Αλγόριθμοι

Abstract

The attempt of science to codify and describe with mathematical models the natural, the social and economic phenomena was mainly based on deterministic and probabilistic ones descriptions. But with the advent of computing machines, a third approach occupied him scientific world. It is about the development of algorithms and standardization in a scientific discipline of Algorithmic theory. Designing an algorithm can be done in many ways. But one of these he is "better" than others. The definition of the concept "best algorithm", as well determining the methodology that safely leads to the design of the "best" algorithm, are the subject of Algorithmic Theory. In the evolution of algorithms, importance is primary of retrospection. An elementary process of expressing an evolving physical state can be expressed through recursion and recursion relations. These relationships constitute an extremely simple organization of rules that produce sets of results. In the relevant paragraphs, the methodology and the properties of recurrence relations with special reference to Fibonacci sequences and their applications. Special mention is made of the algebraic structure referred to as the "algebra of Fibonacci numbers" and the its relation to the golden ratio. A brief presentation of its individual chapters is also made of a particular algebra of numbers and is reported as concisely and as briefly as possible in appearance of Fibonacci numbers in the surrounding world. Then, its technological applications follow computer science, to digital electronics and finally to the concepts of telecommunications and especially in signal processing. Algorithmic Theory is the area of Computer Science that deals with design efficient algorithms for solving computational problems. Each algorithm must be analyzed mathematically to document its correctness and quantify its performance against various kinds of computing resources, such as time and amount of available memory. The process of the analysis and documentation introduced by the methods of this theory allows the extraction conclusions about the correctness of algorithm results, as well as prior knowledge for the amount of computing resources required to apply the algorithm to specific conditions. The results of analyzing different algorithms for the same or similar problems allow comparison between them and the selection of the most suitable for specific practices applications. The evolution of computing systems has allowed the development of complex and extensive algorithms. THE complexity of modern algorithms gave new impetus to the Theory of Algorithms and led to the development of Computational Complexity Theory, which focuses on the study of different computing models and the effect they have on the ability and amount of computing resources that needed to solve a problem. In relation to the theory of algorithms, the theory of computation complexity provides a complementary perspective on the concept of computation. For each computational model, Computational Complexity Theory studies whether a computational problem can to resolve or not. If the problem is solvable, Complexity Theory studies the minimum amount of computational resources required to solve the problem in that particular model. Based on the answers to the above questions, the computational problems are grouped into classes complexity, which consist of problems that exhibit similar behavior with respect to their solvability in some specific computational model. The evolution of Algorithm Theory itself led to the development of Algorithmic Theory Games. This chapter will not refer to it, but the reader can search for it texts so that it can be found in this special topic. The Algorithmic Game Theory (Algorithmic Game Theory) is an interdisciplinary field of research at the boundaries of the areas of Computer Science (Computer Science) of Game Theory and Economic Theory. In the last 10 years it has experienced enormous growth, because it understands the essence, describes quantitatively and qualitatively and rigorously analyzes complex selfish interaction problems in large-scale networks (Internet, Stock Exchange, International Markets, etc.). Because of their nature, these problems do not controlled by a central authority. For example, we are not only concerned with the existence of balances between entities involved, but also identifying such a balance in an efficient time, how we can converge on some of them as quickly as possible (and possibly with the appropriate incentives for participants), how one may converge on some of them as quickly as possible (and possibly with the appropriate incentives for participants), how can we distinguish and possibly enforce (as administrators of a system) balances that are more desirable for the system than others, how we can design efficient mechanics that affect the game for the benefit of the game as a whole system, etc.

Μια στοιχειώδης διαδικασία έκφρασης μιας εξελισσόμενης φυσικής κατάστασης μπορεί να εκφραστεί μέσα από την αναδρομή και τις αναδρομικές σχέσεις. Οι σχέσεις αυτές αποτελούν μια εξαιρετικά απλή οργάνωση κανόνων που παράγουν σειρές αποτελεσμάτων. Στις σχετικές παραγράφους, αναπτύσσεται η μεθοδολογία και οι ιδιότητες των αναδρομικών σχέσεων με ιδιαίτερη αναφορά στις ακολουθίες Fibonacci και τις εφαρμογές τους. Ιδιαίτερη αναφορά γίνεται στην αλγεβρική δομή που αναφέρεται ως «άλγεβρα των αριθμών Fibonacci» και τη σχέση της με τη χρυσή τομή. Γίνεται επίσης μια σύντομη παρουσίαση των επιμέρους κεφαλαίων της συγκεκριμένης άλγεβρας αριθμών και αναφέρεται περιληπτικά και όσο συντομότερα είναι δυνατόν η εμφάνιση των αριθμών Fibonacci στον περιβάλλοντα κόσμο. Μετά, ακολουθούν οι τεχνολογικές εφαρμογές της στην επιστήμη των ηλεκτρονικών υπολογιστών, στα ψηφιακά ηλεκτρονικά και τέλος στις έννοιες των τηλεπικοινωνιών και ιδιαίτερα στην επεξεργασία σημάτων.