In this work, the method of drawing of nine celebrated Minoan Late Bronze Age wall-paintings, is studied. Four of these frescoes were unearthed at Akrotiri, Thera, while the other five were excavated in Crete; these frescoes have never been thoroughly studied before, concerning their method of drawing. The authors demonstrate that practically all actually drawn contours in these frescoes, optimally fit six geometric stencils -guides, namely, four hyperbolae and two Archimedes' (linear) spirals. It is important to note that the hyperbola and the linear spiral do not exist in nature, definitely not with the precision encountered here. We would like to stress that all wall-paintings the authors have studied so far, i.e. more than twenty-five specialIntscript had been drawn via the use of the same stencils, a fact that renders the eventuality that this was indeed the method of drawing of these frescoes, almost certain. The considered wall-paintings cover a period of at least two centuries, located both before and after the gigantic eruption of Thera volcano. Moreover, here, possible methods of construction of the determined stencils are proposed, as well as the most probable one, according to our opinion. Methodologically, the authors determine couples of the longest contour parts, CP, appearing in the studied frescoes and of a proper sub-curve GP & DEG;rt of one of the aforementioned stencils; these couples are chosen so as they fit with exceptionally low approximation errors. The latter terms are rigorously defined here, for the first time. Special care is taken, mathematically and programmatically, to ensure the maximum possible smoothness between successive optimally fit stencils' parts GP & DEG;rt. Next, the authors employed these modern prerequisites, to introduce the more probable sequence of actions taken by the prehistoric artist(s) for drawing the specific frescoes. Finally, crucial queries and conjectures are set, together with a number of plausible answers.

The main goal of the present work is to determine the hand that has written two newly discovered documents in Romania. For giving the proper answer, the authors introduced the notion of “Ideal Representative”, namely of an object that very well represents the corresponding ideal alphabet symbol that a writer had in his/her mind when writing a document by hand. Moreover, the authors have introduced a novel method, which leads to the optimal evaluation of the Ideal Representative of any alphabet symbol in association with any handwritten document. Furthermore, the authors have introduced methods for comparing these Ideal Representatives, so as a final decision about the hand that has written a document may be obtained with a highly considerable likelihood. The related analysis manifests that the two documents discovered in Romania in 1998, belong to the great personality of Rigas Feraios. The presented method of automatic handwriting Identification seems to be of general applicability.

In the Late Bronze Age, in Second Millennium B. C., very important civilizations had flourished in the Aegean Islands and shores, such as the one of Akrotiri, Thera, of the Minoan Crete and of the Mycenean Boeotia. One of the main characteristics of these civilizations was the widespread Art of Wall Painting. In the present work, the method of drawing of a number of celebrated wall-paintings belonging to the three (3) aforementioned civilizations is studied. The authors have demonstrated that the entire set of the borderlines of all figures appearing in the studied frescoes, have been drawn by six (6) geometric stencils-guides, namely four (4) hyperbolae and two (2) Archimedes (linear) spirals, the conception and construction of which was impressively and extraordinarily advanced for the specific era. The conception and construction of these mathematical curves was, so far, believed that it took place at least one thousand four hundred (1400) years later by “giants of thought, Geometry and Mathematics” in the Classical Era.

In the present work, a general methodology is introduced for the determination of the writer of a given document, provided that texts of the same hand are available. The approach is successfully applied to an unsigned document that has been spotted in the warehouses of Greek Army History Directorate and led, for the first time, to the conclusion that it has been written by the great Greek Politician Eleftherios Venizelos. The content of the paper is important from the historical point of view.

A first aim of the present work is the determination of the actual sources of the “finite precision error” generation and accumulation in two important algorithms: Bernoulli’s map and the folded Baker’s map. These two computational schemes attract the attention of a growing number of researchers, in connection with a wide range of applications. However, both Bernoulli’s and Baker’s maps, when implemented in a contemporary computing machine, suffer from a very serious numerical error due to the finite word length. This error, causally, causes a failure of these two algorithms after a relatively very small number of iterations. In the present manuscript, novel methods for eliminating this numerical error are presented. In fact, the introduced approach succeeds in executing the Bernoulli’s map and the folded Baker’s map in a computing machine for many hundreds of thousands of iterations, offering results practically free of finite precision error. These successful techniques are based on the determination and understanding of the substantial sources of finite precision (round-off) error, which is generated and accumulated in these two important chaotic maps.

In the present paper, we propose a methodology of general applicability for matching, comparing and grouping planar shapes, under a unified framework. This is achieved by interpreting shapes’ grouping as a result of the hypothesis that shapes of the same class come from the same implicit family of curves. In order to render the analysis independent of the functional form of the curves families’ implicit function, we have formalized the shapes’ comparison in terms of the implicit curvature function. The implementation of the methodology targets towards automatic writer identification and the corresponding information system has been applied to the identification of the writer of Byzantine codices that preserve Iliad. The shapes in hand are the alphabet symbols appearing in the documents’ images to be classified. The realizations of each alphabet symbol are compared pairwise, modulo affine transformations. The statistical compatibility of these comparisons inside the same document and between different documents determines the likelihood of attributing different documents to the same hand. By maximizing the joint likelihood for all alphabet symbols, common in all documents we determine the most probable classification of the given documents into writing hands. Application of the methodology to 25 images of Byzantine codices’ pages indicated that these pages have been written by 4 hands in full accordance with experts’ opinion and knowledge.

In this work, a generic framework is developed that simultaneously embeds 2D shapes’ registration, comparison and grouping, via the assumption that shapes of the same class are distorted level-sets of the same implicit function. The corresponding system is developed so as to deal with the automatic classification of documents according to their writers. The data that the system processes are the realizations of the individual characters, which are mutually aligned per document and per character, modulo affine transformations, and then reduced to a single representative shape. Stationarity conditions of these representatives are then used to statistically test the hypothesis that different documents bear a single representative. The considered documents are grouped according to their writer, by determining the maximal groups that also maximize the joint probability of the classification, computed over the characters, which occur in all documents. We have tested the system on 26 pages of Byzantine manuscripts that preserve Iliad. The computed classification of these pages in 4 writers has been independently verified by expert scientists.

In the present paper, a novel approach is introduced for the study, estimation and exact tracking of the finite precision error generated and accumulated during any number of multiplications. It is shown that, as a rule, this operation is very "toxic", in the sense that it may force the finite precision error accumulation to grow arbitrarily large, under specific conditions that are fully described here. First, an ensemble of definitions of general applicability is given for the rigorous determination of the number of erroneous digits accumulated in any quantity of an arbitrary algorithm. Next, the exact number of erroneous digits produced in a single multiplication is given as a function of the involved operands, together with formulae offering the corresponding probabilities. In case the statistical properties of these operands are known, exact evaluation of the aforementioned probabilities takes place. Subsequently, the statistical properties of the accumulated finite precision error during any number of successive multiplications are explicitly analyzed. A method for exact tracking of this accumulated error is presented, together with associated theorems. Moreover, numerous dedicated experiments are developed and the corresponding results that fully support the theoretical analysis are given. Eventually, a number of important, probable and possible applications is proposed, where all of them are based on the methodology and the results introduced in the present work. The proposed methodology is expandable, so as to tackle the round-off error analysis in all arithmetic operations.

Recently, very large-scale decision support systems (DSSs) have been developed, which tackle very complex problems, associated with very extensive and polymorphic information, which probably is geographically highly dispersed. The management, updating, modification and upgrading of the data and program core of such an information system is, as a rule, a very difficult task, which encompasses many hazards and risks. The purpose of the present work was (a) to list the more significant of these hazards and risks and (b) to introduce a new general methodology for designing decision support (DS) systems that are robust and circumvent these risks. The core of this new approach was the introduction of a meta-database, called teleological, on the base of which management, updating, modification, reduction, growth and upgrading of the system may be safely and efficiently achieved. The very same teleological meta-database can be used for the construction of a sound decision support system, incorporating elements of a previous one at a future stage.