and methodology is presented, as well as the results of a pilot study. Possibilities for continuing research on the topic are discussed both in terms of interpretations related to the fractal (for example, using different angles in the steps, different coloring options, using figures other than regular polygons, etc.), and in terms of software products for creation and technological tools for impact (for example, virtual reality, mapping, etc.). The possibility of using such resources in science museums is also emphasized, which will improve the implementation of their educational function.

This paper offers a critical review of the applications of AI in cybersecurity, focusing on the recent trends of automation in threat detection, enhancement of response strategies, and prediction of vulnerabilities. The methodology is based on a thorough analysis of empirical studies up to 2021 as per the efficiency of AI malware detection, insider threat identification, and mitigation of zero-day vulnerabilities. In particular, machine learning- and deep learning-based methodologies of artificial intelligence ensure clear advantages over conventional models concerning the precision in detection and reduction of false positives. However, challenges persist regarding explainability, scalability, and ethical concerns around data bias and quality. Finally, this paper concludes by pointing out some areas of future research with regard to needing XAI techniques and methods related to bias reduction to establish better trust in the efficacy of AI-driven cybersecurity frameworks.

Modern technologies enable the use of visually oriented tools to organize projects, ideas, and tasks in a completely informal manner. Visually oriented software does not impose a structure or method for organizing and presenting user projects. This approach gives users the ability to organize their tasks in a visual format, promoting a more intuitive and adaptable work environment. This article introduces software product designed and developed to effectively organize and plan tasks, projects, or ideas. The product allows users to create boards on which they can mark their ideas by adding certain elements. With these different elements, the workflow of each user can be visually illustrated and parts of the plan can be highlighted. Easy planning is achieved from an easy-to-read format with a variety of visual elements that are more natural to organize a plan with step implementation, and quickly share the plan for all of its development teams.

The article presents data from a survey of the attitudes towards working with intelligent programs based on large linguistic models (LLM) among 218 students and specialists – future teachers educated at Shumen University. The survey was conducted within various disciplines and students were encouraged to use chatbots in the preparation of coursework, didactic materials, presentations, solving and composing tasks. Despite the presence of certain experience in working with a chatbot among some of the respondents, the survey was conducted in two stages. In the first stage, the survey was asked directly to the students. It was followed by a brief presentation of some possibilities for working with a chatbot. We mainly used ChatGPT for demo. A demonstration was made with examples typical for the relevant discipline. Pedagogically oriented projects were assigned for independent implementation with ChatGPT. When the majority of students gained experience using the chatbot to handle didactic tasks, the second survey was conducted. There were also final discussions in the individual groups about the possibilities and the place of the LLM in the work of the teacher and the student. The results show positive attitudes towards the implementation of artificial intelligence approaches and tools, both in the students’ self-training and in the teacher’s work. The received responses and shared ideas show that future teachers are responsible, ethical and critical in implementing learning innovations. We clarify the necessity of supplementing current academic disciplines with activities aimed at responsible application of LLM-based programs, both for self-training and for the creation of didactic materials for practical classes in schools and kindergartens.

Dynamic computer models of a geometry problem from the 23rd Balkan Mathematical Olympiad in 2006 are presented. Ideas for their use are described: for providing conditions for research work and formulating hypotheses, for discovering an idea for solving, for presenting a solution, for learning the software through tasks. A reference is made to dynamic computer models of other Olympiad tasks provided in the Virtual Mathematics Laboratory developed in the Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences. The possibility of their usage in both informal and in formal education in mathematics and information technologies is discussed.