Prof.Asoc.Dr. Orgest ZAKA

The foundations for the study of the connections between axiomatic geometry and algebraic structures were set forth by D. Hilbert [6]. And some classic for this are, E. Artin [1], D.R. Huges and F.C. Piper [4], H. S. M Coxeter [3]. Marcel Berger in [2], Robin Hartshorne in [5], etc. In this presentation we will talk about for some: Reciprocal relations of the Desargues Affine Plan with some Algebraic structures. More specifically, we will present a step-by-step transformation of the set of points of a line of the Desargues affine plane up to a skew-field. For this construction we will use only the axioms of the affine plane, and the meaning of the parallelism of the lines in it. In this my research in "this line": all results based in geometric intuition, in axiomatic of Desargues affine plane, and in skew-field properties, we utilize a method that is naive and direct, without requiring the concept of coordinates. In this presentation we will define addition and multiplication of points in a line on Desargues Affine Plane, and we have proven that on each line on Desargues affine plane, we can construct a skew-field related to these two actions, this construction has been achieved, simply and constructively, using simple elements of elementary geometry, and only the basic axioms of Desargues affine plane. Even earlier, in we work's [7-21] we have brought up quite a few interesting facts about the association of algebraic structures with affine planes and with 'Desargues affine planes', and vice versa. We are going to list some of the results of our research in Desargues affine plane and some new projects that we have in process. And finally, we will present an application of the affine plan in experiment modeling.

In this summary, my scientific research activity is presented: • 14-Published Book's in Albanian-Language [approximately (more than) 3000 pages in B5-format, and in writing 11pt]. • 13-papers in process [which will be completed within the year 2023]. • 5-Preprints which are currently under review. • 20-Published papers. • 22-Conference papers. More details: In references [1-14] are my published books, which are in Albanian Language. List of References [15-27] presented my non-published papers (preprints and projects, which are currently being completed). In list of references [28-32] present my papers which are currently under review. Reference-List [33-52]present my published papers and research results, whereas, references [53-74] present my references in various conferences.

This paper introduces advances in the geometry of the transforms for cross ratio of four points in a line in the Desargues affine plane. The results given here have a clean, based Desargues affine plan axiomatics and definitions of addition and multiplication of points on a line in this plane, and for skew field properties. In this paper are studied, properties and results related to the some transforms for cross ratio for 4-points, in a line, which we divide into two categories, Invariant and Preserving transforms for cross ratio. The results in this paper are (1) the cross-ratio of four points is Invariant under transforms: Inversion, Natural Translation, Natural Dilation, Mobiüs Transform, in a line of Desargues affine plane. (2) the cross-ratio of four points is Preserved under transforms: parallel projection, translations and dilation's in the Desargues affine plane.

Based on the following very interesting work in the past [2], [3], [4], [9], [12], this article becomes a description of collineations in the affine plane [10]. We are focusing at the description of translations and dilatations, and we make a detailed description of them. We describe the translation group and dilatation group in affine plane [11]. A detailed description we have given also for traces of a dilatation. We have proved that translation group is a normal subgroup of the group of dilatations, wherein the translation group is a commutative group and the dilatation group is just a group. We think that in this article have brings about an innovation in the treatment of detailed algebraic structures in affine plane.

In this paper we present an application possibility of the affine plane of order $n$, in the planning experiment, taking samples as his point. In this case are needed $n^2$ samples. The usefulness of the support of experimental planning in a finite affine plane consists in avoiding the partial repetition combinations within a proof. Reviewed when planning cannot directly drawn over an affine plane. In this case indicated how the problem can be completed, and when completed can he, with intent to drawn on an affine plane.

In this paper we will show how to constructed an Skew-Field with trace-preserving endomorphisms of the affine plane. Earlier in my paper, we doing a detailed description of endomorphisms algebra and trace-preserving endomorphisms algebra in an affine plane, and we have constructed an associative unitary ring for which trace-preserving endomorphisms. In this paper we formulate and prove an important Lemma, which enables us to construct a particular trace-preserving endomorphism, with the help of which we can construct the inverse trace-preserving endomorphisms of every trace-preserving endomorphism. At the end of this paper we have proven that the set of trace-preserving endomorphisms together with the actions of 'addition' and 'composition' (which is in the role of 'multiplication') forms a skew-field.

This paper introduces a description of Endomorphisms of the translation group in an affine plane, will define the addition and composition of the set of endomorphisms and specify the neutral elements associated with these two actions and present the Endomorphism algebra thereof will distinguish the Trace-preserving endomorphism algebra in affine plane, and prove that the set of Trace-preserving endomorphism associated with the 'addition' action forms a commutative group. We also try to prove that the set of trace-preserving endomorphism, together with the two actions, in it, 'addition' and 'composition' forms an associative and unitary ring.

This paper introduces a description of Endomorphisms of the translation group in an affine plane, will define the addition and composition of the set of endomorphisms and specify the neutral elements associated with these two actions and present the Endomorphism algebra thereof will distinguish the Trace-preserving endomorphism algebra in affine plane, and prove that the set of Trace-preserving endomorphism associated with the 'addition' action forms a commutative group. We also try to prove that the set of trace-preserving endomorphism, together with the two actions, in it, 'addition' and 'composition' forms an associative and unitary ring.

In this paper we will give some applications of group theory. The first application makes use of the observation that computing in Z can be replaced by computing in Zn, if n is suficiently large. Zn can be decomposed into a direct product of groups with prime power order, so we can do the computations in parallel in the smaller components. Group theory have interesting applications in the design of computer software, they result in computing techniques which speed up calculations considerably, one such example is bringing in this paper. Group theory is the main tool to study symmetries, revolution and many geometric transactions, this will be presented in this article and furthermore we will show intersting applications of group theory in chemistry.

In this article we give the meaning of the determinant for 3D matrices with elements from a field F, and the meaning of 3D inverse matrix. Based on my previous work titled '3D Matrix Rings', we want to constructed the 'general linear group of degree $n$ for 3D matrices, which i mark with $GL(n,n,p;F)$' for 3D-matrices, analog to 'general linear group of degree $n$' known.

In this article, we are giving the meaning of a ’New Multiplication’ for the matrices. I have studied the properties of this multiplication in two cases, in the case of 2-D matrices and in the case of 3-D matrices, with elements from over whatever field $F$.

This paper considers dilations and translations of lines in the Desargues affine plane. A dilation of a line transforms each line into a parallel line whose length is a multiple of the length of the original line. In addition to the usual Playfair axiom for parallel lines in an affine plane, further conditions are given for distinct lines to be parallel in the Desargues affine plane. This paper introduces the dilation of parallel lines in a finite Desargues affine plane that is a bijection of the lines. Two main results are given in this paper, namely, each dilation in a finite Desarguesian plane is an isomorphism between skew fields constructed over isomorphic lines and each dilation in a finite Desarguesian plane occurs in a Pappian space.

This paper introduces ordered skew fields that result from the construction of a skew field over an ordered line in a Desargues affine plane. A special case of a finite ordered skew field in the construction of a skew field over an ordered line in a Desargues affine plane in Euclidean space, is also considered. Two main results are given in this paper: (1) every skew field constructed over a skew field over an ordered line in a Desargues affine plane is an ordered skew field and (2) every finite skew field constructed over a skew field over an ordered line in a Desargues affine plane in $\mathbb{R}^2$ is a finite ordered skew field.

In this paper we present a set transformation of points in a line of the Desargues affine plane in a additive group. For this, the first stop on the meaning of the Desargues affine plane, formulating first axiom of his that show proposition D1. Afterwards we show that little Pappus theorem, which we use in the construction of group proofs in additions of points on a line on desargues plane, also applies in the Desargues affine plane.

In this paper, based on several meanings and statements discussed in the literature, we intend constuction a affine plane about a of whatsoever corps (K,+,*). His points conceive as ordered pairs (α,β), where α and β are elements of corps (K,+,*). Whereas straight-line in corps, the conceptualize by equations of the type x*a+y*b=c, where a≠0_K or b≠0_K the variables and coefficients are elements of that body. To achieve this construction we prove some theorems which show that the incidence structure A=(P, L, I) connected to the corps K satisfies axioms A1, A2, A3 definition of affine plane. In all proofs rely on the sense of the corps as his ring and properties derived from that definition. Keywords: The unitary ring, integral domain, zero division, corps, incdence structurse, point connected to a corp, straight line connected to a corp, affine plane.

Based on the following very interesting work in the past [2], [3], [4], [9], [12], this article becomes a description of collineations in the affine plane [10]. We are focusing at the description of translations and dilatations, and we make a detailed description of them. We describe the translation group and dilatation group in affine plane [11]. A detailed description we have given also for traces of a dilatation. We have proved that translation group is a normal subgroup of the group of dilatations, wherein the translation group is a commutative group and the dilatation group is just a group. We think that in this article have brings about an innovation in the treatment of detailed algebraic structures in affine plane.

In this article will do a' concept generalization n-gon. By renouncing the metrics in much axiomatic geometry, the need arises for a new label to this concept. In this paper will use the meaning of n-vertexes. As you know in affine and projective plane simply set of points, blocks and incidence relation, which is argued in [1], [2], [3]. In this paper will focus on affine plane. Will describe the meaning of the similarity n-vertexes. Will determine the addition of similar three-vertexes in Desargues affine plane, which is argued in [1], [2], [3], and show that this set of three-vertexes forms an commutative group associated with additions of three-vertexes. At the end of this paperare making a generalization of the meeting of similarity n-vertexes in Desargues affine plane, also here it turns out to have a commutative group, associated with additions of similarity n-vertexes.

In this article, starting from geometrical considerations, he was born with the idea of 3D matrices, which have developed in this article. A problem here was the definition of multiplication, which we have given in analogy with the usual 2D matrices. The goal here is 3D matrices to be a generalization of 2D matrices. Work initially we started with 3×3×3 matrix, and then we extended to m×n×p matrices. In this article, we give the meaning of 3D matrices. We also defined two actions in this set. As a result, in this article, we have reached to present 3-dimensional unitary ring matrices with elements from a field F.

In this paper we present an application possibilities of the affine plane of order n, in the planning experiment, taking samples as his point. In this case are needed $n^2$ samples. The usefulness of the support of experimental planning in a finite affin plan consists in avoiding the partial repetition combinations within a proof. Reviewed when planning can not directly drawn over an affine plane. In this case indicated how the problem can be completed, and when completed can he, with intent to drawn on a affine plane..

ARTICLE INFO ABSTRACT In this paper we present a set transformation additive group. For this, the first stop on the meaning of the Desargues affine plane, formulating first axiom of his that show proposition D1. Afterwards we show that little Pappus theorem, which we in the construction of group proofs in additions of points on a line on desargues plane, also applies in the Desargues affine plane. Copyright©2016, Orgest Zaka and Kristaq Filipi. This unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, based on several meanings and statements discussed in the literature, we intend constuction a affine plane about a of whatsoever corps (K,+,*). His points conceive as ordered pairs (α,β), where α and β are elements of corps (K,+,*). Whereas straight-line in corps, the conceptualize by equations of the type x*a+y*b=c, where a≠0_K or b≠0_K the variables and coefficients are elements of that body. To achieve this construction we prove some theorems which show that the incidence structure A=(P, L, I) connected to the corps K satisfies axioms A1, A2, A3 definition of affine plane. In all proofs rely on the sense of the corps as his ring and properties derived from that definition. Keywords: The unitary ring, integral domain, zero division, corps, incdence structurse, point connected to a corp, straight line connected to a corp, affine plane.

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In this summary, my scientific research activity is presented: • 14-Published Book's in Albanian-Language [approximately (more than) 3000 pages in B5-format, and in writing 11pt]. • 13-papers in process [which will be completed within the year 2023]. • 5-Preprints which are currently under review. • 20-Published papers. • 22-Conference papers. More details: In references [1-14] are my published books, which are in Albanian Language. List of References [15-27] presented my non-published papers (preprints and projects, which are currently being completed). In list of references [28-32] present my papers which are currently under review. Reference-List [33-52]present my published papers and research results, whereas, references [53-74] present my references in various conferences.