Institute of Mathematics
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Item type:Publication, 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 TrenčevskiThe 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. - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Asymptotic behavior of central order statistics under monotone normalization(Elsevier, SIAM, 2014)Pancheva, E.I. Gacovska, A.Smirnov [Trudy Mat. Inst. Steklov., 25 (1949), pp. 3-60 (in Russian); Amer. Math. Transl., 67 (1952) (in English)] derived four limit types of distributions for linearly normalized central order statistics under the weak convergence. In this paper we investigate the possible limit distributions of the kth upper order statistics with central rank using monotone regular norming sequences and obtain 13 possible types. - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Randomly Indexed Central Order Statistics(Bulgarian Academy of Sciences, 2013)Gacovska, Aneta Pancheva, Elisaveta I.In our paper from 2012 we have considered the upper order statistics with central rank of sample with deterministic size. Here we investigate the asymptotic behaviour of randomly indexed upper order statistics using regular norming time-space changes. - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Attitudes Towards Reproduction and Creating a Family Among Albanian Women—The Case of Arachinovo Municipality, North Macedonia(Springer, 2024-03-17)Biljana Apostolovska Toshevska, Mirjanka Madjevikj, Marija Ljakoska, Aneta Gacovska BarandovskaThe place people live can determine whom they meet, the partners they choose, the level of education they accomplish, the income they earn, and the options available to them and their families. It also affects life habits, customs, and affects attitudes regarding reproduction, and creating a family. The municipality of Arachinovo with all its characteristics and specifics, especially the attitudes of women toward reproduction and creating a family is very challenging to study. Located around 12 km from the capital city center, this municipality stands out in terms of birth rates, and ethnic and religious homogeneity. Data for the main analyses in the paper were obtained by a field survey conducted in April and May of 2020. The survey included a little over 60 women of reproductive age, divided into three age groups. For the analysis of the variance of the means of the three chosen groups, standard statistical tests were used, while the one-way ANOVA test was used for testing seven hypotheses for the equality of the means of the three independent specimens. A chi-squared test of independence of a pair of random variables based on observations was made, for the whole sample, according to the considered questions. The research and the gained results are expected to contribute to the creation of appropriate economic, social, and demographic policies, which will enable greater inclusion of women in the educational process, the labor force, and society, and will raise awareness about their role and place in the family. - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Students’ ability to use geometry knowledge in solving problems of geometrical optics(Taylor & Francis, International Journal of Mathematical Education in Science and Technology, 2023-03-20)Aneta Gacovska Barandovska, Boce Mitrevski, Lambe BarandovskiProblem-solving is an essential part of teaching, learning, and assessment of physics and mathematics. The continuing educational reforms have a deep impact on everyday teaching as well as working with talented students. In the Macedonian educational system, the curricula do not explicitly point out the connection between mathematics and physics, but findings show that if the students do not understand the connection, they have problems adopting the knowledge. The gap that appears in applying geometry subsists even among talented students. When solving problems of geometrical optics, besides physics, it is inevitable that students use the standard geometrical apparatus. Even when the basics of physics are well known, possible mathematical mistakes often lead to wrong results in the given problems in physics. Here we demonstrate such an example of a selected problem from Macedonian state competitions in physics. The case study investigates two samples of students, 18 primary school students and 67 high school students, involved in State physics competitions in the year 2016. The selected problems have similar demands. - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Global Scheme of the basic interactions and their geometrical interpretations(Faculty of Natural Sciences and Mathematics, University of Nis, Serbia, 2023)Kostadin TrenchevskiOur space-time consists of three 3-dimensional spaces: space $S$, space rotations $SR$ and time $T$. First are considered the basic possible 4 cases for exchange among them: 1. $r\rightarrow s$, 2. $s\rightarrow r$, 3. $r\rightarrow t$, and 4. $s\rightarrow t$, where $s\in S$, $r\in SR$, and $t\in T$. Analogous to the affine group of translations and rotations ${\cal A}$, it is considered a space group $G_s$ of $6\times 6$ matrices, which is isomorphic to the group $Spin(4)$. The space metric observed by the particles is found. Further are considered 4 generalized exchanges $1^*$, $2^*$, $3^*$ and $4^*$, induced by the cases 1, 2, 3, and 4. The case $1^*$ leads to the electro-weak interaction, and it is a consequence of non-commutativity between one translation and one rotation in the space group $G_s$. The case $2^*$ leads to the strong interaction, and it is a consequence of non-commutativity between two translations in the space group $G_s$. It leads also to the galactic acceleration which is observed at the periphery of each galaxy, and now we do not need dark matter in order to explain the motion of the distant stars in the galaxies. The case $3^*$ leads to electromagnetic interaction, and it is a consequence of non-commutativity between one translation and one rotation in the affine group ${\cal A}$. The case $4^*$ leads to gravitational interaction and it is a consequence of non-commutativity between one translation and one "radial translation" in the affine group ${\cal A}$. The corresponding accelerations are deduced and for a fixed space positions they are of type ${\bf a}={\rm rot}(\vec{\varphi})$ (gauge invariant), but the quantum and wave effects are neglected. It is also predicted a new gravity-weak interaction, which belongs to the case $2^*$.