Comparing Entire Prototypes at Semigroup Homomorphisms and Specifically at Regular Languages Substitutions

The following statements are proven: A correspondence of a semigroup in another one is a homomorphism if and only if when the entire prototype of the product of images contains (always) the product of their entire prototypes. The Kleene closure of the maximal rewriting of a regular language at a regular language substitution contains in the maximal rewriting of the Kleene closure of the initial regular language at the same substitution. Let the image of the maximal rewriting of a regular language at a regular language substitution covers the entire given regular language. Then the image of any word from the maximal rewriting of the Kleene closure of the initial regular language covers by the image of a set of some words from the Kleene closure of the maximal rewriting of this given regular language everything at the same given regular language substitution. The purposefulness of the ¯rst statement is substantiated philosophically and epistemologically connected with the spirit of previous mathematical results of the author. A corollary of its is indicated about the membership problem at a regular substitution.

A Statistical Convergence Aplication for the Hopfield Networks

When Recurrent Neural Networks (RNN) are going to be used as Pattern Recognition systems, the problem to be considered is how to impose prescribed prototype vectors ξ^1,ξ^2,...,ξ^p as fixed points. The synaptic matrix W should be interpreted as a sort of sign correlation matrix of the prototypes, In the classical approach. The weak point in this approach, comes from the fact that it does not have the appropriate tools to deal efficiently with the correlation between the state vectors and the prototype vectors The capacity of the net is very poor because one can only know if one given vector is adequately correlated with the prototypes or not and we are not able to know what its exact correlation degree. The interest of our approach lies precisely in the fact that it provides these tools. In this paper, a geometrical vision of the dynamic of states is explained. A fixed point is viewed as a point in the Euclidean plane R2. The retrieving procedure is analyzed trough statistical frequency distribution of the prototypes. The capacity of the net is improved and the spurious states are reduced. In order to clarify and corroborate the theoretical results, together with the formal theory, an application is presented

Digital Libraries and Portals Saving National Cultural Heritage (IMI–BAS Experience)

The current research activities of the Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences (IMI—BAS) include the study and application of knowledge-based methods for the creation, integration and development of multimedia digital libraries with applications in cultural heritage. This report presents IMI-BAS’s developments at the digital library management systems and portals, i.e. the Bulgarian Iconographical Digital Library, the Bulgarian Folklore Digital Library and the Bulgarian Folklore Artery, etc. developed during the several national and international projects: - "Digital Libraries with Multimedia Content and its Application in Bulgarian Cultural Heritage" (contract 8/21.07.2005 between the IMI–BAS, and the State Agency for Information Technologies and Communications; - FP6/IST/P-027451 PROJECT LOGOS "Knowledge-on-Demand for Ubiquitous Learning", EU FP6, IST, Priority 2.4.13 "Strengthening the Integration of the ICT research effort in an Enlarged Europe" - NSF project D-002-189 SINUS "Semantic Technologies for Web Services and Technology Enhanced Learning". - NSF project IO-03-03/2006 ―Development of Digital Libraries and Information Portal with Virtual Exposition "Bulgarian Folklore Heritage". The presented prototypes aims to provide flexible and effective access to the multimedia presentation of the cultural heritage artefacts and collections, maintaining different forms and format of the digitized information content and rich functionality for interaction. The developments are a result of long- standing interests and work in the technological developments in information systems, knowledge processing and content management systems. The current research activities aims at creating innovative solutions for assembling multimedia digital libraries for collaborative use in specific cultural heritage context, maintaining their semantic interoperability and creating new services for dynamic aggregation of their resources, access improvement, personification, intelligent curation of content, and content protection. The investigations are directed towards the development of distributed tools for aggregating heterogeneous content and ensuring semantic compatibility with the European digital library EUROPEANA, thus providing possibilities for pan- European access to rich digitalised collections of Bulgarian cultural heritage.