Institute of Mathematics

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Space-time description of atoms, part I: Electronic structures, dark matter, and g-factors of electron, muon and tau
(World Scientific Publishing Company, 2024-06-17)
Kostadin Trenčevski
The description of atoms is based on 3D time and some relativistic effects about spinning bodies have been published previously. The time displacement of the electrons also plays an important role. While the principal quantum number n refers to the angular momentum nℏ observed by external observer, the azimuth quantum number l refers to the angular momentum lℏ observed from the electron itself. The intrinsic angular momentum observed by the electron is ±ℏ/2 according to Stern–Gerlach experiment, but the angular momentum observed by external observer is about α^2ℏ/4. The magnetic quantum numbers are deduced from the mentioned effects and the trajectories of electrons are non-probabilistic and geometrically well determined. The spin quantum number indicates the time arrow toward the future or toward the past. So, the electrons with opposite time arrows can be grouped in pairs, where the nucleus is in the middle. Descriptions of the dark matter particles and the electrons are given. Using the value α=1/137.035999166 suggested from the QED, the g-factors of the electron and muon are ge=−2.002319304361166; gμ=−2.002331839354934, which give excellent agreements with the experiments. So, if we equalize the formulas for g-factor from QED and this approach, it determines the theoretical value of α, without experiment.
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Statistical Literacy of Undergraduate Students Before Taking the Introductory Statistics Course
(Сојуз на математичари на Македонија, 2023)
Erblina Zeqiri
;
Stevo Gjorgiev
;
Irena Stojkovska
Statistical literacy is an essential knowledge for all citizens in today’s data‐driven society. There are numerous definitions and descriptions for statistical literacy, statistical reasoning and statistical thinking. All of them are based on a common principal that the statistical literacy develops starting in schools and it relates mainly to the processes of evaluating, interpreting and communicating data. If the students do not meet these skills on time, the lack of statistical knowledge may interfere with the process of proper decision making further in their lives. A research for statistical literacy of undergraduate mathematics students at Faculty of Natural Sciences and Mathematics in Skopje, before taking the Statistics course is conducted. The research questionnaire is based on Gal (2002) and Watson and Callingham (2003) models for statistical literacy. Besides the statistical literacy components, student gave their attitudes towards statistics and its application in everyday life. The results show that the basic statistical literacy skills like reading graphical charts or tables are met, statistical knowledge needed for understanding sampling, randomness and sense of making statistical conclusions as well as mathematical knowledge are on satisfactory level, but contextual understanding and critical reasoning are below average. The results of the research will be used to improve teaching statistics at university level.
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SCALE INVARIANT STOCHASTIC GRADIENT METHOD WITH MOMENTUM
(Matematichki Bilten, Union of Mathematicians of Macedonia, 2023)
Nikolovski, Filip
;
Optimization in noisy environments arises frequently in applications. Solving this problem quickly, efficiently, and accurately is therefore of great importance. The stochastic gradient descent (SGD) method has proven to be a fundamental and an effective tool which is flexible enough to allow modifications for improving its convergence properties. In this paper we propose a new algorithm for solving an unconstrained optimization problems in noisy environments which combines the SGD with a modified momentum term using a twopoint step size estimation in the Barzilai-Borwein (BB) framework. We perform a high probability analysis for the proposed algorithm and we establish its convergence under the standard assumptions. Numerical experiments demonstrate a promising behavior of the proposed method compared to the "vanilla" SGD with momentum in noise-free and in noisy environment when the objective function is scaled.
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Assessment of the Environmental Gamma Dose Rate in Macedonia
(Technical Faculty “Mihajlo Pupin”, Zrenjanin & Faculty of Sciences and Mathematics, Niš, 2024-10)
Lambe Barandovski, Robert Šajn, Trajče Stafilov, Aneta Gacovska – Barandovska
Assessment of the environmental gamma dose rate in Macedonia was conducted in 2010, 2015, and 2020 to establish baseline data of the gamma dose rate and observe potential changes over time. For the purpose, the gamma dose rate was measured at 68 locations in 2010, and 72 location in 2015 and 2020 using a portable dosimeter positioned 1 m above the ground surface. The gamma dose rates in the country were in the range 53 - 340 nSv/h in 2010, 38 - 360 nSv/h in 2015 and 41 - 300 nSv/h in 2020. It was found that the reason for a variation in the measured values is due to the geological factors and geological formations which are different in different parts of the country.
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Topological transitivity of algebraically recurrent sets
(Academy of Sciences and Arts of Bosnia and Herzegovina, 2024)
Shoptrajanov, Martin
In this paper we will discuss the connection between topological transitivity and recurrence of G-flows acting on a compact metric space X. We will prove that the T T -property of the set of all algebraically recurrent points AR(ϕ) implies chain recurrent properties of the whole space and hence improve some of the results from [6].
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FUZZY IDEALS IN (n + k, n)-SEMIGROUPS
(Union of Mathematicians of Macedonia, 2022)
Valentina Miovska, Delcho Leshkovski, Vesna Celakoska-Jordanova
In this paper we introduce the concepts of fuzzy subset of vector valued groupoid (semigroup), as well as fuzzy subgroupoid, fuzzy subsemigroup, fuzzy i-ideal (ideal) and bi-ideal of vector valued groupoid (semigroup), investigate their properties and present suitable examples. Prime and fuzzy prime, semiprime and fuzzy semiprime subsets of vector valued groupoids are defined and their properties are investigated. We characterize the Green's relations J_{i} on a vector valued semigroup S in terms of fuzzy subsets. Green's relations J_{i}^{F} on S are suitably defined and it is shown that they coincide with the relation J_{i} on S.
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Методот на разгледување на случаи во теореми и проблеми од геометрија
(DFRM, SMM, 2024)
Анета Гацовска-Барандовска, Валентина Миовска
;
Aneta Gacovska-Barandovska, Valentina Miovska
Методот на разгледување на случаи (proof by cases), како тип на директен доказ во математиката, во некои математички кругови се смета за неелегантен и предолг. Но, од аспект на наставата по математика, во смисла на развивање на критичкото мислење, согледување на ширината и сеопфатноста на одреден проблем, посебно во одредување на сите потенцијални можности (потслучаи), методот е корисен и можеби и наједноставен за воведување на доказот во наставата по математика, посебно во теоријата на броеви, комбинаториката и геометријата. Во геометријата, голем број на основни теореми се докажани токму на овој начин. Уште во доказите на Архимед, овој метод бил она што денес елегантно се покажува со користење на гранични вредности. Овде, ќе разгледаме примери за примена на методот во содржините од геометрија во основното и средно образование, кои помагаат и во подобра визуелизација на проблемот, но и во воочување на специјалните случаи наспроти доказот во општ случај. Илустрираните примери можат да се користат како во редовната така и во додатната настава по математика. / The method of considering cases (proof by cases), as a type of direct proof in mathematics, is considered in some mathematical circles to be inelegant and too long. But from the aspect of teaching mathematics, in terms of developing a critical opinion, understanding the breadth and comprehensiveness of a certain problem, especially in determining all potential possibilities (subcases), the method is useful and perhaps the simplest for introducing the proof in teaching, especially in number theory, combinatorics and geometry. In geometry, a number of fundamental theorems have been proved exactly in this way. As far back as Archimedes' proofs, this method was what is now elegantly demonstrated by using boundary values. Here, we will consider examples of the application of the method in geometry content in primary and secondary education, which help both in better visualization of the problem, but also in perceiving the special cases against the proof in the general case. The illustrated examples can be used both in the regular and in the additional teaching of mathematics.
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Конструкција на множествата на цели и позитивни рационални броеви во наставата по математика
(DFRM, SMM, 2024)
Дончо Димовски, Анета Гацовска Барандовска
;
Dončo Dimovski, Aneta Gacovska-Barandovska
Природните броеви во наставата по математика, уште на најмала возраст, се воведуваат целосно интуитивно. На различна возраст, целите и позитивните рационални броеви се воведуваат различно. Така, целите броеви се воведуваат преку дискусија, како на пример, за негативни воздушни температури, а позитивните рационални броеви со дискусии за дропки и децимални броеви. За целосно разбирање на множествата на целите и позитивните рационални броеви од страна на учениците, неопходна е и одредена математичка прецизност и воведување на причините за конструкција на овие множества. Во наставните програми, повеќе ниту во средното образование, не се среќава проширувањето на природните броеви од аспект на конструкција на одредени алгебарски структури, ниту пак од аспект на добивање на решенија на одредени равенки. Во овој труд е дадена целосна конструкција на множествата на целите и позитивните рационални броеви, преку проширување на комутативна полугрупа со кратење до комутативна група. /Natural numbers in mathematics teaching, at an early age, are introduced completely intuitively. At different ages, integers and positive rational numbers are introduced differently. Thus, whole numbers are introduced by discussing, for example, negative air temperatures, and positive rational numbers by discussing fractions and decimal numbers. For a complete understanding of the sets of integers and positive rational numbers by the students, a certain mathematical precision and the introduction of the reasons for the construction of these sets are also necessary. In the curricula, not even in secondary education, the expansion of natural numbers from the aspect of construction of certain algebraic structures, nor from the aspect of obtaining solutions of certain equations, is found. In this paper, a complete construction of the sets of integers and positive rational numbers is given, through the extension of a commutative cancellative semigroup to a commutative group.
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Математички јазик
(Union of Mathematicians of Macedonia, 2023-01)
Doncho Dimovski, Jasmina Markoska, Aneta Gacovska - Barandovska, Gjorgji Markoski
Трудот се осврнува на конструктивните целини, лексиката, граматиката, карактеристиките и својствата на математичкиот јазик. Илустрирани ќе бидат сличностите и различностите со македонскиот јазик, односно говорниот јазик. Посочена е неизбежната врска на математичкиот јазик со говорниот јазик и значењето при формулација на задачите во образованието по математика. Дадени се примери на непрецизни формулации на математички задачи, како и предлози за нивно правилно формулирање. /
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The need of studying probability and statistics in High school education, as a preparation for the university courses
(Union of Mathematicians of Macedonia, 2023-01)
Delcho Leshkovski, Aneta Gacovska - Barandovska
In a time of rapid technological and technical development, the areas of probability and statistics are an inevitable part of everyday life and professional work. They are widely used in the natural sciences, as well as in the technical and engineering professions, information technologies, economics and finance. The curricula of the High school education and some of the technical faculties have been reviewed and for better understanding of certain subjects studied at the technical faculties, the need of studying certain notions of probability and statistics in the High school education has been recognized. / Во време на забрзан технолошко‐технички развој, областите веројатност и статистика претставуваат неизбежен дел од секојдневието и професионалното работење. Тие имаат широка примена, како во природните науки, така и во техничките и инженерските струки, информациските технологии, економијата и финансиите. Во трудот е направен преглед на наставните програми во гимназиското средно образование и некои од техничките факултети, и согледана е потребата од изучување на одредени поими од веројатност и статистика во гимназиското образование, за полесно совладување на одредени предмети на техничките факултети.

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