Aneta Velkoska - Academia.edu (original) (raw)

Papers by Aneta Velkoska

Research paper thumbnail of Method for solving system of linear equations mxn, m=1, 2, 3; Based on its Geometrical Interpretations

Research paper thumbnail of IoT and IoT blockchain market analyzes and benefits in Europe

AIP conference proceedings, 2024

Research paper thumbnail of Теорија на веројатност и статистика

Research paper thumbnail of Статистика за бизнис

Research paper thumbnail of Теорија на веројатност и статистика

Оваа книга ја содржи предвидената материја со наставниот план и програма на факултет по Информати... more Оваа книга ја содржи предвидената материја со наставниот план и програма на факултет по Информатика во третта година на Првиот Приватен Европски Универзитет на Република Македонија, Скопје. Но, истата можат да ја користат и студентите од други сродни факултети, бидејќи се опфатени неопходните методи на Статистичката анализа и заклучување. Материјалот во книгата е поделен во 8 глави: Во првата глава дефинирани се основните поими од теорија на веро-јатност, со решени примери и задачи за да понатаму во втората глава можат да се дефинираат поимите на теориската статистика: случајна променлива и распределба на веројатноста на случајната променлива. Во третата глава е разработен историскиот развој на статистиката, пред-метот и начините на статистичкото истражување, како и основните поими на статистиката: популација, обележје и примерок. Во четвртата глава е разработена првата група на статистички методи на истражувањето на масовните појави: прибирање, групирање и претставување на статисти...

Research paper thumbnail of Статистика за бизнис

Оваа книга ја содржи предвидената материја со наставниот план и програма на економскиот факултет ... more Оваа книга ја содржи предвидената материја со наставниот план и програма на економскиот факултет во прва година на Првиот Приватен Европски Универзитет на Република Македонија, Скопје. Но, истата можат да ја користат и студентите од други сродни факултети, бидејќи се опфатени неопходните методи на Статистичката анализа и заклучување. Материјалот во книгата е поделен во 9 глави: Во првата глава дефинирани се основните поими од теорија на веројатност, со решени примери и задачи за да понатаму во втората глава можат да се дефинираат поимите на теориската статистика: случајна променлива и распределба на веројатноста на случајната променлива. Во третата глава е разработен историскиот развој на статистиката, предметот и начините на статистичкото истражување, како и основните поими на статистиката: популација, обележје и примерок. Во четвртата глава е разработена првата група на статистички методи на истражувањето на масовните појави: прибирање, групирање и претставување на статистичките п...

Research paper thumbnail of Theorem of De Rham on Tlli Sets

institutzamatematika.com

Зборник на трудови од IV конгрес на математичарите на Македонија Струга, Македонија, 19-22.10. 20... more Зборник на трудови од IV конгрес на математичарите на Македонија Струга, Македонија, 19-22.10. 2008 Стр. 300-317 ... Proceedings of IV congress of mathematicians of Republic of Macedonia Struga, Macedonia, 19-22.10. 2008 Pages 300-317

Research paper thumbnail of Totally Weakly Chain Separated Sets in a Topological Space

Kragujevac journal of mathematics, Dec 31, 2022

In this article, by using the notion of chain, we give some characterizations of totally separate... more In this article, by using the notion of chain, we give some characterizations of totally separated spaces. Then, we give some examples, study the properties of those spaces and give new proofs. Furthermore, by using the notion of chain, we introduce the notions of totally weakly chain separated and totally chain separated sets in a topological space, we state some useful aspects of these sets as well as the various relationships between them and by using these notions we give some characterizations of discrete and totally separated spaces.

Research paper thumbnail of Inner differentiability and differential forms on tangentially locally linearly independent sets

Filomat

The de Rham theorem gives a natural isomorphism between De Rham cohomology and singular cohomolog... more The de Rham theorem gives a natural isomorphism between De Rham cohomology and singular cohomology on a paracompact differentiable manifold. We proved this theorem on a wider family of subsets of Euclidean space, on which we can define inner differentiability. Here we define this family of sets called tangentially locally linearly independent sets, propose inner differentiability on them, postulate usual properties of differentiable real functions and show that the integration over sets that are wider than manifolds is possible.

Research paper thumbnail of Weakly chain separated sets in a topological space

Mathematica Montisnigri, 2021

In this paper we introduce the notion of pair of weakly chain separated sets in a topological spa... more In this paper we introduce the notion of pair of weakly chain separated sets in a topological space. If two sets are chain separated in the topological space, then they are weakly chain separated in the same space. We give an example of weakly chain separated sets in a topological space that are not chain separated in the space. Then we study the properties of these sets. Also we mention the criteria for two kind of topological spaces by using the notion of chain. The topological space is totally separated if and only if any two different singletons (unit subsets) are weakly chain separated in the space, and it is the discrete if and only if any pair of different nonempty subsets are chain separated. Moreover we give a criterion for chain connected set in a topological space by using the notion of weakly chain separateness. This criterion seems to be better than the criterion of chain connectedness by using the notion of pair of chain separated sets. Then we prove the properties of ...

Research paper thumbnail of Increasing Information Secrecy by Jamming with Uniformly Distributed Eavesdroppers

Abstract: Transmission of confidential messages over wireless networks between a transmitter and ... more Abstract: Transmission of confidential messages over wireless networks between a transmitter and a receiver in the presence of illegitimate receivers called eavesdroppers is an active area of research. Cooperation by jamming can improve information-theoretic secrecy in wireless networks. However, randomly chosen jammers can have a negative impact on the secrecy capacity. Our goal is to provide a closed form for positioning the friendly jammers in two-dimensional wireless network with uniformly distributed eavesdroppers while two legitimate partners are communicating.

Research paper thumbnail of Improved bounds on secrecy capacity in twin-code cloud storage framework

2016 8th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), 2016

Security and reliability in cloud storage networks are important features for achieving system ef... more Security and reliability in cloud storage networks are important features for achieving system efficiency. An explicit construction of regenerating codes that perform reliability, availability and security of the stored data, and at the same time perform efficient node repair is already provided. Understanding and achieving the optimal tradeoff among the above listed features is of great importance for improving the functioning in all cloud storage systems. Therefore in this work, we propose a new code construction that includes an outer and an inner code in the process. The outer code, Gabidulin code, as a pre-coding step achieves improved security level in the storage system and the inner one, Twin-code framework, performs efficient distribution, reconstruction and repair process. The proposed code is resilient to an eavesdropper model and gives better results regarding the achievements when are used the minimum bandwidth regenerating (MBR) and minimum storage regenerating (MSR) c...

Research paper thumbnail of One invariant of intrinsic shape

Filomat, 2015

Based on the intrinsic definition of shape by functions continuous over a covering and correspond... more Based on the intrinsic definition of shape by functions continuous over a covering and corresponding homotopy we will define proximate fundamental group. We prove that proximate fundamental group is an invariant of pointed intrinsic shape of a space.

Research paper thumbnail of Security in Twin-code framework

2015 7th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), 2015

The distributed data storage systems are constructed by large number of nodes which are interconn... more The distributed data storage systems are constructed by large number of nodes which are interconnected over a network. Each node in such peer-to-peer network is vulnerable and at a potential risk for attack. The attackers can eavesdrop the nodes and possibly modify their data. Hence distributed storage systems should be secure apart from satisfying the reconstruction and repair requirements. We constructed a distributed storage system, Twin MDS code framework which is more efficient than the regenerating codes based storage systems. We prove that this Twin MDS code framework gives better performance than MBR codes and equal with MSR codes in the distribution process and investigate its security performance comparing with the security of the MBR and MSR codes. Such Twin MDS code framework is examined in an eavesdropper model where passive attackers can access to the stored data or/and downloaded data during the repair process. We demonstrate that the Twin MDS code framework manages better results than MBR and MSR codes regarding the security in the system. twin-code framework, distributed storage systems (DSS), eavesdropper, informationtheoretic secrecy, security in DSS, minimum bandwidth regenerating codes, minimum storage regenerating codes * N. Marina and A. Velkoska are with the Faculty of Communication Networks and Security,

Research paper thumbnail of De Rham’s Theorem on Tlli Sets

The De Rham theorem answers one of the first questions which appear when discussing singular and ... more The De Rham theorem answers one of the first questions which appear when discussing singular and De Rham cohomology . Are they related in any way? The De Rham theorem gives a natural isomorphism between De Rham cohomology and singular cohomology on a paracompact differentiable manifold . In other words The De Rham theorem says that they will produce the same group to within an isomorphism. This is very important fact as singular cohomology is very topological theory. De Rham cohomology is much more analytical. It comes from differential forms on manifolds and the exterior derivate. Perhaps it is strange to see that analysis and topology can be linked in such a nice way, however Stoke’s theorem also shows this relationship with the left hand side of the equation coming from topology and the right hand side coming from analysis. The differentiable mappings are usually defined on open sets. On arbitrary set a function is differentiable, if there is a bigger open set containing that set...

Research paper thumbnail of Increasing Information Secrecy with Poisson Distributed Jammers

ABSTRACT Transmission of confidential messages over wireless networks between a transmitter and a... more ABSTRACT Transmission of confidential messages over wireless networks between a transmitter and a receiver in the presence of illegitimate receivers called eavesdroppers is an active area of research. Cooperation by jamming can improve informationtheoretic secrecy in wireless networks. However, randomly chosen jammers can have a negative impact on the secrecy capacity. This raises the question about the configuration of the jammers which minimizes the negative impact on the secrecy capacity. For this purpose, we consider a network configuration with two legitimate communicating parties, one eavesdropper and randomly positioned jammers. The jammers follow a spatial Poisson process. Only the jammer closest to the eavesdropper is active. For a simplified model of this network configuration, we derive analytical results on the size of the secrecy region.

Research paper thumbnail of Proximate fundamental group

The intrinsic definition of shape based on proximate sequences for compact metric spaces and for ... more The intrinsic definition of shape based on proximate sequences for compact metric spaces and for paracompact spaces based on proxi-mate nets indexed by open coverings directed by refinement of cover-ings are presented in the papers “Intrinsic definition of strong shape for compact metric spaces”, N. Shekutkovski, (2012) and “Maps of quasicomponents induced by a shape morphism”N. Shekutkovski, T.A. Pachemska, G. Markoski, (2012). Using this approach we define proximate fundamental group, an in-variant of pointed shape of a space.

Research paper thumbnail of Adversarial attacks in the twin-code framework

2016 8th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), 2016

Ensuring the security and reliability of the stored data in a Distributed Storage System (DSS) is... more Ensuring the security and reliability of the stored data in a Distributed Storage System (DSS) is a pivotal issue in a distributed storage network. The storage system must be secured against passive and active types of attacks for efficient protection of the stored data. The aim is achieving satisfactory level of security of the storage system and at the same time maintaining the desired level of reliability, which arises when there are node failures. In our work we apply this concept in a Twin-code framework, in presence of an intruder who can observe and modify the stored data on the networks nodes. We are making a new code construction, which is simultaneously secure against passive and active attacks and at the same time satisfies particular bound for reliability. Moreover, for this construction we are proving the level of security under the above mentioned constraints.

Research paper thumbnail of Математика за бизнис

Оваа книга ја содржи предвидената материја со наставниот план и програма на економскиот факултет ... more Оваа книга ја содржи предвидената материја со наставниот план и програма на економскиот факултет во прва година на Првиот Приватен Европски Универзитет на Република Македонија, Скопје, по предметот Математика за бизнис. Истата можат да ја користат и студентите од други сродни факултети, бидејќи се опфатени неопходните поими од Елементарна алгебра, Калкулус и Финансиска математика. Материјалот во книгата е поделен во 11 глави: Во првата глава дефинирани се основните поими од множества и релации, со решени примери и задачи. Понатаму во втората глава, почнуваме да се запознаваме со елементарните алгебарски поими, како што се матрици и детерминанти, за да во третата се потсетуваме на системи линеарни равенки, проширувајќи ги нашите знаења за нивно решавање со нови методи на решавања на системите линеарни равенки. Во четвртата глава се разработени низите од реални броеви, при што се дефинира поимот за граница на низа и некои карактеристични граници, за да накрај се воведат поимите за ари...

Research paper thumbnail of GeoGebra as e-Learning Resource for Teaching and Learning Statistical Concepts

Understanding probability and statistics is essential in the modern world, where the print and el... more Understanding probability and statistics is essential in the modern world, where the print and electronic media are full of statistical information and interpretation. The probability and statistics lessons should provide to the students the ability to collect, organize and analyze numerical data, and to understand chance. Appropriate use of technology allows more students access to mathematical concepts in general and also to access statistical concepts. A number of software tools are available for solving and visualization of mathematical problems. GeoGebra, as dynamic mathematics open-source software, is attracting a lot of interest in the mathematical community. Spreadsheet, which enable statistics calculations, and probability calculator are features of GeoGebra that is not found in other dynamic mathematics software. In this paper, using GeoGebra will be created instructional materials that will solve several practical problems from the area of probability and statistics. Use ...

Research paper thumbnail of Method for solving system of linear equations mxn, m=1, 2, 3; Based on its Geometrical Interpretations

Research paper thumbnail of IoT and IoT blockchain market analyzes and benefits in Europe

AIP conference proceedings, 2024

Research paper thumbnail of Теорија на веројатност и статистика

Research paper thumbnail of Статистика за бизнис

Research paper thumbnail of Теорија на веројатност и статистика

Оваа книга ја содржи предвидената материја со наставниот план и програма на факултет по Информати... more Оваа книга ја содржи предвидената материја со наставниот план и програма на факултет по Информатика во третта година на Првиот Приватен Европски Универзитет на Република Македонија, Скопје. Но, истата можат да ја користат и студентите од други сродни факултети, бидејќи се опфатени неопходните методи на Статистичката анализа и заклучување. Материјалот во книгата е поделен во 8 глави: Во првата глава дефинирани се основните поими од теорија на веро-јатност, со решени примери и задачи за да понатаму во втората глава можат да се дефинираат поимите на теориската статистика: случајна променлива и распределба на веројатноста на случајната променлива. Во третата глава е разработен историскиот развој на статистиката, пред-метот и начините на статистичкото истражување, како и основните поими на статистиката: популација, обележје и примерок. Во четвртата глава е разработена првата група на статистички методи на истражувањето на масовните појави: прибирање, групирање и претставување на статисти...

Research paper thumbnail of Статистика за бизнис

Оваа книга ја содржи предвидената материја со наставниот план и програма на економскиот факултет ... more Оваа книга ја содржи предвидената материја со наставниот план и програма на економскиот факултет во прва година на Првиот Приватен Европски Универзитет на Република Македонија, Скопје. Но, истата можат да ја користат и студентите од други сродни факултети, бидејќи се опфатени неопходните методи на Статистичката анализа и заклучување. Материјалот во книгата е поделен во 9 глави: Во првата глава дефинирани се основните поими од теорија на веројатност, со решени примери и задачи за да понатаму во втората глава можат да се дефинираат поимите на теориската статистика: случајна променлива и распределба на веројатноста на случајната променлива. Во третата глава е разработен историскиот развој на статистиката, предметот и начините на статистичкото истражување, како и основните поими на статистиката: популација, обележје и примерок. Во четвртата глава е разработена првата група на статистички методи на истражувањето на масовните појави: прибирање, групирање и претставување на статистичките п...

Research paper thumbnail of Theorem of De Rham on Tlli Sets

institutzamatematika.com

Зборник на трудови од IV конгрес на математичарите на Македонија Струга, Македонија, 19-22.10. 20... more Зборник на трудови од IV конгрес на математичарите на Македонија Струга, Македонија, 19-22.10. 2008 Стр. 300-317 ... Proceedings of IV congress of mathematicians of Republic of Macedonia Struga, Macedonia, 19-22.10. 2008 Pages 300-317

Research paper thumbnail of Totally Weakly Chain Separated Sets in a Topological Space

Kragujevac journal of mathematics, Dec 31, 2022

In this article, by using the notion of chain, we give some characterizations of totally separate... more In this article, by using the notion of chain, we give some characterizations of totally separated spaces. Then, we give some examples, study the properties of those spaces and give new proofs. Furthermore, by using the notion of chain, we introduce the notions of totally weakly chain separated and totally chain separated sets in a topological space, we state some useful aspects of these sets as well as the various relationships between them and by using these notions we give some characterizations of discrete and totally separated spaces.

Research paper thumbnail of Inner differentiability and differential forms on tangentially locally linearly independent sets

Filomat

The de Rham theorem gives a natural isomorphism between De Rham cohomology and singular cohomolog... more The de Rham theorem gives a natural isomorphism between De Rham cohomology and singular cohomology on a paracompact differentiable manifold. We proved this theorem on a wider family of subsets of Euclidean space, on which we can define inner differentiability. Here we define this family of sets called tangentially locally linearly independent sets, propose inner differentiability on them, postulate usual properties of differentiable real functions and show that the integration over sets that are wider than manifolds is possible.

Research paper thumbnail of Weakly chain separated sets in a topological space

Mathematica Montisnigri, 2021

In this paper we introduce the notion of pair of weakly chain separated sets in a topological spa... more In this paper we introduce the notion of pair of weakly chain separated sets in a topological space. If two sets are chain separated in the topological space, then they are weakly chain separated in the same space. We give an example of weakly chain separated sets in a topological space that are not chain separated in the space. Then we study the properties of these sets. Also we mention the criteria for two kind of topological spaces by using the notion of chain. The topological space is totally separated if and only if any two different singletons (unit subsets) are weakly chain separated in the space, and it is the discrete if and only if any pair of different nonempty subsets are chain separated. Moreover we give a criterion for chain connected set in a topological space by using the notion of weakly chain separateness. This criterion seems to be better than the criterion of chain connectedness by using the notion of pair of chain separated sets. Then we prove the properties of ...

Research paper thumbnail of Increasing Information Secrecy by Jamming with Uniformly Distributed Eavesdroppers

Abstract: Transmission of confidential messages over wireless networks between a transmitter and ... more Abstract: Transmission of confidential messages over wireless networks between a transmitter and a receiver in the presence of illegitimate receivers called eavesdroppers is an active area of research. Cooperation by jamming can improve information-theoretic secrecy in wireless networks. However, randomly chosen jammers can have a negative impact on the secrecy capacity. Our goal is to provide a closed form for positioning the friendly jammers in two-dimensional wireless network with uniformly distributed eavesdroppers while two legitimate partners are communicating.

Research paper thumbnail of Improved bounds on secrecy capacity in twin-code cloud storage framework

2016 8th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), 2016

Security and reliability in cloud storage networks are important features for achieving system ef... more Security and reliability in cloud storage networks are important features for achieving system efficiency. An explicit construction of regenerating codes that perform reliability, availability and security of the stored data, and at the same time perform efficient node repair is already provided. Understanding and achieving the optimal tradeoff among the above listed features is of great importance for improving the functioning in all cloud storage systems. Therefore in this work, we propose a new code construction that includes an outer and an inner code in the process. The outer code, Gabidulin code, as a pre-coding step achieves improved security level in the storage system and the inner one, Twin-code framework, performs efficient distribution, reconstruction and repair process. The proposed code is resilient to an eavesdropper model and gives better results regarding the achievements when are used the minimum bandwidth regenerating (MBR) and minimum storage regenerating (MSR) c...

Research paper thumbnail of One invariant of intrinsic shape

Filomat, 2015

Based on the intrinsic definition of shape by functions continuous over a covering and correspond... more Based on the intrinsic definition of shape by functions continuous over a covering and corresponding homotopy we will define proximate fundamental group. We prove that proximate fundamental group is an invariant of pointed intrinsic shape of a space.

Research paper thumbnail of Security in Twin-code framework

2015 7th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), 2015

The distributed data storage systems are constructed by large number of nodes which are interconn... more The distributed data storage systems are constructed by large number of nodes which are interconnected over a network. Each node in such peer-to-peer network is vulnerable and at a potential risk for attack. The attackers can eavesdrop the nodes and possibly modify their data. Hence distributed storage systems should be secure apart from satisfying the reconstruction and repair requirements. We constructed a distributed storage system, Twin MDS code framework which is more efficient than the regenerating codes based storage systems. We prove that this Twin MDS code framework gives better performance than MBR codes and equal with MSR codes in the distribution process and investigate its security performance comparing with the security of the MBR and MSR codes. Such Twin MDS code framework is examined in an eavesdropper model where passive attackers can access to the stored data or/and downloaded data during the repair process. We demonstrate that the Twin MDS code framework manages better results than MBR and MSR codes regarding the security in the system. twin-code framework, distributed storage systems (DSS), eavesdropper, informationtheoretic secrecy, security in DSS, minimum bandwidth regenerating codes, minimum storage regenerating codes * N. Marina and A. Velkoska are with the Faculty of Communication Networks and Security,

Research paper thumbnail of De Rham’s Theorem on Tlli Sets

The De Rham theorem answers one of the first questions which appear when discussing singular and ... more The De Rham theorem answers one of the first questions which appear when discussing singular and De Rham cohomology . Are they related in any way? The De Rham theorem gives a natural isomorphism between De Rham cohomology and singular cohomology on a paracompact differentiable manifold . In other words The De Rham theorem says that they will produce the same group to within an isomorphism. This is very important fact as singular cohomology is very topological theory. De Rham cohomology is much more analytical. It comes from differential forms on manifolds and the exterior derivate. Perhaps it is strange to see that analysis and topology can be linked in such a nice way, however Stoke’s theorem also shows this relationship with the left hand side of the equation coming from topology and the right hand side coming from analysis. The differentiable mappings are usually defined on open sets. On arbitrary set a function is differentiable, if there is a bigger open set containing that set...

Research paper thumbnail of Increasing Information Secrecy with Poisson Distributed Jammers

ABSTRACT Transmission of confidential messages over wireless networks between a transmitter and a... more ABSTRACT Transmission of confidential messages over wireless networks between a transmitter and a receiver in the presence of illegitimate receivers called eavesdroppers is an active area of research. Cooperation by jamming can improve informationtheoretic secrecy in wireless networks. However, randomly chosen jammers can have a negative impact on the secrecy capacity. This raises the question about the configuration of the jammers which minimizes the negative impact on the secrecy capacity. For this purpose, we consider a network configuration with two legitimate communicating parties, one eavesdropper and randomly positioned jammers. The jammers follow a spatial Poisson process. Only the jammer closest to the eavesdropper is active. For a simplified model of this network configuration, we derive analytical results on the size of the secrecy region.

Research paper thumbnail of Proximate fundamental group

The intrinsic definition of shape based on proximate sequences for compact metric spaces and for ... more The intrinsic definition of shape based on proximate sequences for compact metric spaces and for paracompact spaces based on proxi-mate nets indexed by open coverings directed by refinement of cover-ings are presented in the papers “Intrinsic definition of strong shape for compact metric spaces”, N. Shekutkovski, (2012) and “Maps of quasicomponents induced by a shape morphism”N. Shekutkovski, T.A. Pachemska, G. Markoski, (2012). Using this approach we define proximate fundamental group, an in-variant of pointed shape of a space.

Research paper thumbnail of Adversarial attacks in the twin-code framework

2016 8th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), 2016

Ensuring the security and reliability of the stored data in a Distributed Storage System (DSS) is... more Ensuring the security and reliability of the stored data in a Distributed Storage System (DSS) is a pivotal issue in a distributed storage network. The storage system must be secured against passive and active types of attacks for efficient protection of the stored data. The aim is achieving satisfactory level of security of the storage system and at the same time maintaining the desired level of reliability, which arises when there are node failures. In our work we apply this concept in a Twin-code framework, in presence of an intruder who can observe and modify the stored data on the networks nodes. We are making a new code construction, which is simultaneously secure against passive and active attacks and at the same time satisfies particular bound for reliability. Moreover, for this construction we are proving the level of security under the above mentioned constraints.

Research paper thumbnail of Математика за бизнис

Оваа книга ја содржи предвидената материја со наставниот план и програма на економскиот факултет ... more Оваа книга ја содржи предвидената материја со наставниот план и програма на економскиот факултет во прва година на Првиот Приватен Европски Универзитет на Република Македонија, Скопје, по предметот Математика за бизнис. Истата можат да ја користат и студентите од други сродни факултети, бидејќи се опфатени неопходните поими од Елементарна алгебра, Калкулус и Финансиска математика. Материјалот во книгата е поделен во 11 глави: Во првата глава дефинирани се основните поими од множества и релации, со решени примери и задачи. Понатаму во втората глава, почнуваме да се запознаваме со елементарните алгебарски поими, како што се матрици и детерминанти, за да во третата се потсетуваме на системи линеарни равенки, проширувајќи ги нашите знаења за нивно решавање со нови методи на решавања на системите линеарни равенки. Во четвртата глава се разработени низите од реални броеви, при што се дефинира поимот за граница на низа и некои карактеристични граници, за да накрај се воведат поимите за ари...

Research paper thumbnail of GeoGebra as e-Learning Resource for Teaching and Learning Statistical Concepts

Understanding probability and statistics is essential in the modern world, where the print and el... more Understanding probability and statistics is essential in the modern world, where the print and electronic media are full of statistical information and interpretation. The probability and statistics lessons should provide to the students the ability to collect, organize and analyze numerical data, and to understand chance. Appropriate use of technology allows more students access to mathematical concepts in general and also to access statistical concepts. A number of software tools are available for solving and visualization of mathematical problems. GeoGebra, as dynamic mathematics open-source software, is attracting a lot of interest in the mathematical community. Spreadsheet, which enable statistics calculations, and probability calculator are features of GeoGebra that is not found in other dynamic mathematics software. In this paper, using GeoGebra will be created instructional materials that will solve several practical problems from the area of probability and statistics. Use ...