LINEAR RECURRENCE RELATONS AND ORDINARY GENERATING FUNCTIONS APPLIED ON MODELING PROCESSES IN CONTROL THEORY

In this paper we apply multistep recurrence relations, as one of very simple and useful mathematical models. It is an efficient tool for solving many problems in mathematics, science, and technics. We also use generating functions, as a connection between real number sequences and real functions, and as a very smooth and efficient connection between the discrete mathematics and (continual) mathematical analysis. We present an application of multistep homogenous linear recurrence relations for modelling some processes in the control theory. Further on, we use the ordinary generating function aiming to find appropriate formulae for calculating members of an appropriate recurrence sequence. Finally, we show the application of this novel mathematical approach on one real example in the control theory

Grading as factor of student motivation: theoretical principles and legal regulation

Predmet ovog istraživanja je motivacija učenika u školskom ocenjivanju sa stanovišta teorija motivacionih potreba i važećih zakonskih akata. Cilj rada je da se razmotri da li su učenici motivisani tokom školskog ocenjivanja. Motivaciju razmatramo u skladu sa teorijom samoodređenja (integrativna motivaciona teorija) čime se omogućava istraživanje uloge nastavnika. Metodom analize sadržaja istraženi su: a) Zakon o osnovama sistema obrazovanja i vaspitanja, Zakon o osnovnoj i srednjoj školi, Zakon o prosvetnoj inspekciji; b) pravilnici o ocenjivanju učenika u osnovnoj i srednjoj školi; c) pravilnici koji uređuju rad nastavnika i ustanova (Pravilnik o standardima kompetencija za profesiju nastavnika i njihovog profesionalnog razvoja, Pravilnik o stalnom stručnom usavršavanju i napredovanju u zvanja nastavnika, vaspitača i stručnih saradnika, Pravilnik o stručno-pedagoškom nadzoru, Pravilnik o standardima kvaliteta rada ustanova, Pravilnik o vrednovanju kvaliteta rada ustanova). Praćenje razvoja i postignućaučenika u toku školske godine ostvaruje se formativnim i sumativnim ocenjivanjem (Zakon o osnovama sistema obrazovanja i vaspitanja, Zakon o osnovnoj školi), a Zakon o srednjoj školi definiše formativno i sumativno ocenjivanje u srednjоškolskom obrazovanju. Pravilnici o ocenjivanju uređuju svrhu, principe, vrste, kriterijume, način i postupak ocenjivanja. Za razliku od osnovne škole, u srednjoj školi se ocenom, pored ostvarenosti ciljeva, ishoda i angažovanja učenika, izražava i napredovanje i preporuka za dalje napredovanje. Motivacija učenika u ocenjivanju pominje se u više standarda za nastavničku profesiju. Najširi okvir dat je u standardima kvaliteta rada ustanova, oblast Nastava i učenje, standard 2.4. Postupci vrednovanja su u funkciji daljeg učenja. Odnos motivacije učenika, ocenjivanja i učenja bi trebalo precizno regulisati podzakonskim aktima. Načelne kriterijume u oba pravilnika o ocenjivanju treba menjati uvođenjem razumljivih, konkretnih i obaveznih kriterijuma ocenjivanja. U standarde kompetencija za profesiju nastavnik trebalo bi uvesti indikator o motivaciji učenika tokom ocenjivanja i usaglasiti ostale pravilnike. Da motivacija učenika ne bi ostalalična strategija uspešnih nastavnika, potrebno je zakonski definisati balans između priznavanja angažovanja učenika i evaluacije znanja sa stanovišta naučne discipline.The subject of this research is analysis of student motivation in school grading from the standpoint of theories of motivational needs and legal acts. The aim of this paper is to consider whether the students are motivated in school grading. We appraise motivation in accordance with the theory of self-determination (integrative motivational theory), which enables appraising the role of teachers as well. The method of content analysis was used on: a) Law on the fundamentals of the education system, Laws on primary and secondary education, and the Law on Educational Inspection; b) rulebooks on student assessment in primary and secondary education; c) rulebooks governing the work of teachers and institutions (Rulebook on the standards of competencies for the profession of teacher and their professional development, Rulebook on continuous in-service training and advancement to titles of teachers, educators and professional associates, Rulebook on professional pedagogical supervision, Rulebook on quality standards of institutions, Rulebook on assessing the quality of work of educational institutions). Monitoring development and achievement of students during the school year is achieved by formative and summative assessment (Law on fundamentals of the education system, Law on Primary Education); the Law on Secondary Education defines formative and summative assessment in secondary education. Grading rulebooks regulate the purpose, principles, types, criteria, manner and procedure of grading. In secondary school, unlike the primary school, in addition to achievement of goals, the grade expresses outcomes and engagement of students, as well as the progress and recommendations for further progress. Student motivation in grading is mentioned in several standards for the profession of a teacher. The broadest framework is given in the standards regarding quality of work of educational institutions, section Teaching and Learning, standard 2.4. Evaluation procedures are in function of further learning. Relation between student motivation, grading and learning should be precisely regulated by by-laws. Basic criteria in both grading rulebooks should be changed by introducing understandable, concrete and mandatory grading criteria. The indicator of student motivation in grading should be included in the standards of teaching profession competencies, and the other rulebooks should conform. In order for the student motivation not to remain a personal strategy of successful teachers, it is necessary to legally define balance between the recognition of student engagement and evaluation of knowledge from the scientific discipline point of view.Knjiga rezimea, 25. Međunarodna naučna konferencija "Pedagoška istraživanja i školska praksa"Book of abstracts / 25th International Scientific Conference "Educational Research and School Practice

Thermal parameters defined with graph theory approach in synthetized diamonds

The Nanocrystaline diamonds are very important biomedical material with variety of applications. The experimental procedures and results have been done in the Institute of Functional Nanosystems at the University Ulm, Germany. There is an existing biocompatibility of the diamond layers, selectively improved by biomimetic 3-D patterns structuring. Based on that, we have been inspired to apply the graph theory approach in analysing and defining the physical parameters within the structure of materials structure samples. Instead the parameters values, characteristic at the samples surface, we penetrate the graphs deeply in the bulk structure. These values could be only, with some probability, distributed through the micro-structure what defines not enough precious parameters values between the micro-structure constituents, grains and pores. So, we originally applied the graph theory to get defined the physical parameters at the grains and pores levels. This novelty, in our paper, we applied for thermophysical parameters, like thermoconductiviy. By graph approach we open new frontiers in controlling and defining the processes at micro-structure relations. In this way, we can easily predict and design the structure with proposed parameters

Fractal reconstruction of fiber-reinforced polymer composites

Polymers offer the possibility of different reinforcement incorporation due to a broad range of chemical structures. Along with this feature, their light weight and processing ease made them a class of materials that have been applied in construction parts, drug delivery agents or electronic devices. Epoxy-based composites have used as insulators in microelectronic devices due to its chemical resistance, good adhesion properties and endurance. As epoxies have low fracture resistance, they are often reinforced with different kinds of fibers. With thorough knowledge of the structure, physical properties can be predicted and included in the processing of future composites, especially that electronic materials minituarization brought micro- and nanoscale level properties at spotlight. Fractal nature analysis is a mathematical method that has proved to be efficient in grain interface properties applied on perovskite ceramic materials. In our study, fiber shape reconstruction and determination of Hausdorff dimension have been achieved with the application of fractal regression model employed in software Fractal Real Finder opening a new path for the prediction of reinforcement shape and size, all with the aim of processing composite materials with desired propertie

The ceramics materials density defined by artificial neural networks

Predicting the ceramic materials properties and designing the desired microstructures characteristics are very important objectives in ceramic samples consolidating process. The goal of our research is to calculate the density within consolidated BaTiO3-ceramic samples for different consolidation parameters, like sintering temperature, using obtained experimental data from the material’s surface, by applying back propagation neural network (BP). This method, as a very powerful tool, provides the possibility to calculate the exact values of desired microelectronic parameter at the level of the grains’ coating layers. The artificial neural networks, which have biomimetic similarities with biological neural networks, propagate the input signal forward, unlike the output signal, designated as error, which is propagated backwards spreading throughout the whole network, from output to input neuron layers. Between these two neuron layers, there are usually one or more hidden layers, where the grains of the sintered material are represented by network neurons. Adjustable coefficients, called weights, are forward propagated, like input signals, but they modify the calculated output error, so the neural network training procedure is necessary for reducing the error. Different consolidated samples density values, measured on the bulk, substituted the errors, which are calculated as contribution of all network elements, thus enabling the density calculation of all constituents of ceramic structure presented by neural network. In our future research we plan to increase the number of neurons and hidden layers in order to improve this method to become even more accurate and precise

Approximation and Error Prediction in Electrochemical Parameters Calculation Using Neural Networks

Various interesting results have been achieved in calculation of electrochemical parameters in nanomaterials, using neural networks. There appear some error, during those calculations, and it varies depending on number of neurons in layers. In this research we deal with errors, calculated for neural networks with n=1,2…10, neurons in first or second layer. We applied mean square approximation method, in order to get explicite formula for predicton of error, for other cases

Fractals, Graphs and Neural Networks: The Holly Trinity of Nanostructures - An Overview and Comparison of Methods

There are a lot of recently published research papers regarding representing nanostructures and biomimetic materials, using simple but powerful mathematical methods. In most of them, fractal theory, graph theory and neural networks are used. Having in mind variety of those methods, but in the same time complementarity and compatibility, they became very useful tool, and we named it “Holly Trinity” of mathematical approach in nanostructures. In this research we give an overview on interesting results in modelling nanostructures and their electrochemical and magnetic parameters, using those very actual and “easy to use” methods: fractal theory, graph theory and neural networks. We also compare them, in order to conclude about areas of their most useful applications

Reconstruction of fiber reinforcement in epoxy-based composite

Polymer matrix composites (PMCs) are very attractive materials due to a possibility to achieve versatile properties by combining with ceramic or metal reinforcement in different shapes and sizes. As a result, PMCs have found application in nearly every field, from household appliances to aerospace industry. Modern microelectronic devices contain conductive polymers with fillers that enhance their electrical properties. In addition, PMCs are being used as insulators and adhesives, contributing to the long life of electronic devices. Epoxy resins are the most commonly used insulators and adhesives. In order to improve their fracture toughness, glass fibers can be used as an efficient reinforcement. However, with the purpose of designing a composite with good mechanical properties and durability, deep knowledge of microstructure is required. In addition, microstructural analysis can be used to connect shape and size of pores or reinforcement with various physical properties. Fractal nature analysis is a valuable mathematical tool that can be employed for different shapes and forms rendering. In this manner, successful design and prediction of composite’s properties could be obtained. In this research, field emission scanning electron microscopy (FESEM) images were used for fractal analysis of glass fibers, with the aim of reconstructing the shape

Projektovanje 3D koordinatnih usmerenih grafova na ravan i pravu i primena u procesu sinteze sistoličkih polja

Each algorithm uniquely corresponds to a directed graph, and each parallel computer system, also uniquely corresponds to a directed graph. Therefore, solving many problems regarding algorithms and parallel computer systems, is reduced to solving problems on the graph level. In this thesis, we deal with the problem how to determine a mapping of graph of algorithm, which corresponds to a given problem, so that the result of mapping is a graph of corresponding parallel computer system, on which the problem is going to be effectively implemented. Practically, on the basis of the algorithm of problem, we design a parallel computer system of special purpose, suitable for solving the problem. The main topic of interest in this thesis are 3D coordinate graphs, as well as possibilities of their mapping to the plane (2D) and line (1D). The geometrical relationship between three-dimensional objects and their two-dimensional images, ie. projections in some planes, is very important and has many applications in various fields of science and technology. During determination of those mappings, principle of neighborhoods between nodes in a given graph and in its image has to be preserved, and also minimal number of nodes in a graph, which corresponds to a parallel computer system. Without loss of generality, we deal with graphs that can be presented in a three-dimensional coordinate system. This thesis gives an answer to the question how the three-dimensional mesh, ie. coordinate graph, maps to a plane and a line. First, we determine all possible projection directions for design, with respect to certain limitations. For each possible projection direction we define the mapping, which ensures that the resulting image have the optimal number of nodes, taking into account that principle of neighborhood between nodes is preserved. Beside compilation of previous results, the main results and contributions of this thesis has been provided via formal mathematical forms through 1 lemma, 7 theorems and 3 corollaries. The most important theorems, that are base of whole process of projection of coordinate graphs are given with complete proofs. These theorems are still not published and they represents an original contribution of this thesis. In other parts of the dissertation, using introduced mapping, we design all possible 1D systolic arrays for solving one of the basic and most used problems in linear algebra, matrix product. Then, we implement the same methods for synthesis of 1D systolic arrays for solving some standard problems in graph theory: finding the transitive closure of directed graph, finding all shortest paths between nodes in graph, finding all possible minimum cost spanning trees in a given graph. Several years of research in this area, showed us the path for solving the complete set of tasks. It is shown, first on the theoretical level, and then through all those applications, that method of projection of graphs, introduced in this thesis, is the best, and also enables an automatic synthesis of arrays, through explicit formulas, and also gives optimality of the obtained arrays, in terms of space, or time or space-time

Projektovanje 3D koordinatnih usmerenih grafova na ravan i pravu i primena u procesu sinteze sistoličkih polja

Each algorithm uniquely corresponds to a directed graph, and each parallel computer system, also uniquely corresponds to a directed graph. Therefore, solving many problems regarding algorithms and parallel computer systems, is reduced to solving problems on the graph level. In this thesis, we deal with the problem how to determine a mapping of graph of algorithm, which corresponds to a given problem, so that the result of mapping is a graph of corresponding parallel computer system, on which the problem is going to be effectively implemented. Practically, on the basis of the algorithm of problem, we design a parallel computer system of special purpose, suitable for solving the problem. The main topic of interest in this thesis are 3D coordinate graphs, as well as possibilities of their mapping to the plane (2D) and line (1D). The geometrical relationship between three-dimensional objects and their two-dimensional images, ie. projections in some planes, is very important and has many applications in various fields of science and technology. During determination of those mappings, principle of neighborhoods between nodes in a given graph and in its image has to be preserved, and also minimal number of nodes in a graph, which corresponds to a parallel computer system. Without loss of generality, we deal with graphs that can be presented in a three-dimensional coordinate system. This thesis gives an answer to the question how the three-dimensional mesh, ie. coordinate graph, maps to a plane and a line. First, we determine all possible projection directions for design, with respect to certain limitations. For each possible projection direction we define the mapping, which ensures that the resulting image have the optimal number of nodes, taking into account that principle of neighborhood between nodes is preserved. Beside compilation of previous results, the main results and contributions of this thesis has been provided via formal mathematical forms through 1 lemma, 7 theorems and 3 corollaries. The most important theorems, that are base of whole process of projection of coordinate graphs are given with complete proofs. These theorems are still not published and they represents an original contribution of this thesis. In other parts of the dissertation, using introduced mapping, we design all possible 1D systolic arrays for solving one of the basic and most used problems in linear algebra, matrix product. Then, we implement the same methods for synthesis of 1D systolic arrays for solving some standard problems in graph theory: finding the transitive closure of directed graph, finding all shortest paths between nodes in graph, finding all possible minimum cost spanning trees in a given graph. Several years of research in this area, showed us the path for solving the complete set of tasks. It is shown, first on the theoretical level, and then through all those applications, that method of projection of graphs, introduced in this thesis, is the best, and also enables an automatic synthesis of arrays, through explicit formulas, and also gives optimality of the obtained arrays, in terms of space, or time or space-time