Extending the Class of Mathematical Problems Solvable in School

The problems of practical importance which are considered in school today necessarily have to lead to a mathematical model that can be solved by school mathematics knowledge. This includes systems of equations of at most second degree, some simple trigonometry and/or some basic geometry. This restricts severely the class of such problems and conveys the impression that mathematics is not applicable enough. We provide examples of problems related to practice which are difficult to solve by means of traditional school mathematics but are amenable for solving (at least with a certain precision) with the use of software systems dealing with mathematical problems. We also present the results of an experiment with such problems that were given to school students participating in the second round of the competition “VIVA Mathematics with Computer”. ACM Computing Classification System (1998): K.3.1

Mathematics competitions: an integral part of the educational process

For a century and a half, the scene of mathematics competitions underwent a remarkable transformation from isolated and geographically scattered events to a full-scale and a full-featured vibrant global ecosystem comprising an impressive variety of competitions, school students, university students, teachers, mentors, scientists, schools, universities, research institutions, journals, websites, civil society organizations, educational authorities, parents, etc. The evolution, the current state, the functioning of this ecosystem as well as its role for the identification and development of talent, its impact on the educational process, and the institutions involved with it, is briefly reflected on. Some relatively new online competitions are presented that cultivate the use of dynamic geometry software systems for a deeper understanding of mathematical facts and phenomena, and for finding approximate numerical solutions to problems that are not part of a typical school curriculum, but often arise from real-life practice.National Scientific Program "Information and Communication Technologies for Unified Digital Market in Science, Education and Security” realized with the support of the Bulgarian Ministry of Education and Science

Dense Continuity and Selections of Set-Valued Mappings

∗ The first and third author were partially supported by National Fund for Scientific Research at the Bulgarian Ministry of Science and Education under grant MM-701/97.A theorem proved by Fort in 1951 says that an upper or lower semi-continuous set-valued mapping from a Baire space A into non-empty compact subsets of a metric space is both lower and upper semi-continuous at the points of a dense Gδ -subset of A. In this paper we show that the conclusion of Fort’s theorem holds under the weaker hypothesis of either upper or lower quasi-continuity. The existence of densely defined continuous selections for lower quasi-continuous mappings is also proved

Achievements and Problems in the In-service Teacher Education in Inquiry Based Style

Report published in the Proceedings of the National Conference on "Education and Research in the Information Society", Plovdiv, May, 2015The paper deals with the experience of the authors in promoting the Inquiry Based Learning (IBL) in mathematics and science education within international and national projects. The emphasis is on in-service teacher education. Various types of activities and resources in support of all levels of IBL are considered, e.g. professional development courses, seminars, workshop, and performances; implementation of resources stimulating students to behave like working mathematicians. The first visible positive effects and potential problems in the implementation of IBL in the Bulgarian schools are discussed.Association for the Development of the Information Society, Institute of Mathematics and Informatics Bulgarian Academy of Sciences, Plovdiv University "Paisii Hilendarski

Uniqueness on a Residual Part of Best Approximations in Banach Spaces

[Kenderov Petar S.; Кендеров Петър С.]The paper presents some general results concerning continuity on a residual part of every semi-continuous multivalued mapping. As an application, some results are proved asserting (under some conditions) that every metric projection is single-valued on a residual part of the space. The results were announced in [6]

Псевдокомпактни полу-топологични групи

Митрофан М. Чобан, Петър Ст. Кендеров, Уорън Б. Муурс - Полу-топологична група (съответно, топологична група) е група, снабдена с топология, относно която груповата оперция произведение е частично непрекъсната по всяка от променливите (съответно, непрекъсната по съвкупност от променливите и обратната операция е също непрекъсната). В настоящата работа ние даваме условия, от топологичен характер, една полу-топологична група да е всъщност топологична група. Например, ние показваме, че всяка сепарабелна псевдокомпактна полу-топологична група е топологична група. Показваме също, че всяка локално псевдокомпактна полу-топологична група, чиято групова операция е непрекъсната по съвкупност от променливите е топологична група.A semitopological group (topological group) is a group endowed with a topology for which multiplication is separately continuous (multiplication is jointly continuous and inversion is continuous). In this paper we give some topological conditions on a semitopological group that imply that it is a topological group. For example, we show that every separable pseudocompact group is a topological group. We also show that every locally pseudocompact group whose multiplication is jointly continuous is a topological group. *2010 Mathematics Subject Classification: Primary 22A10, 54E52, 54D30