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This paper presents the modeling of mathematical tasks which includes their mathematical expression (their mathematical form) and then the usage of mathematical tools to analyze and solve those tasks. To do this, first, we will try to understand the problem (task) which involves the identification of all its important elements and the selection of parameters associated with them. Then we will present mathematical expressions that describe various relations. We will choose the model depending on the nature of the problem and its complexity. In tasks that involve geometry, we will use the GeoGebra application. With this application, we have the opportunity to visualize, explore, and analyze the given model interactively. While, for models that involve analysis and algebra, we will use Wolfram Mathematica, which includes Wolfram Language as its programming language. After using these applications for different kinds of tasks, we will analyze and compare their solution by also solving them in other forms (classical forms for solving mathematical tasks). The purpose of using different applications is to improve the process of mathematical modeling, providing an interactive way to solve tasks, and often allowing faster and more advanced methods that lead to our task solution. [ABSTRACT FROM AUTHOR]

In this paper, firstly, we show that there can be constructed an affine plane A(F) from ternary ring in natural way. Firstly, we present the basic properties of affine and projective planes including their completion with each other, respectively, then we continue with their definition over a skew-field. Considering that not all affine planes are of the form A ² (F), we use the Desargues properties to characterize them. Mathematically projective geometry if even more natural than its affine version. The work continues by obtaining the projective planes P(F) by "completing" the plane constructed from ternary system (..., F), by means of projective completion and then constructing affine planes from projective planes by means of affine restriction. One should add a new point "at infinity" for each direction, there will also be a line "at infinity". Affine lines l are too short, we must force the projective line to contain the direction ... = l U {[l]}. The concepts are equivalent, if you have got one, you have got the other. In the end we show the process of affinization and projectivization of the projective and affine plane. Affinization of projectivization of an affine plane A(F) may depend on the choise of line removed from Â(F), and need not be isomorfiphic to A(F). [ABSTRACT FROM AUTHOR]

Universities, like many other organizations and institutions, process a great deal of documentation at multiple spectrums, which are very important to all concerned. The processing of documentation at the University of Tetova has conventionally been retained by manual or semi-automated means. The natural growth in the student population and the constant influx of requests for certain documents led to difficulties in due time, as all the documents were stored in one big collection; thus, it frustrated everybody. To bridge these gaps, an automated document management system was designed and put into operation by the professional software engineers' team at the University of Tetova. It aimed to resolve all the challenges thrown up using manual methods and effectively integrate the work. The automated document management system ensures better access, fast processing time, and efficiency within the university. Its seamless integration with the LMS says a lot about its cohesive and integrated approach towards data and document management, reflecting that commitment to a modern and technologically advanced environment for academic and administrative staff and students. The automated document management system at the University of Tetova marked a great leap forward: it overcame the existing weaknesses with fully automated processes, meeting the current needs, and aiming for further growth and development during the next years within the frame of the digital era. [ABSTRACT FROM AUTHOR]

The dynamics of a generalized form of fractional-order Lorenz system are investigated by employing a modified version of the Runge-Kutta 4 method (RK4). The method is very simple and very effective for solving differential equations of fractional order, it may be used. To illustrate the new technique, the numerical algorithm is applied in the 3D solution of the Lorenz system by adding the fourth varied parameter, considered as a highly simplified model for the weather. Parameter fixed dynamical analysis method and chaos diagram are used. Results show that the fractional order Lorenz system has rich dynamical behavior and it is a potential model for application. Investigation of dynamics is realized by fixing the parameters a=10, b= 8/ 3, c = 24.74 (system has chaotic behavior, numerically illustrated), for d ∈ [-1,10] implemented with the aid of Mathematica symbolic package. The fractional derivatives are described in the Caputo sense. Based on RK4 and Vectorial RK4 algorithms, is shown that the system has rich dynamical characteristics, it changes from a non-chaotic system to a chaotic one, which is more complex for smaller fractional derivative order v ∈(0,1) closer to 0. [ABSTRACT FROM AUTHOR]

In this paper, we study the connections between linear codes and projective geometries over finite fields. Each of these two topics is interesting by itself and has been subject to substantial research. In the last decade, a lot of progress has been made in both areas. We introduce some of the basic ideas and connections between finite projective spaces and coding theory. We begin by studying projective geometries, from this, we introduce a very interesting action in projective planes which lead to many other interesting areas of finite geometry, coordination of the plane. We will coordinate the lines using point coordination. Our focus then shifts to coding theory and in particular three-dimensional linear codes. The linear code Cs,t n, q of s-spaces and t-spaces in a projective space PG (n, q), q = pd, p prime, is defined as the vector space spanned over Fqby the rows of the incidence matrix of s-spaces and t-spaces. Three-dimensional code applied on the constructed projective model: Fano Plane, like a3-dimensional vector space over F2. The Fano plane, like a model, occurs in algebraic geometry and geometric algebra in a number of cases, constructing a link between such important mathematical concepts. There are given different ways of constructing the model, taking in consider that it is impossible to label the Fano plane in such a way that all or just five of its lines would be ordinary. [ABSTRACT FROM AUTHOR]

In this paper, we are going to analyze a wide class of topological spaces - the so-called metric spaces. These are sets in which is defined the notion of distance between any two points. It is well known that the distance function or metric defined on a metric space X induces a topology on that space X. In the second part of the paper, we are going to study well-known characterizations of the class of topological spaces, the topology of which is determined with the help of metrics - the so-called metrizable spaces. The question here is when is a topological space metrizable? The answer of this question is the main result of this research, which are some important criteria, who are necessary and sufficient, that topological spaces must possess in order to be metrizable. [ABSTRACT FROM AUTHOR]

By the standard definition, a point x is an isolated point in a topological space if its corresponding one-element set is open. Here it is characterized the notion of isolated point using chains and open covers of the space. Namely, x is an isolated point in a T1 topological space if there exists an open covering of the space such that for any point of the space different from x, there is no chain in the covering joining it with the point x. Equivalently, it is provided a characterization of isolated point using the notion of pair of chain separated sets relatively a T1 space, while when using the notion of pair of weakly chain separated sets relatively a T1 space it is shown that only the sufficiency of the claim holds. Using these results, it is provided a characterization of discrete topological spaces and it is reproved that every compact Hausdorff space without isolated points is uncountable. [ABSTRACT FROM AUTHOR]

In this paper, we are going to analyze the relations between different types of convergence of a random sequence, such as almost sure convergence, convergence in mean square, convergence in distribution and convergence in probability. The convergence in distributions says nothing about the relationship between the random variables Xn and X and X, while for convergence in probability, the joint distribution of Xn and X is relevant. In the main part of the paper, we are going to prove the theorem which argues that the convergence in probability implies convergence in distribution, and the opposite is not true. But if ..., where c is a constant, then ... which mean that convergence in probability to a constant is equivalent to convergence in distributions. Also, we give some interesting examples. [ABSTRACT FROM AUTHOR]

In this article we present the basic properties of projective geometry and coding theory and show how finite geometry can contribute to coding theory. Often good codes come from interesting structures in projective geometries. For example, MDS codes come from arcs (i.e. sets of points which are extremal in the sense that they admit no other than the obvious dependencies). We concentrate on introducing the basic concepts of these two research areas and give standard references for all these research areas. We also mention particular results involving ideas from finite geometry, and particular results in coding theory. [ABSTRACT FROM AUTHOR]

Differential equations are used to model problems in science and engineering that involve the change of some variable with respect to another. In real-life problems, the differential equation that models the problem is too complicated to solve exactly for this reason, in recent years much attention has been devoted to deriving numerical methods for approximating their solution. In particular, in this paper we consider the use of Adams-Bashforth like a multi-step method. Adams-Bashforth method of different steps is constructed, and then we use the fourth-step one, on Mathematica Package for numerical approaches. The numerical results are compared with analytical ones, shown in different ways, tables and graphics, accompanied by examples of their use. [ABSTRACT FROM AUTHOR]

Inspired by the Euler identity, there have been a lot of attempts to approximate particular integers by applying some algebraic operations to some non-integer numbers. Here it is used a method of generating some almost integers with the help of Pisot numbers (real algebraic integers greater than one with all Galois conjugates located in the open unit disc in the complex plane). The method consists of obtaining values that are very close to whole numbers by taking high powers of Pisot numbers. There are used some terms of three polynomial sequences to generate some Pisot numbers. A desired approximation to the integer depending on the power of the Pisot number and the distance of its Galois conjugates from the origin will be examined. [ABSTRACT FROM AUTHOR]

This paper is a short survey of projective geometry, history of Hamming codes and the relationship between them and projective planes. The projective plane of order n, over a finite field (Fp ={0,1,2,..., p -1},+,*) p, which has arithmetic donemod p, denoted by PG(n, p), is the set of all subspaces of vector space Fnp. It can be endowed with the distance function f (U,V) = dim(U)+ dim(V) - 2dim(U V) which turns PGn, p into a metric space. A (n,M,d) code in projective space is a subset of PG(n, p) of size M such that the distance between any two code-words is at least d. The first error correction code, the Hamming code, is intrinsic to the projective plane of order 2 over F2 . A connection between planes and codes is given by construction of Hamming code related to PG(2,2) and we generalize them to PG(3,2) using Hamming and Generator matrix. The codes constructed in this way are called projective codes. [ABSTRACT FROM AUTHOR]

In this paper, we will show that based on the concept of multialgebras and relying on the power structures of these algebras, which we could also consider as relational structures, and adding the set of operations ... we get a Boolean algebra with operator. The power structure of relational structure = (A,) is an algebra with the set (A) and the set of all basic operations defined. [ABSTRACT FROM AUTHOR]