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Muskhelishvili Institute of Computational Mathematics
Email: micm@gtu.ge, Phone: (+995) 555 12 97 50
Address: Georgia, Tbilisi, Grigol Peradze str., 4, 0159
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Muskhelishvili Institute of Computational Mathematics of the Georgian Technical University is a scientificresearch and informationanalytical institution, which aims to acquire, disseminate and apply new knowledge in the process of formation of an innovative economy in Georgia to solve fundamental scientific, technological and socioeconomic problems. The purpose is to carry out scientific research and innovative projects of various complexity, based on the efficient use of the Institute's intellectual resources, with the aim of conducting fundamental and applied research for various government and nonstate clients, including research projects funded by local and foreign institutions. An important part of the activities is to preserve the traditions established by the scientists of the previous generation. The Institute perceives its mission in maintaining and strengthening of its role as one of the leading scientificresearch institutes, which contributes to the development of science. In order to achieve the set goals, the Institute studies and solves the following problems:  Applied problems of mathematical statistics, data analysis (environment, agriculture, medicine), mathematical modeling and imitation, development of new computer technologies, system analysis (environment, water pollution);  Development of numerical solution methods for engineering mechanics problems related to the determination of deformations causing breakdowns of various structures and equipment;  Construction of parallel algorithms, creation of relevant software, their research and implementation on a parallel computing system;  Mathematical modeling of social, economic and mathematical physics problems and development of appropriate computational methods and optimal algorithms;  Probabilistic measures and random sequences in topological vector spaces and topological groups; Compact vector summation and its uses; problems related with the rearrangements of series;  Cauchytype singular integrals approximation schemes and their application for the boundary problems of function theory, as well as the elasticity theory, nuclear physics and other related problems;  to study the problem of approximate solution of generalized harmonic threedimensional problems in the case of finite and infinite areas bounded by one or more surfaces;  Study and solve problems related to rotary shell calculations.
Structural Units
Department of Computational Methods Department of Probabilistic and Statistical Methods Department of Informatics Department of Mathematical Modeling
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Title of the equipment/device  Technical characteristics  Date of issue  Exploitation staring year  Usage/application  Purpose of usage/application  Technical condition 
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International Scientific Works
Project number/ID  Project title  Name of the grant call  Funding organization  Grant budget (total)  Start/end dates  Principal investigator  Key personnel  Project Summary  Detailed description  Achieved results 

101078950  Georgian Artificial Intelligence Networking and Twinning Initiative (GAIN)  HORIZON EUROPE  WIDERA2021ACCESS03 (Twinning )  European Union  1319250 EUR  2022  2025  G. Giorgobiani  V. Kvaratskhelia, Z. Sanikidze, K. Kachiashvili, Z. Tabagari, M. Menteshashvili, G. Ghlonti, T. Saginadze, T. Javakhishvili, B. Oikashvili, I. Kachiashvili, V. Berikashvili (and others)  The project started on October 1, 2022. Our partners are distinguished European institutions in the field: German Research Centre for Artificial Intelligence (DFKI); National Institute for Research in Digital Science and Technology (INRIA); EXOLAUNCH GmbH (EXO). Within the GAIN project the Artificial Intelligence (AI) related scientific and innovation excellence of the major Georgian research institute working in the area of Computer Sciences, Muskhelishvili Institute of Computational Mathematics at Georgian Technical University (MICM), will be radically strengthen and networked with the leading European research communities. Therefore, the entire ICT research community of Georgia will improve its integration with the concerted European efforts aimed at preserving the European leadership and research capacities in a such strategically important area as AI. The importance of the project’s specific focus on this area can be justified by the common understanding that AI represents one of the greatest opportunities for the global societal and economic progress.    As the project has just started, only two tasks are processed at the moment: The project Kick Off Meeting was held at Muskhelishvili Institute of Computational Mathematics (MICM). It was attended by the representatives of partner organizations from Europe, heads of departments and employees of GTU and the Institute, guests from other organizations (up to 70 participants in total); Deliverable 1.1 (related to the ethical issues) was submitted to EC. In the frames of the project "MICM AI Lab" is functioning, which includes about 20 researchers, mostly young. The lab will be supported financially as well as scientifically by the project for 3 years. After that we plan to maintain created research capacities by the help of the government of Georgia. 
DI181429  Application of probabilistic methods in discrete optimization and scheduling problems  Research with participation of compatriots residing abroad  Shota Rustaveli National Science Foundation of Georgia  200900 GEL  13/12/201812/12/2021  Nodar Vakhania  V. Tarieladze (coinvestigator), B. Mamporia (manager), Z. Sanikidze, V. Berikashvili, A. Chakhvadze  The project deals with the theoretical and practical aspects of determining the relationship between probability theory and combinatorial optimization.  The project deals with the theoretical and practical aspects of determining the relationship between probability theory and combinatorial optimization. The analysis in this area revealed perspectives on the application of various probabilistic processes and their associated random elements and stochastic integrals, as well as the application of a number of known results in scheduling problems. Unification of the two named fields of mathematics in the development of methodologies for solving practically important tasks of production management and optimization, as well as the economicsocial and nonlocal nature of these tasks, implies the need for interdisciplinary research in the context of international scientific cooperation which, to some extent, was implemented by the participants while working on the project. The project introduces new approaches that involve the use of probabilistic estimates and random processes in the algorithmic implementation of scheduling problems.  The following results were obtained by the project participants:  the number of optimal solutions is determined, the corresponding optimal schedules are presented and the magnitude of the complete optimal completion time for specific cases in singleprocessor scheduling problems is determined. The probability of the occurrence that the randomly taken schedule from the possible allowable schedule is optimized is calculated.  has been explored the possibility of using certain types of probabilistic distributions in scheduling problems where the processing times by the processor are random values which may lead to incorrect results. The concept of symmetrically sliced normally distributed random quantities has been introduced, the application of which, by selecting the slicing level, avoids the development of irregular processes in solving these problems.  theorems have been formulated and validated for the application of the probabilistic analogy of a set of optimal solutions for some scheduling problems.  For multiprocessor scheduling problems, the issues related to the mathematical processing of the process of efficient distribution of tasks on identical processors have been studied, in case of different job execution times. a practical case from the theory of scheduling is considered when, under certain conditions, a new structural algorithmic scheme of the process is proposed to minimize the total importance of deliveries in continuous batches and the corresponding delays, including the online scenario. The presented main results are realized in the form of 7 articles published in international refereed scientific journals, and reports (or conference theses, publications) presented at 13 international and 1 local scientific forums. 
FR/539/5100/13  150000 GEL  20142016  S. Chobanyan  N. Vakhania, V. Kvaratskelia, G. Chelidze, V. Tarieladze, G. Giorgobiani  The project deals with the problems of analysis related to the rearrangements of summands. The famous Riemann theorem and the concepts of conditionally and unconditionally convergent series are closely related to the topic of the project.  The project deals with fundamental and applied problems related to mathematical methods of research of rearrangements of vector summands. The following tasks were planned in the project: (i) find new maximal inequalities of Leventhal type, as well as maximal inequalities for fixed (invariant) multipliers (scalars or operators). Also, rewrite the inequalities in terms of rearrangements of random variables. (ii) prove the Steinitztype theorem for nonlocally convex spaces (maximally expand the class of spaces where the "sigmatheta" condition is sufficient). (iii) a complete generalization of the Nikishintype theorem, where the convergence of series holds for almost all simple permutations. Prove similar theorems from probability theory: the strong law of large numbers and the law of repeated logarithm. (iv) Investigate the methods around the Kolmogorov hypothesis and implementing the following approach: using the transference lemma, prove Garcia's hypothesis first for equalmodule coefficients, then for the general case, using the Maurey Pisier idea. (v) Study of optimal algorithms in the compact vector summation problem. Clarify the relationship between "greedy" and optimal algorithms. Applications in scheduling theory: volume calendar planning problem, Sevastianov's approach.  Our contribution to the topic of the project can be formulated as follows: (i) We derived a number of maximal inequalities: the basic transfer lemma, the twosided maximal inequality for rearrangements in terms of means and distributions. These inequalities include, in particular, the wellknown Garcia inequalities for arbitrary sequence of numbers and orthonormal systems. (ii) We solved M. Kadetz's longstanding problem about the structure of the set of sums: what conditions must be satisfied by a conditionally convergent series in order to satisfy Steinitz's theorem (namely, its set of sums be affine and closed). We have shown that the socalled ""sigmatheta"" condition ensures the validity of Steinitz's theorem in metrizable locally convex and locally bounded spaces. (iii) We have found a methodology for proving Nikishintype theorems for rearrangement convergent series. Using it, we generalized and improved Nikishin's theorems, which today represent the strongest theorems in this direction. (iv) We have found an ""almost"" solution to Kolmogorov's famous conjecture about the existence of a rearrangement of an orthonormal system that transforms the system into a convergence system. We have also shown that the fulfillment of the ""sigmatheta"" condition ensures that the rearranged Fourier series of a continuous periodic function is uniformly convergent; The result is related to Ulyanov's open problem. (v ) Our permutation lemma suggests an algorithm for finding the optimal permutation in the compact vector summation problem (this fact was first noted by Mackay). The algorithm reduces the permutations algorithm to a simpler sign algorithm; This idea is used in scheduling theory. Shortly:  In the fundamental problems: we proved a number of new maximal inequalities, which are used in the problems of the convergence of Fourier series (the long unsolved hypothesis of KolmogorovGarcia and Ulyanov's problem), as well as in the traditional problem of characterizing the set of sums of conditional series in a normed space.  In Applications: We improved the algorithm for finding the nearbest rearrangement in the compact vector summation problem. Our method, which is based on the transference inequality [S.Chobanyan and G.Giorgobiani, Lecture Notes in Math., 1391, 1989, 3346], has already found many applications in scheduling theory, machine learning, pattern recognition and discrepancy theory. The results are reflected in 7 scientific articles and 1 monograph published in international scientific refereed journals: 1. S. Chobanyan, G. Giorgobiani, V. Kvaratskhelia, S. Levental, V. Tarieladze. On rearrangement theorems in Banach spaces. Georgian Math. Journal 21(2), 2014, p. 157163. https://doi.org/10.1515/gmj20140016 2. Chobanyan S. Levental S., Salehi H. Maximum inequalities for rearrangements of summands and assignments of signs. Teor. Veroyatnost. i Primenen (English version  Th. Prob. Appl.), 59 (4), 2014, p. 800807. https://doi.org/10.1137/S0040585X97T987399 3. A. Figula, V. Kvaratskhelia. Some numerical characteristics of Sylvester and Hadamard matrices. Publ. Math. Debrecen, 86/12, 2015, p. 149168. 4. Elena MartinPeinador, Vaja Tarieladze. On dually cMackey spaces. Proceedings of A. Razmadze Mathe¬matical Institute, 168, 2015, p. 79–86. 5. N. Vakhania, V. Kvaratskhelia, V. Tarieladze. Some remarks on unconditional convergence of series in Banach spaces. Proceedings of A. Razmadze Mathematical Institute, 168, 2015, p. 149160. 6. G. Chelidze, G. Giorgobiani, V. Tarieladze. Sum Range of Quaternion Series. J. Math. Sci. v. 216, 4, 2016, p. 519521. 7. Elena MartinPeinador, Anatolij Plichko and Vaja Tarieladze. Compatible locally convex topologies on normed spaces: cardinality aspects . Bull. Aust. Math. Soc. (2017). doi:10.1017/S0004972717000090 8. G. Giorgobiani. Rearrangements of series. Journal of Mathematical Sciences, Vol. 239, No. 4, 2019, p. 438 543. DOI 10.1007/s10958019043159 One article was published later: [Chobanyan S., Levental S. Maximum Inequalities in Rearrangements of Orthogonal Series. Georgian Math. Journal, 2022, https://doi.org/10.1515/gmj20222181]. Also, 9 reports were made at international conferences.  
#609531  Knowledge Transfer Community to bridge the gap between research, innovation and business creation (NoGAP)  FP7. FP7INCO20139  European Union    20132016  D. Chiran  G. Giorgobiani (and others)  NoGAP bridges the gap between research and innovation and contributes to taking advantage of the innovation potential of SMEs based on a better cooperation with researchers, transferring and using knowledge resulting from research. The overall objective of the project is to reinforce cooperation with Eastern Partnership countries to develop a “Common Knowledge and Innovation Space” on societal challenge ”secure, clean and efficient energy”. The NoGAP consortium is composed of 13 organizations from 6 countries of which 3 are EU members (Germany, Romania, Slovakia) and 3 are members of the Eastern Partnership (Belarus, Ukraine, Georgia). In order to improve mobility between research, business and innovation, interrelated tandem relations between research organizations and innovation support services are established.  In order to stay competitive and to keep a leading role at world level, the European Union (EU) has developed a strong and coherent international science and technology policy. The integration of the neighbouring countries of the EU into the European Research Area and their possible association to Horizon 2020 is a prominent objective in this respect. NoGAP addresses this global vision by targeting selected Eastern Partnership Countries (EPC). The overall objective of the project is therefore to reinforce cooperation with Eastern Partnership countries to develop a “Common Knowledge and Innovation Space” on societal challenge ”secure, clean and efficient energy”. The NoGAP consortium is composed of 13 organizations from six countries of which three are EU members (Germany, Romania, Slovakia) and three are members of the Eastern Partnership (Belarus, Ukraine, Georgia). In order to improve mobility between research, business and innovation, interrelated tandem relations between research organizations and innovation support services are established. Within the NoGAP project we aspired to: • identify the main drivers and obstacles of closer links between academia and the market in the field of secure, clean and efficient energy in the Eastern Partnership Region, • develop a best practice methodology to enhance successful commercialization of research results and to improve the management of these results; • develop innovation support services to foster existing and establish new strategic partnerships; • assess the opportunities for the establishment of sustainable Technology Transfer Centres (TTC) in the participating partner countries on the basis of existing structures and good practice, • improve the competencies of researchers, entrepreneurs and multiplicators by organizing trainings; • develop a list of pilot activities to foster mutually beneficial publicprivatepartnerships between EU and Eastern Partnership countries in the energy sector; • create and organize twinning sessions between the regions, • promote networking between EU and Eastern Partnership countries. Indeed, NoGAP was an efficient instrument to bridge the gap between research and innovation. The current EU research and innovation programme Horizon 2020 in combination with the EPC sets the frame for closer collaboration with the neighbours in research and innovation, especially in the field of societal challenges. It opens the door for their stronger participation and will be the main instrument for implementing the EU international research and innovation cooperation actions. The focus in the countries covered by the EPC will be on fostering integration into the European Research Area, including through their possible association to Horizon 2020. For these countries, our project contributed significantly to developing a “Common Knowledge and Innovation Space”, including improving the research and innovation competences of the EPC. NoGAP developed between the consortium members and the other stakeholders, involved in or informed about the project, a systematic and much stronger interaction between research and innovation. This comprises better alignment with the international cooperation priorities of actors such as industry, universities and research organizations, but also with the priorities of the Joint Programming initiatives, European Platforms and European Innovation Partnerships. Project Context and Objectives: In order to stay competitive and to keep a leading role at world level, the European Union (EU) has developed a strong and coherent international science and technology policy. The integration of the neighbouring countries of the EU into the European Research Area and their possible association to Horizon 2020 is a prominent objective in this respect. NoGAP addresses this global vision by targeting selected Eastern Partnership Countries (EPC). It aims at creating a common “Knowledge Transfer Community to bridge the gap between research, innovation and business creation”. As such, the project strives to foster a systematic and strong interaction between research and innovation partners from EU and EPC. Its aim is to build sustainable innovation partnerships between actors from all steps of the value chain in both regions in order to develop effective instruments for successful technology transfer and innovation (I&TT). The NoGAP consortium is composed of 13 organizations from 6 countries of which 3 are EU members (Germany, Romania, Slovakia) and 3 are members of the Eastern Partnership (Belarus, Ukraine, Georgia). Within the NoGAP project we wanted to: • identify the main drivers and obstacles of closer links between academia and the market in the field of secure, clean and efficient energy in the Eastern Partnership Region; • develop a best practice methodology to enhance successful commercialization of research results and to improve the management of these results; • develop innovation support services to foster existing and establish new strategic partnerships; • assess the opportunities for the establishment of sustainable Technology Transfer Centres (TTC) in the participating partner countries on the basis of existing structures and good practice, • improve the competencies of researchers, entrepreneurs and multiplicators by organizing trainings; • develop a list of pilot activities to foster mutually beneficial publicprivatepartnerships between EU and Eastern Partnership countries in the energy sector; • create and organize twinning sessions between the regions, • promote networking between EU and Eastern Partnership countries.  The overall objective of the project is to reinforce cooperation with Eastern Partnership countries to develop a “Common knowledge and innovation space” on societal challenge ”secure, clean and efficient energy”. The main S&T results will help the European Commission and other stakeholders to boost innovation & technology transfer within the Eastern Partnership Countries (EPC) as well as between these countries and the EU. 1. Identification of BOTTLENECKS & OPPORTUNITIES as regards innovation & technology transfer in the EPC between public research and industry in the societal challenge of secure, clean and efficient energy. These insights could be gained through a sophisticated survey in the six target countries. a) Unexperienced, uninformed stakeholders: almost half of the stakeholders in the societal challenge of secure, clean and efficient energy in the six EPC are not experienced in technology transfer; b) Unfavorable environment (lack of legislation, strategy, funding, education): The current ineffectiveness of cooperation and technology transfer in EPC seems to be due to two crucial problems. The lack of financing for projects and lack of legislation for innovation & technology transfer are the most virulent barriers to technology transfer. c) Poor interaction at all levels (national, regional, international): cooperation between public and private is perceived as poor by a great majority of respondents; only few interviewees participate in dual education programs; despite the fact that almost all organisations have staff speaking foreign languages, only 40% act at an international level; direct interaction seems to be missing between research and business as Internet is the dominant source of information for people interested in technology transfer; this backlog becomes particularly visible when it comes to clusters in EPC. 2. BEST PRACTICES METHODOLOGY Based upon the insights above and existent general knowledge, our project developed a best practice handbook in this context. This document looks at the relevance gap in the management of the knowledge transfer (KT) to the market. Its focus is the nature of the knowledge created by research at the interface between business and academia in the context of major changes likely to affect the nature of demand for such knowledge. Our identified best practices:  Knowledge transfer as a strategic mission of Public Research Organisations (PRO): Ensure that all PROs define knowledge transfer as a strategic mission. Plus: Green and White Papers on KT and IP Management:  Policies for managing intellectual property  Supporting PROs’ intellectual property policy and procedure development: Encourage public research organisations to establish and publicise policies and procedures for the management of intellectual property.  Improving knowledge transfer capacities and skills: Support the development of knowledge transfer capacity and skills in public research organisations, as well as measures to raise the awareness and skills of students  in particular in the area of science and technology  regarding intellectual property, knowledge transfer and entrepreneurship.  Promoting broad dissemination of knowledge while protecting intellectual property: Promote the broad dissemination of knowledge created with public funds, by taking steps to encourage open access to research results, while enabling, where appropriate, the related intellectual property to be protected. Moreover, it is necessary to plan, implement intellectual property policy, licensing/startup policy, to consider the awareness of the importance of monitoring institutional performance and progress.  Facilitating transnational cooperation, research and KT: This also includes the globalisation of research collaboration and knowledge transfer. It is highly recommended to cooperate and take steps to improve the coherence of their respective ownership regimes as regards intellectual property rights in such a way as to facilitate crossborder collaborations and knowledge transfer in the field of research and development.  Introducing or adapting national KT guidelines and legislation: Use the principles outlined in this ecommendation as a basis for introducing or adapting national guidelines andlegislation. The harmonisation of intellectual property ownership may bring several benefits, such as easier collaborative research and the reduction of information and other transaction costs.  Improved monitoring of policy measures and KT performance: The list of best practices includes also that the necessary mechanisms should be put in place to monitor and review the progress made by national public research organisations in knowledge transfer activities, e.g. through annual reports of the individual public research organisations. 3. SERVICE DOCUMENTS Based upon the insights from the survey (see 1) and existent general knowledge, our project developed service documents, especially to respond to the backlog of uninformed and unskilled stakeholders. All service documents are available both in Russian and English language on the website of NoGAP. Besides, they were printed and distributed among stakeholders.  Handbook for services in IPR and Innovation Management: It emphasises the protection of inventions, trade marks and patents, the conflict between intellectual property rights and the principle of free movement and between the protection and enforcement of intellectual property and the need of technology transfer.  Handbook / Business Plan in Innovation Environment: It will help stakeholders to understand not only the importance of a properly prepared business plan but also the issues generated by mistakes and omissions, when starting a project, no matter of the nature (research or industry).  A brochure related to financing issues in Technology transfer and Innovation was prepared in order to help stakeholders to understand financing issues and to forecast the cash flow needed.  Another brochure related to technical assistance services related to market access (i.e. how to draw a technology offer and a technology request, an express of interest letter and a company profile) was prepared.  Handbook on on innovation and technology transfer adapted to the needs of three different target groups:  researchers: the material that you are reading is aimed at covering the most important topics of interest for researchers, in order to stimulate the approach of the academic and economic environments for producing concrete results pertaining to the generation and use of renewable energy. Without the intention of being exhaustive, the training material tries to pinpoint the main topics and discuss upon their contribution towards successful endeavors. As a consequence, subjects such as the use and exploitation of knowledge, legal framework for technology transfer or licensing are discussed in a regional context, which is relevant for the participants. 
543868TEMPUS120131DETEMPUSJPCR  Modernization of Mathematics and Statistics curricula for Engineering and Natural Sciences studies in Georgian and Armenian Universities by introducing modern educational technologies (MATHGEAR)  TEMPUS IV6  European Union    20132016  S. Sosnovsky  G. Giorgobiani (and others)  The goal of the project was to improve the quality and effectiveness of Math studies within STEM curricula by transferring best European practices in this field, first of all  modern educational technologies and technologyenabled pedagogical strategies in Georgian and Armenian higher education institutions. The Project Coordinator was the Saarland University (Germany). Participants of the project were higher education institutions from Georgia, Armenia, France, Germany and Finland. Georgian National Center for Educational Quality Enhancement, Georgian Technical University, association Grena, Batumi Shota Rustaveli State University, Akaki Tsereteli State University and the University of Georgia were the partners of the project from Georgian side.  There were 6 work packages in the project: WP1. Math Curricula Comparative Case Study. Activity 1.1 Case Studies methodology development The consortium members will select and appoint the experts who will participate in the case studies and work on the final book. Activity 1.2 Learning European experience The working group of expert formed within activity 1.1 extended with additional specialists will visit EU Universities Activity 1.3 National math curricula workshops The best practice workshops will be organized in Tbilisi and Yerevan. Each country case will be presented and discussed (partner Universities). Activity 1.4 Case Studies writing and evaluation Each case description shall be ca. 3040 pages with necessary graphics and other supporting materials. Overall editing and harmonization will be done by USAAR (English). In order to evaluate the Case Studies result (the Book) the special evaluation workshop will be organized in Tbilisi. The book “Europeanization of mathematical curricula for STEM studies in Caucasian countries” will be published. WP 2. Math Curricula Modernization Activity 2.1 Curricula modernization Each partner University will select one BSc and MSci curricula from the STEM spectrum. Activity 2.2 Math teaching materials selection Identification of the topics in the revised curricula most appropriate for blended learning. Activity 2.3 European expert evaluation 6 peerview visits to the partner Universities (TUT, USAAR and UCBL carry out 2 visits each). 2 workshops (in Lyon and Tampere) with presentations of the plans for modernization Activity 2.4 National expert evaluation 2 national workshops "Technology intensive European Mathematics and Statistics curricula for STEM studies". The materials will be published on the project website before the workshops. WP3. Tools and capacity building Activity 3.1 Technical capacity building Purchase and installation of the equipment necessary for local hosting of MathBridge system. Training of local technical personnel on installation and maintenance of MathBridge. Activity 3.2 Content localization Training of the partner university specialists involved in localization (both linguists and math teachers) on the technology of localization. Activity 3.3 MathBridge localization, installation and testing Symmetrical translation of the MathBridge interface (incl. help service) into Georgian and Armenian. Activity 3.4 University teachers training A series of training workshops for University teachers teaching math within STEM curricula on the use of TEL tools and methods. WP4. Math Curricula Pilot Implementation and Evaluation Activity 4.1 Evaluation methodology development and adoption The work will start from the methodology workshop to be held in Saarbruecken. The goals, development process, tangible results will be defined. Activity 4.2 Two semester pilot implementation The groups of BSc & MSci STEM students in all partner Universities (at least 20+5 students per location) will go through the modernized curricula and use the respective eLearning tools Activity 4.3 Processing and presentation of the evaluation results Statistical processing of the gathered data using evaluation instruments provided by the European partners will be done. The evaluation report will be composed. The Evaluation workshop will be organized in Tbilisi. Activity 4.4 Postevaluation update for the curricula and accreditation preparation The feedback gathered through the evaluation will be used to improve the resulting curricula. All partner Universities supervised by the national quality assurance bodies will prepare necessary documentation for the subsequent accreditation of the courses. WP5. Dissemination and Sustainability Activity 5.1 Regional website "Mathematics and Statistics for Technical and Engineering Education in South Caucasian countries" Activity 5.2 Annual project workshops GTU and SEUA will be responsible for organizing 2 annual workshops bringing together professional community from all South Caucasian countries. The project activities and results will be presented. Activity 5.3 Traditional dissemination All partners will be use all appropriate means to promote/present/publish the project. Set of promotional materials will be developed. Activity 5.4 Final Project Conference The final project results will be presented and disseminated at the Conference ""Mathematics and Statistics for Technical and Engineering Education in South Caucasian countries"" Activity 5.5 Revision of national standards and accreditation NCEQE and ANQA will use the project results in order to consider the revision of the national standards in the field. WP6. Quality control, Management and Coordination Activity 6.1 Quality Control Framework. At the very beginning the project will set up a Quality Control Work Force (QCWF) as a group of 3 experts (from USAAR, GTU and SEUA), which will be responsible for developing quality monitoring and reporting procedures:  development and control of the system of internal indicators of progress (included into annual reports)  quality control of the principle deliverables (peer review by Individual experts upfront their submission, peerview visits by appointed experts; internal reporting to the project decision making bodies).  design and implementation of interproject surveys (feedback questionnaires design, gathering, processing and analytical reports provision)  event organization quality control procedures (development of blueprint documents for all project events, control of performance) Activity 6.2 Project Management Framework. The coordinator appoints a manager who will be responsible for daily management:  collective decision making bodies establishment and operation (Project Coordination Board)  internal 3monthly reporting in accordance with the developed templates  centralized documentation maintenance (supported by the project website's restricted area)  financial monitoring and payments management  external reporting in accordance with the contract  interproject communication and coordination of tasks (the key for success)  coordination meetings organization and followup (4 meetings)  risk and crisis management  The working group of experts formed within activity 1.1 visited EU Universities; The best practice workshops were organized in Tbilisi and Yerevan; Each partner University selected one BSc and MSci curricula from the STEM spectrum, the topics were identification of in the revised curricula most appropriate for blended learning; 6 peerview visits to the partner Universities (TUT, USAAR and UCBL carry out 2 visits each) and 2 workshops (in Lyon and Tampere) with presentations of the plans for modernization were held; 2 national workshops "Technology intensive European Mathematics and Statistics curricula for STEM studies" were held; Training of local technical personnel on installation and maintenance of MathBridge was held in Saarbruecken; The groups of BSc & MSci STEM students in all partner Universities went through the modernized curricula and used the respective eLearning tools, the Evaluation workshop was organized in Tbilisi in order to present and compare the results obtained at different locations; Regional websites "Mathematics and Statistics for Technical and Engineering Education in South Caucasian countries" were designed (see e.g. http://math.grena.ge/eng/main/index/3); GTU and SEUA organized 2 annual workshops (after 1st and 2nd project years) bringing together professional community from all South Caucasian countries. The project activities and results were presented; The final project results were presented and disseminated at the Conference "Mathematics and Statistics for Technical and Engineering Education in South Caucasian countries"; NCEQE and ANQA used the project results in order to consider the revision of the national standards in the field; The book "Modern Mathematics Education for Engineering Curricula in Europe" was published (ISBN : 9783319714158). 
#266155  Recreation and building of capacities in Georgian ICT Research Institutes (GEORECAP)  FP7  INCO. 20106.1: Eastern Europe and South Caucasus  European Union  397719 EUR  20102012  G. Giorgobiani  G. Giorgobiani, V. Kvaratkhelia, Z. Sanikidze (and others)  The GEORECAP project is an EU FP7 funded INCO project which aims to recreate and build capacities in Georgian ICT Research Institutes. Modernization of research system is one of the most important issues in the country. Though the research institutes have its own profile and are highly specialized in particular research areas, diversification of the scientific fields imply the problems of management, difficulties to create synergies, etc. Mainstream of GEORECAP to improve the profile of ICT scientific institutes and to elaborate new funding models, in particular with regard to international programmes and projects is in line with the appropriate reforms in Georgia, where ICT is regarded as a national priority area. GEORECAP consortium includes two leading Georgian ICT research centers GTUMICM and GTUIC. They are supported by the European partners DFKI GmbH, GIRAF PM Services GmbH, GEIE ERCIM and the local partner ICARTI, which are distinguished and experienced organizations in their own fields of expertise. By bringing together research teams from ERA and Georgia, GEORECAP allows both sides to extend their knowledge and expertise in ICT.  Up to recently Georgia’s research system comprised 7 public universities and more than 60 research institutes. Research was carried out at the research institutes and at the universities. Universities additionally fulfilled the education and training function. The research institutes were established as legal entities of public law, each with its own profile and highly specialized in a particular research area. Diversification of the scientific fields at the institutes implied the problems of management. Moreover, it was very difficult for each individual institute to generate a significant impact on national, regional, or European and International level. Due to the conducted reforms, in the beginning of 2011 most of the research institutes joined the State Universities. According to this process MICM and IC merged to the Georgian Technical University (GTU), as independent structural units (ICT cluster). As part of the continuing reform of the research system, Georgia tries not only to improve the number and profile of its scientific research institutions but also to elaborate new funding models in particular with regard to International Programmes and projects. Hereby, the area of information and communication technologies (ICT) is considered the national priority area. Thus, supporting research in ICT will imply significant positive socioeconomic effects and positive impacts. Due to the trend of consolidation and reorganization and an important role of ICT, GEORECAP consortium includes two leading Georgian research centers in this area  GTUMICM and GTUIC. Georgian researchers are supported by the European partners DFKI GmbH, GIRAF PM Services GmbH, GEIE ERCIM and the local partner ICARTI, which are distinguished and experienced organizations in their own fields of expertise such as: innovative software technologies, Informatics and Mathematics, training and consultancy etc. According to its central goal to initiate and increase scientific cooperation between the ERA and both GTUMICM and GTUIC, GEORECAP will systematically support Georgian institutes in the venture to enhance the cooperation capacities. The impact of the project on the Georgian ICT research community will be maximized as both centers will act as trendsetters with regard to international research cooperation. The strategy developed within GEORECAP directly contributes to the ongoing and wanted general reorganization. Considering that ICT is of central importance both to ERA and Georgia and that through GTUMICM and GTUIC Georgia offers leading ICT centers, GEORECAP will not only be beneficial to Georgia, but also to ERA: By bringing together research teams from ERA and Georgia, GEORECAP allows both sides to extend their knowledge and expertise in ICT, to minimize costs (through cooperation and collaboration) and to maximize outputs of the research activities. Overall, GEORECAP will implement a set of focused activities – networking, training and coaching, strategy development – in order to build and maximize the cooperation capacities of the Georgian research centers. Exceptionally, GEORECAP will be designed as a practical “trainingonthe job” and supported “learningbydoing” action that will empower the Georgian research centers and allow them not only to learn but also put into practice everything that is relevant to sustaining and actively integrating themselves into the international research community. In all their tasks, they will be guided, advised and directly supported by the respective European expert(s) and the local player ICARTI. Project pursues the following specific objectives: Objective O. 1: To conduct two networking events – one in Georgia and one in ERA. The events enable exchange and partnering between Georgian researchers and their matching European counterparts. DFKI’s and ERCIM' contacts and dissemination channels ensure the recruitment of best and most relevant toplevel European experts. The main selection criteria are (i) research expertise that matches the research capacities and interests of GTUMICM and GTUIC, and (ii) capacity and experience to initiate a joint project. Networking events include: networking session, poster session, technical session. Objective O.2: To conduct two training workshops and to design the coaching scheme for Georgian researchers. The training workshops comprise sciencerelated training to raise awareness among the participating Georgian researchers on EU ICT priority research areas and cooperative opportunities in FP7. They also include actual EU project training. The coaching scheme will allow a number of Georgian researchers to meet and directly work with suitable European partners for actual practice and implementation of joint projects. Basis for the mobility of researchers is a mobility scheme. Each mobility grant is intended to contribute to travel and subsistence costs for one participant of the networking event or the coaching scheme. The scheme includes an online submission tool on the project website. Evaluation and allocation of grants is done by the project steering committee. Objective O.3: Development of a sustainable strategy to increase the research centre’s regional coverage and to improve its response to the existing socioeconomic conditions. Objective is based on the conduct of a SWOT analysis. The strategy will give clear recommendations as to how the institutes can change their structures with regard to links with other institutes/universities, innovation and links with industry. The required information will be obtained through interviews with GTUMICM and GTUIC directors, heads of departments and leaders of research groups. Based on the outcomes of this analysis, a corresponding strategy will be suggested to the Georgian research centers. The details of the SWOT analysis as well as the development strategy will be formally laid down in the Strategy papers.  WP1 – Networking: 1st Networking Event was held in Budapest, Hungary, in parallel with European Future Technologies Conference and Exhibition 2011 (fet11), 46 May. Publication of the 1st Networking Event: "Supporting Georgia in Enhancing the Cooperation Capacities of its ICT Research Centers” (ERCIM News, #84, January, 2011, pg. 9; http://ercimnews.ercim.eu/ images/stories/EN84/EN84web.pdf); 2nd Networking Event was held in Tbilisi, Georgia, at the Meeting Room of the Georgian Technical University (GTU), 27  28 June, 2012. Event was organized by GTUMICM, GTUIC and ICARTI. Publication of the networking event: the websites of GTU ttp://news.gtu.ge/index.php?newsid=1887, the websites of GTUMICM and GTUIC (http://www.compmath.ge, http://www.cybernet.ge/) and of GHN News Agency http://ghn.ge/news70064.html; G. Giorgobiani, G. Kochoradze. Cooperation with Georgia’s ICT Research Centers: The Second GEORECAP Networking Event. ERCIM News #91, October, 2012, pg. 4 http://ercimnews.ercim.eu/images/stories/EN91/EN91web.pdf WP2 – Training and Coaching: The 1st Training Event was held in Tbilisi, GTU, 1819 June, 2011. The preparations started right after the return of the Georgian team from Budapest. The responsible for this Event was GIRAF PM Services GmbH. 21 participants for the 1st Training Event were selected by GTUMICM and GTUIC based on (i) their research expertise and (ii) on the relevance of their research to the upcoming FP7 calls (in particular ICT). The training event included the following basic modules: 1. Theoretical training: How to identify funding opportunities, in particular in FP7; How to shape a project idea;How to write a proposal (in particular Part B); How to find partners; Overview of Evaluation and Submission Process; Basics of Financial Regime in FP7. 2. Practical Training: Presentation of individual research ideas/proposals by Georgian scientists; Identification of funding opportunities; Evaluation of proposal with recommendations for change and mediation of partners. Overall 21 Georgian researchers received theoretical training and relevant practical exercises based on real successful projects and the upcoming ICT and other calls. 2nd training event was conducted at GTU in Tbilisi (Georgia), on 1516 October, 2012. The objectives of the training were to raise awareness on EU ICT priority research areas and cooperative opportunities in FP7 among Georgian researchers. Overall 14 researchers participated in the Coaching Scheme, which allowed them to meet and directly work with suitable European partners for actual practice and implementation of jointprojects. Outcomes: 1. Submitted 1 TEMPUS proposal (TEMPUS IV  5th Call for Proposals). 2. Submitted 3 research FP7 proposal (FP7INCO20139) 3. Prepared 1 research FP7 proposal (FP7ICT201311; ICT2013.3.2 Photonics). 4. 2 proposals for the Alain Bensoussan Fellowship Programme (ERCIM). 5. Signature of collaboration agreements GTU – HTW, GTU UDS, TSUUDS. 6. Joint research project between DFKI’s IWI and GTU. 7. Grant # YS/38/6210/12 for Training at Zaragoza University, Spain: The Program of Governmental Scientific Grants for Young Scientists – Research Training in World Scientific Centers. 8. 5 Georgian researchers attended International Conference in Rome (ICECECE 2012). WP3 – Mobility Overall, during the two years of GEORECAP project implementation 14 Georgian scientists have been supported with 25 Mobility Grants. 
GNSF/ST08/3390  On Application of a Singular Integral Approximation and a Method of Fundamental Solutions to Approximate Solution of Certain Boundary Problems  Fundamental Research Grants  Georgian National Science Foundation (GNSF)  131200 GEL  01/03/200928/02/2011  J. Sanikidze  M. Zakradze (manager), M. Mirianashvili, G. Kutateladze, Z. Sanikidze, K. Kupatadze  The main goal of the implemented project was the construction of computational schemes for the numerical solution of a certain class of boundary value problems of the theory of complex variable functions. The main goal of the implemented project was the development of effectively realizable computational schemes, their foundation and use for the numerical solution of a certain class of boundary value problems of the theory of complex variable functions.  The main goal of the project was the development of effectively realizable computational schemes, their foundation and use for the numerical solution of a certain class of boundary value problems of the theory of complex variable functions. Research in this direction was mainly based on the use of approximation of singular integrals of the Cauchy type and methods of socalled fundamental solutions. In the process of research, a fairly wide class of different types of boundary value problems from the theory of analytic, harmonic and biharmonic functions was considered. In the project, significant attention was also paid to the study of current problems of mathematical physics, quantum mechanics, the theory of elasticity and their approximate solution.  The following results were obtained by the project participants:  Constructive schemes of approximate calculation of Cauchytype singular integrals with corresponding error estimation are built;  established criteria of resolution and convergence of obtained schemes;  obtained quadratic formulas of high accuracy for integrals of a special type;  for effectively solving problems in the theory of cracks, exact solutions in Cauchytype integrals are obtained;  solved some plane and threedimensional direct dynamic problems of the theory of elasticity;  an approximate process for the singular integral equation of LippmannSchwinger, known in quantum field theory, is built ;  with the help of conformal mapping, some boundary problems of the theory of harmonic functions, in the case of domains of different configurations, are solved;  the possibility of using the method of conformal mapping for the solution of some dynamic problems is explored;  for built approximation schemes, corresponding computational algorithms, with corresponding software, are compiled. The presented main results are realized in the form of 11 articles published in international refereed scientific journals and reports presented at 7 international scientific forums. 
GNSF/FT09_99_3104  Characterization Problems of Probability Distributions and their Applications  Fundamental Research Grants  Georgian National Science Foundation (GNSF)  149250 GEL  20092011  V. Tarieladze  V. Kvaratskhelia (manager), N. Vakhania, B. Mamporia, G. Chelidze  The characterization problems posed in the project have the fundamental role for the Probability theory and Mathematical statistics. These problems are mainly solved for real random variables. The technique of the proofs of these results uses the highly developed methods of the theory of analytic functions which are not typical for probabilisticstatistical problems. The approach developed in our project to these problems is interesting from the following two points of view: a) it can give the possibility to prove the known results using the traditional methods of Probability theory and Mathematical statistics and b) by the same methods the new results can be obtained for complex and quaternion random variables. As the recent investigations show (see References of the Project), the characterization problems have wide applications for the research in ICA (Independent Component Analysis) models. It is to be mentioned that this direction is actual not only for real but also for complex and quaternion scalars. After obtaining SkitovichDarmois’ and Polya’s theorems for quaternions it would be possible to consider quaternion linear models analogously to ICA complex models. Recent publications show that, subgaussian measures have a significant role not only in probabilisticstatistical problems but also, generally, in infinite dimensional linear analysis. The study of subgaussian measures in infinite dimensional linear spaces gives a new motivation for their applications in the theory of random processes and Functional analysis. The stochastic calculus created by K. Ito has a wide spectrum of applications. The infinite dimensional version of this calculus for general Banach spaces is not created yet. The approach to this problem proposed by the project can give a basis for the construction of unified theory of stochastic calculus in Banach spaces.  One of the goals of this project was the characterization of the Gaussian distribution, which has a long history. Purely probabilistic investigations in this direction were first conducted by wellknown mathematicians D. Poya, M. Kats and S. Bernshtein. The culmination was V. Skitovich's result. At about the same time, the same result was obtained by G. Darmois, and this theorem is now called the SkitovichDarmois theorem. The mentioned theorem was obtained using nonelementary and nonprobabilistic methods of the theory of analytic functions. The project planned to investigate the quaternion versions of the Poya and SkitovichDarmois theorems and obtain new results in this direction. The project also planned a study of the characterization problem for subGaussian distributions given in the Banach space. The concept of subGaussian distribution in infinitedimensional vector spaces allows for several nontrivial generalizations. In particular, the natural generalization of this concept to the general Banach space leads to definitions of weak subGaussianity, in Talagran's (Tsubgaussian) and Fukuda's opinion (Fsubgaussian) sense. One of the goals of the project was to study Tsubgaussian and Fsubgaussian random elements; description of the class of Banach spaces where the norm of a Fsubgaussian random element is exponentially integrable. In addition, one of the tasks of the project was to find the necessary and sufficient conditions for T gaussianity in the Hilbert space in terms of the Gaussian standard, and also to study the almost everywhere unconditional convergence of a series consisting of Tsubgauss or Fsubgauss random elements. Finally, the project envisaged the characterization and study of a special Gaussian process  a Wiener process with values in the Banach space on the basis of weak independence. The study of weak independence itself has practical value. It was planned to present Wiener processes in the weak sense as the sum of series consisting of weakly independent Gaussian random elements and to study the properties of its realization. In addition, one of the goals of the project was to study the existence of a stochastic integral when the integrand is a Gaussian or, more generally, a subGaussian process. Along with this, further research of the existence of solutions of stochastic differential equations, study of linear stochastic equations and filtering problems were planned. In connection with the abovementioned goals of the project, the following works were performed: 1. Taking into account theoretical and applied aspects, Poya's and SkitovichDarmois theorems occupy a noteworthy place among the characterizations of Gaussian distribution. These theorems and the tasks related to them do not lose their relevance even today and many papers are written both in the direction of simplifying their proofs and their possible generalizations. Our group has made a certain contribution to this work and published the results in the high impact scientific journals. In the quaternion variant of Poya's theorem, interesting principal difficulties appear, some of which we were able to overcome in terms of covariance operators. We characterized joint quaternion systems. We gave a negative answer to the following natural question  does the analogue of Poya's theorem hold for quaternion Gaussian random variables in the case when the coefficient stands on the left with some random variables, and with others  on the right. In addition, we collected materials related to the SkitovichDarmois theorem and quaternion random variables and began to study and analyze them. This provided us with significant help in proving an analogue of the SkitovichDarmois theorem for quaternion random variables, which we proved for complex and quaternion random variables. The method of proof is original and is based on the analog of Poya's theorem we proved earlier for complex and quaternion random variables. 2. Subgaussian random variables occupy an important place from both theoretical and applied points of view. Random variables of this type were introduced by the famous French mathematician J.P. Kakhane in 1960 and a lot of works are devoted to their study. We considered in detail and studied the properties of subGaussian random variables, analyzed the question of the existence of their moments and their alternative characterizations, obtained inequalities of the Khinchin type. We proved a number of results about the convergence of subGaussian sequences and series. We proved the exponential integrability of T  and Fsubgaussian random elements. 3. The conditions ensuring the representability of the generalized solution of the stochastic differential equation are obtained. Also, the solution of the linear stochastic differential equation using the corresponding Itotype stochastic integral is obtained and given explicitly. Continuity of trajectories of stochastic processes with subGaussian increments and existence of stochastic integral were proved. Representation similar to KarunenLoev was obtained for the weak Wiener process. For weakly independent random elements in the Hilbert space, we proved the strong law of large numbers, which generalizes one of the results obtained by the famous Russian mathematician Khinchin in the twenties of the last century.  • The quaternion variant of Poya's theorem is proved; • We characterized jointly quaternion systems; • We gave a negative answer to the following question  is the analogue of Poya's theorem true for the quaternion Gaussian random variables in the case when the coefficient is on the left with some random variables, and with others  on the right; • We studied the properties of subGaussian random variables, analyzed the existence of their moments and their alternative characterizations, obtained inequalities of the Khinchin type; • Some results on the convergence of subGaussian sequences and series is proved; • The exponential integrability of T and Fsubgaussian random elements is proved; • The conditions ensuring the representability of the generalized solution of the stochastic differential equation are obtained; • The solution of the linear stochastic differential equation using the corresponding Itotype stochastic integral is obtained and explicitly given; • Continuity of trajectories of stochastic processes with subGaussian increments and the existence of stochastic integral were proved; • Representation, similar to KarunenLoev was obtained for the weak Wiener process; • For weakly independent random elements in the Hilbert space, we proved the strong law of large numbers, which generalizes one of the results obtained by the famous Russian mathematician Khinchin. 
GNSF/ST08/3384  Maximum Inequalities for Rearrangements with Applications to Functional Analysis and Scheduling Theory  Fundamental Research Grants  Georgian National Science Foundation (GNSF)    20092011  S. Chobanyan  A. Shangua (manager), N. Vakhania, V. Kvaratskelia, G. Chelidze, V. Tarieladze, G. Giorgobiani  The project consists of five main problems: a) compact vector summation; b) Kolmogorov's hypothesis; c) description of the set of sums of a conditionally convergent series; d) problems about rearrangements and Cesaro summation; e) Application of the obtained results in scheduling theory. The basis of the project is the GNSF project GNSF/ST06/3009, which ended in 2008.  The first maximal inequalities for vector summands and related compact vector summations were obtained in the works of Levy, 1905 and Steinitz, 1913. The problem can be formulated as follows: for a finite set of vectors in the normed space, find such a rearrangement π that will give a minimum to the radius R_π of the sphere containing all partial sums, that is, estimate the minimum of R_π with respect to the rearrangements. A comprehensive bibliography on these and related issues can be found in the monograph of M.I. Kadets and V.M. Kadets, 1997, and I. Halperin and Ts. Ando's Bibliographical Booklet, 1989. The technique for finding the minimizer π or evaluating R_π for the minimizer π is sometimes called compact vector summation. Among the wellknown tasks of functional analysis, we can name at least two such, for the solution of which the method of compact vector summation is used. One of them is the description of the set of sums of conditionally convergent series in a topological vector space, which is the subject of Levi and Steinitz's pioneering works. The second problem, which is also closely related to compact vector summation, is the famous Kolmogorov conjecture (so far unsolved) about systems of convergence for the rearrangements of orthonormal systems. In the late 80s we found a new approach to the problem of compact vector summation, which allows us to obtain both old, wellknown and new maximal inequalities in a simple way. This method is effective for both finite and infinite dimensions. This method gave us the opportunity to find general conditions for the set of sums of conditionally convergent series in locally bounded and locally convex vector spaces to be linear and closed (Chobanyan and Giorgobiani, 1991, Chobanyan, 1994, Chasko and Chobanyan, 1997). These results combine all known results on the linearity of sum sets. Furthermore, the functional variant of our maximal inequality allowed us to prove in an elementary way two wellknown results of Garcia (Garcia, 1970) about the almost everywhere convergence of rearranged orthogonal series. In the grant project GNSF/ST06/3009 we have obtained a general functional maximum inequality that can lead to the validity of the Kolmogorov hypothesis under some restrictions on an orthonormal system or/and for a (quite rich) set of coefficients. The next interesting direction in the use of rearrangements of vector summands is the study of the of rearrangement BanachSachs property of the Banach space. This is dictated by such related ideas as of rearrangement Cesaro summability or, in the language of probability, of rearrangement law of large numbers (Shangua, Tarieladze, 2006). Our maximal inequality not only proves the existence of the desired π, but also gives us an algorithmic way to find it. An appropriate algorithm is important in problems of scheduling theory (Makai, 2004, Chobanian et al. 2006). Many other applied problems that reduce to the minimization of π are discussed in the seminal work of Sevastianov, 1994. In this project, we plan to conduct the following studies: in maximal inequalities: finding new forms of of rearrangement maximal inequalities in function spaces; For the Kolmogorov hypothesis: finding the maximal inequality corresponding to a finite orthonormal system and proving the Kolmogorov hypothesis for particular orthonormal systems and sets of coefficients; use our maximum inequality to implement the Garcia program, 1970: For orthogonal systems, transfer Zigmund’s sign maximal inequalities to rearrangements; to develop the idea of Cesaro rearrangement summability in Banach space and the BanachSachs property; to find new possibilities of using the rearrangement maximal inequality in scheduling theory; create software for the algorithm of Chobanyan, 2006; Volume calendar scheduling and Johnson problems of scheduling theory: reduce these problems to the minimization problem of π; Optimal and "Greedy" Algorithms for the permutations: how often is a "Greedy" algorithm close to optimal? Is a "greedy" algorithm always optimal in scheduling theory? If not, then build counterexamples. Use computer modeling if the theory turns out to be difficult.  18 papers were published: 1. A. Shangua , V. Tarieladze. Some variants of the BanachSaks property. Transactions of Sukhumi State University, v. 4, 2008, 100109. 2. S. Chobanyan, S. Levental, V. Mandrekar. Equivalence of convergence for almost all signs and almost all rearrangements of functional series. Bulletin of Georgian National Academy of Sci. 3, 2, 2009, 2329. 3. V. Tarieladze, R. Vidal. Convergence Properties of ϕSumming Operators. Bulletin of Georgian National Academy of Sci. 3, 3, 2009, 1624. 4. E. Corbacho, V. Tarieladze, R. Vidal. Observations about equicontinuity and related concepts. Topology and its Applications, 156, 18, 2009, 30623069. 5. G. Chelidze, S. Chobanyan, G. Giorgobiani, V. Kvaratskhelia. Greedy algorithm fails in Compact Vector Summation. Bulletin of Georgian National Academy of Sciences, v. 4, no. 2, 2010, 57. 6. V. Tarieladze. An operator version of Abel's continuity Theorem. Georgian Mathematical J. 17, 2010, n. 4, 787794. 7. L. Chobanyan, S. Chobanyan G. Giorgobiani. A maximum inequality for rearrangements of summands and its applications to orthogonal series and scheduling theory processes. Bulletin of Georgian National Academy of Sci., v.5. no.1, 2011. 1620. 8. S. Chobanyan, G. Giorgobiani, V. Kvaratskhelia, V. Tarieladze. A note on the rearrangement theorem in a Banach space. Informational and Communication Technologies – Theory and Practice: Proceedings of the International Scientific Conference ICTMC2010 Devoted to the 80th Anniversary of I.V. Prangishvili. Nova Science Publishers, 2011. 9. S. Chobanyan, S. Levental, V. Mandrekar. Towards Nikishin’s theorem on the almost sure convergence of rearrangements of functional series. Funct. Anal. Its Appl. 45, 33–45 (2011). 10. S. Chobanyan, S. Levental, V. Mandrekar. Almost surely convergent summands of a random sum. Statistics and Probability Letters 82(2012) 212216. 11. S. Chobanyan, S. Levental, H. Salehi. A distribution maximum inequality for rearrangements of summands. Bull. Georgian Nat. Acad. Sci. 5, no. 1, 2011, p. 2530. 12. V. Tarieladze. M. J. Chasco, X. Dominguez. Convergence of characters in Schwartz groups. Topology and its Applications, 158(2011), 484491. 13. V. Tarieladze. UMAP classes of groups. Journal of Mathematical Sciences, Vol. 197, No. 6, March, 2014 14. G. Giorgobiani, V. Tarieladze. Special universal series. Several Problems of Applied Mathematics and Mechanics. Nova Science Publishers; Mathematics Research Developments, 2012. 15. G. Giorgobiani, V. Tarieladze. On Complex Universal Series. Proceedings of A. Razmadze Mathematical Institute. V. 160 (2012), 5363. 16. S. Chobanyan, G. Giorgobiani, V. Tarieladze. Signs and Permutations: Two Problems of the Function Theory. Proceedings of A. Razmadze Mathematical Institute. V. 160 (2012), 2434. 17. Tarieladze V., Domiguez X. Metrizable TAP, HTAP and STAP groups. Topology and its Applications, 159(2012), 2338–2352. 18. Tarieladze V., Dikranjan D. and MartinPeinador E. Group valued null sequences and metrizable nonMackey groups. Forum Mathematicum, vol. 26, no. 3, 2014, pp. 723757. Also, 3 articles were prepared, 14 reports were made at international conferences. 
GNSF/ST06/3009  Fundamental Research Grants  Georgian National Science Foundation    20062008  N. Vakhania  A. Shangua, S.Chobanyan, V. Kvaratskelia, V. Tarieladze, G. Giorgobiani  The project deals with wellknown problems in mathematics and applications that are united around the idea of rearrangements. The problems posed are studied using the results obtained earlier and new methods.  The following tasks are considered in the project: 1. To find statements of the form of previously obtained rearrangement maximum inequalities in abstract normed spaces for function spaces. 2. Using the obtained results, we study Kolmogorov's still unsolved hypothesis and Garcia's hypothesis directly related to it, which refer to the rearrangement convergence system. We find a complete or a partial solution to these hypotheses. 3. In the case of nonlocally convex spaces, in particular for the space of measurable functions, we study the LevySteinitz problem for the set of sums of conditionally convergent series. 4. Specifying the inequality related to compact vector summation in terms of its application in scheduling theory. The problem of compact vector summation in a conic setting. 5. Searching for new problems where our methods of compact vector summation can be used. 6. Creation of algorithms, such that it will be possible to solve the tasks of the previous sections in polynomial time. 7. Comparison of efficient and optimal algorithms in the reroute sequence problrm. 8. Application of the idea of permutations in the law of large numbers of probability theory. Elaboration of appropriate foundations for the general formulation of the problem, namely the study of Banach spaces that have the rearrangement BanachSachs property.  An analogue of the fact of convergence of the series for all signs for the convergenc for only almost all signs is found. An application of the result to the Nikishin problem is considered [1] It is shown that a Gaussian measure in a given infinitedimensional Banach space always admits an essentially unique Gaussian disintegration with respect to a given continuous linear operator [2]. The permutational and the weak permutational BanachSaks properties are introduced and their relationship with the usual BanachSaks and Weak BanachSaks properties is clarified [3]. UMAPclass of groups is introduced. Some of its properties are proved [4]. 1. S. Chobanyan, S. Levental, V. Mandrekar. Equivalence of Convergence for Almost all Signs and Almost all Rearrangements of Functional Series. Bulletin of Georgian National Academy of Sci. 3, 2, 2009, 2329. 2. V. Tarieladze, N. Vakhania. Disintegration of Gaussian measures and averagecase optimal algorithms. Journal of Complexity V. 23, Issues 4–6, 2007, 851866. 3. A. Shangua. Two permutational versions of the BanachSaks property. Bull. Georgian Acad. Sci. Vol. 173 (2006), 29–32. 4. V. Tarieladze. UMAP classes of groups. Journal of Mathematical Sciences, Vol. 197, No. 6, March, 2014.  
GEM13328TB03  Fundamental Research  Georgian Research and Development Fund (GRDF)    20052006  N. Vakhania    The project deals with the permutations of vector summands and related maximal inequalities. Theoretic and as well applied problems are considered.  The following problems are studied: 1. Maximal inequalities in abstract normed spaces; 2. The study of the KolmogorovGarsia conjecture related with the convergent systems under rearrangements by use of obtained maximal inequalities; 3. Rearrangment SLLN for the random elemnts in a Banach space; 4. Unconditional convergence of random series; 5. Compact vector summation method for the reroute sequece problem in computer networks.  By use of the maximum inequality obtained by us:  A knew proof of the MeńshovRademacher inequality, which gives the better constant, is established [1];  We prove a strong law of large numbers (SLLN) for sequences of pth order random variables. The inequality enables to extend the general result to pth order random variables, as well as to the case of Banachspacevalued random variables [2].  For any sequence of random variables, maximal inequalities from which we can derive conditions for the a.s. convergence to zero of the normalized differences is obtained. In the special case of quasistationary sequences, we obtain a sufficient condition for the SLLN; this condition is an improvement on the wellknown Móricz conditions [3]; Characterizations of the class of Banach spaces isomorphing to the space $c_0$, as well as to the class of Banach spaces not containing $l_\infty^n$'s uniformly, are obtained in terms of almost surely unconditional convergence of weakly subGaussian random series. Under almost surely unconditional convergence of random series, convergence of all permutations on the same set of full probability is understood. The questions of almost surely unconditional and weak absolute convergence in the spaces isomorphing to $c_0$ are investigated as well. 1. S. Chobanyan, S. Levental, H. Salehi . On the constant in MeńshovRademacher inequality. J. Inequal. Appl., 2006, 68969 (2006). https://doi.org/10.1155/JIA/2006/68969. 2. S. Chobanyan, S. Levental, H. Salehi. Strong Law of Large Numbers Under a General Moment Condition. Electronic Communications in Probability, 10, p. 218222, 200. DOI: 10.1214/ECP.v101156. 3. S. Levental, H. Salehi, S. Chobanyan. General maximal inequalities related to the strong law of large numbers. Math. Notes 81, 85–96 (2007). https://doi.org/10.1134/S0001434607010087 4. N. N. Vakhania, V. V. Kvaratskhelia. Unconditional Convergence of Weakly SubGaussian Series in Banach Spaces. Theory of Probability & Its Applications, 51, 2, 2007. https://doi.org/10.1137/S0040585X97982311  
1.17.02  Functional Equations in Numerical Schemes of the Theory of Analytic Functions  Grant of The Georgian National Academy of Sciences  Georgian National Academy of Sciences    01. 2002 – 12. 2003  J. Sanikidze  G. Khatiashvili, G. Silagadze, G. Kutateladze, M. Zakradze, E. Abramidze, Z. Sanikidze, M. Mirianashvili, Z. Khukhunashvili, K. Ninidze  For some boundary value problems and some types of equations of the theory of functions, the issues of constructing numerical schemes are studied.  In the project's boundaries:  Numerical schemes of the type of discrete singularities have been developed and investigated for one class of singular integral equations, to which some important problems of the theory of analytic functions are reduced. The corresponding application with numerical realizatation is given;  Theoretical issues of using nonlinear equations (for ordinary and in partial derivatives) in field theory have been developed. In this direction, both these equations and their algebraically adjoint equations are described. The results obtained, in particular, give some theoretical characterization of the existence of a charge;  Numerical schemes for the Dirichlet problem based on the method of fundamental solutions are contructed and realized;  In the case of binary mixtures, expressions for the components of stresses and displacements are constructed in an explicit form;  For noncanonical domains of a certain class (with rectilinear and arcuate cuts), some boundary value problems of the theory of analytic functions are studied. Solutions are constructed explicitly in Cauchytype integrals;  The problems of deformation of elastic shells, for the case when the equilibrium equations are represented by parameters describing the deformed state of the shells, are considered and investigated;  The contact problem of the interaction of two rigid stamps (as well as a system of stamps) of an elastic halfplane are studied. The solution in cases of different stamp bases is given with the help of integrals of Cauchy type and singular integrals of the same type. Numerical results (software and computer realizatation) are obtained on the basis of the corresponding quadratic formulas.  The results are published in the following papers:  Sanikidze J.G., Mirianashvili M.G. Singular integral equations in numerical conformal mappings, 2003, KNU, Bull. Kharkov National University, No. 590, 1, pp. 213218 (Rus);  Sanikidze J.G., Mirianashvili M.G. Approximation schemes for singular integrals and their application to some boundary problems, 2004, De Gruyter Publishing; Computational Methods in Applied Mathematics, v.4, #1, 94104;  Sanikidze J.G., Mirianashvili M.G. On some schemes of discrete vortices type for numerical solution of one class of singular integral equations with closed contours, 2004, Pleiades Pub; Differential Equations, v. 40, No. 9, pp. 12801289 (Rus);  Koblishvili N., Tabagari Z., Zakradze M. On Reduction of the Dirichlet Generalized Boundary Value Problem to on Ordinary Problem for Harmonic Function. Proc. A. Razmadze Math. Inst., 132, 2003, 93106;  Abramidze E. The stress state of multilayer flexible cylindrical shells of variable stiffness, 2003, Springer Verlag, NY, International Applied Mechanics, Vol. 39, No. 2, p. 211216;  Mikadze O.I., Bulia B.P., Maisuradze N.I., Kvatadze Z.A., Sanikidze Z.J. Hightemperature oxidation kinetics of lowalloyed chromium. Georgian Engineering News, Tbilisi, 1, 2003, 145147 (Rus);  Iamanidze T., Losaberidze M., Ninidze K., Silagadze G. Schemes of numerical realization of the problems of the interaction between a system of two rigid stamps and elastic semiplane, Bull. Georgian Acad. Sci., 168, 1, 2003, 2427;  Ninidze K., Losaberidze M., Silagadze G. On the schemes for the numerical realization of some problems of pressure of a system of stamps on the elastic halfplane, 2003, KNU, Bull. Kharkov National University, No. 590, pp. 155160 (Rus). 
1.21.02  Generalized spline algorithms and their construction for illposed problems  Grant of The Georgian National Academy of Sciences  Georgian National Academy of Sciences    01. 2002 – 12. 2003  D. Zarnadze    Approximate solutions of problems such that approximate solutions are sought in the form of the best approximation element (Best approximate solution) in a finitedimensional subspace or in a finitecodimensional subspace by means of the best approximation element are studed. In classical cases, such are the methods of least squares and Ritz, where we look for approximate solutions in finitedimensional subspaces, and in the case of spline algorithms in a finitedimensional subspace by means of the best approximation element.  The concept of generalized spline algorithm and generalized central algorithm are introduced and such algorithms for various tasks are studied. Namely, in order to construct a generalized spline algorithm, it is necessary to prove the existence of a strong best approximation element in a finitedimensional subspace of the Frechet space, that is, strong proximity of such subspaces. Also, in order to build a generalized central algorithm, it is necessary that this subspace has an orthogonal complement. Such problems are studied in the space of differentiable locally square integrable functions. In particular, the existence of a generalized spline algorithm for various equations in the Frechet space is proved. For example, for the equation Au=f, where A: H > H is a selfadjoint and positive definite operator equation in the Hilbert space H with a pure point spectrum, it is shown that for reduction of this equation in the Frechet space D(A^{\infty}) the extended Ritz method and a generalized spline algorithm gives the same result, and such an algorithm is generalized central. A generalized central algorithm is also constructed for noncorrect selfadjoint and positive definite compact operator equations in Hilbert space H with a pure point spectrum and for such equations that admit singular decomposition.  Submitted for publication the article: D. Zarnadze. Selfadjoint operators in Frechet spaces and some problems. 
1.22.02  Matrix representation of Clifford algebras and its applications  Grant of The Georgian National Academy of Sciences  Georgian National Academy of Sciences    01. 2003 – 12. 2003  N. Kandelaki  D. Ugulava, T. Chantladze  As it is known, Clifford algebras represent a functor from the category of quadratic forms in the category of associative algebras, thus, they are successfully used in algebraic topology, in particular in the construction of the Kfunctor. In the 8090s of the last century, academician N. Under Vakhania's initiative and leadership, the foundation was laid for the direct and direct study and application of Clifford algebras in the classification and description of symmetric distributions in functional spaces, in particular in infinitedimensional Hilbert spaces, and important results were obtained in this direction  Gaussian distributions with respect to complex and quaternion symmetries were fully analyzed. The topic is dedicated to exploring these issues even more deeply.  The matrix representations of Clifford algebras in the algebras of Hilbert space operators turned out to be determinant. The study of Gaussian invariant measures of even higher order requires the justification that the operators with respect to which the mentioned distributions are invariant are the generators of Clifford algebras. This issue has been successfully resolved in the presented project. The advantage of the obtained matrix representations is that their structure and construction is clearly visible by means of orthogonal operators in the infinitedimensional Hilbert space. This was done in 2002. In particular, matrix representations of Clifford algebras corresponding to quadratic forms for finite and infinitedimensional modules, for those rings whose characteristic is not equal to 2, were given constructively. The use of constructed representations in relation to the study of symmetries was considered. Gaussian random vectors with values in the separable Hilbert space that satisfy various symmetry conditions are systematically studied. In the study of the problem, the matrix presentations obtained in the previous year played a very important role. For example, it turns out that complex Gaussian random vectors are naturally related to the wellknown Dirichlet Lseries. It is also very interesting to discuss highorder symmetries in this relation, which is an independent research object in itself.  1. D.Ugulava, N. Kandelaki, T. Chantladze. On some matrix Clifford algebras. https://doi.org/10.1515/GMJ.2005.15 2. Chatladze T., Kandelaki N., Ugulava D. Gaussian distribution and Dirichlet series. Proceedings of A.Razmadze Math. Institute, v.135, 2004, pp. 4956. http://rmi.tsu.ge/proceedings/volumes/pdf/v1354.pdf 
1.24.02  Descriptive theory of Denjoy's nfold integral  Grant of The Georgian National Academy of Sciences  Georgian National Academy of Sciences    01. 2002 – 12. 2003    It is well known that the Denjoy Khinchin theorem is of fundamental importance for Denjoy's simple area integral, which asserts that a measure function, which is a VBGfunction on some set, is approximatively differentiable almost everywhere on this set (see S. Saks, Theory of Integral. IIL, Moscow , 1949, p. 321). Then it is proved that any function which is an ACGfunction on a set E is also a VBGfunction on this set. Therefore, according to the DenjoyKhinchin theorem, any function that is an ACGfunction on a measurable set is approximatively differentiable almost everywhere on this set. This theorem is the key to the descriptive theory of Denjoy wide integral.  We have introduced the concept of VBGfunction for the function of nvariable and proved a complete analogue of the DanjoyKhinchin theorem for the function of nvariable. It is then proved that every nvariable ACGfunction on some set is also a VBGfunction on that set. Therefore, every measurable function of nvariable that is an ACGfunction on a set E is approximatively differentiable almost everywhere on this set. Then a theorem is proved, which gives us a criterion for the given function to belong to the ACG class. It follows from this theorem that a continuous function that is approximatively differentiable at all points of an ndimensional segment, except possibly for points located on a countable number of lines parallel to the coordinate axes, necessarily represents the Denjoy nfold indefinite integral of its approximatively derivative. A function f of an nvariable is called integrable on an ndimensional segment I in the broad sense of Denjoy, if there exists such an ACGfunction F that the approximative derivative of the function F is equal to the function F almost everywhere on the ndimensional segment I. Thus, Denjoy space integral completely solves the issue of restoring the primitive function according to its approximative derivative.  1. Goguadze D.F. About the notion of semiring of sets. Math. Notes, 74, 34, 2003, 346351.  
1.23.04  Constructive theory of Danjou's nfold wide integral  Grant of The Georgian National Academy of Sciences  Georgian National Academy of Sciences    2003 2004  N. Goguadze    It is proved that Denjoy's nfold wide integral is a transfinite extension of Lebesgue's nfold integral using transfinite numbers of the second class.  Some problems of construction of open and closed sets in Euclidean space are studied. In particular, new properties of closed sets are established. Namely, the concept of the smallest ndimensional segment containing a closed set is introduced. Based on the studied properties, the theory of the abstract integral of the given function on the ndimensional segment is built. In addition, the transfinite extension of the considered abstract integral using transfinite numbers of the second class is studied. It is proved that the nfold Lebesgue integral is a special case of the constructed abstract integral. Thus, Denjoy's nfold wide integral is a transfinite extension of Lebesgue's nfold integral using transfinite numbers of the second class.  1. D. Goguadze. Generalization and new definition of LebesgueStieltjes abstract integral. Proceeding of the Tbilisi State University, 354, 2005. 
1.16.02  Permutations of Random Vectors, Unconditional convergence of series and the Strong Law of Large Numbers  Grant of The Georgian National Academy of Sciences  Georgian National Academy of Sciences    01. 2003 – 12. 2003  N. Vakhania  V. Kvaratskhelia, V. Tarieladze, S. Chobanyan, A. Shangua, G. Giorgobiani, B. Mamphoria, G. Chelidze  The problems of unconditional convergence of series and the strong law of large numbers for permutations are studied.  The problems of unconditional convergence of series and the strong law of large numbers for permutations are studied. An infinitedimensional analogue of the ProkhorovBrunk theorem is obtained. An algorithm for constructing signinvariant pairs of vectors with norm one in any Banach space has been developed.  1. S. Chobanyan, V. Mandrekar. On Kolmogorov SLLN under rearrangements for “orthogonal” random variables in a Bspace. Journal of Theoretical Probability, Vol. 13, No. 1, 2000. 2. S. Chobanyana, H. Salehi. Exact maximal inequalities for exchangeable systems of random variables. Theory Probab. Appl., 45:3 (2001), 424–35. 3. Chelidze G., Kvaratskhelia V., Ninidze K. On one method of finding of sign invariant pair of elements in normed spaces. Bull. Georgian Acad. Sci., 168, 3, 2003. 
1.18.02  Grant of The Georgian National Academy of Sciences  Georgian National Academy of Sciences    01. 2003 – 12. 2003  M. Pkhovelishvili  E. Dekanosidze, N. Archvadze, M. Papiashvili, L. Tsuladze  The forms of teaching courses presentation on WEB pages are studied. Different teaching courses were created.  The form of presentation of educational courses on WEB pages was determined by three sections: the first section contains the theoretical part, the second section contains the exercise section, and the third section contains the C/C++ teaching course.  1. Archvadze N., Pkhovelishvili M. Universal programming issues. Science and Technology, 79, 2003, 4952 (in Georgian).  
1.19.02  Elaboration of computational methods of solution of deterministic and stochastic problems of operations research  Grant of The Georgian National Academy of Sciences  Georgian National Academy of Sciences    01. 2003 – 12. 2003  J. Giorgobiani, G. Kasradze, A. Toronjadze, Nachkebia M.  An approximate method for finding optimal strategies for some classes of infinite games is developed. Problems of finding an unconditional minimum are discussed.  Approximate method for finding optimal strategies for some classes of infinite games are developed. The dynamic programming method is used for the game of selecting several moments of time. To calculate the value of the profit function of this game, a recurrence relation is drawn. The general method for determining the optimal strategies is known, but the approximate method proposed by us for finding the value of the profit function is an effective tool for this type of tasks (we have in mind multistep games). The discussed tasks of finding the unconditional minimum belong to the socalled SC1 class, i.e. the class of convex problems for which the gradient of the objective function is semismooth but nondifferentiable. The setting of parameters for solving such problems with predetermined accuracy is based on two estimates: first, the method of complex gradients converges at the rate of geometric progression, with the common ratio q=(k1)/(k+1) , where k is the square root of the conditioning number. The second, convergence of the same method is an nstep square convergence, where n is the dimension of the problem, i.e. nstep of conjucate gradients is equal to 1step of Newton's method. The process is realized in the Matlab environment.  1. Toronjadze A., Kipshidze Z., Ananiashvili G., Zhuzhunashvili A. Application of Game theiry to a stock market model. Bull. Georgian Acad. Sci., 168, 1, 2003, 2830. 2. Demetrashvili N., Zhuzhunashvili A. On a twostage method for solving some nonlinear equations. Intelecti, 3, 17, 2003, 1518.  
1.15.04  Laws of large numbers and stochastic equations in infinitedimensional spaces  Grant of The Georgian National Academy of Sciences  Georgian National Academy of Sciences    01. 2004 – 12. 2004  N. Vakhania  V. Kvaratskhelia, v. Tarieladze, S. Chobanian, A. Shangua, G. Giorgobiani, G. Chelidze  Laws of large numbers and stochastic equations in infinitedimensional spaces are considered.  The characterization of spaces with finite cotype is obtained by use of almost everywhere unconditionally convergent series of weakly subGaussian random elements. In the reporting year, the properties of convergence in measure related to the rearrangements were studied. Several analogs of the BanachSachs property have been defined using rearrangements instead of subsequences. After clarifying the relationship between BanachSachs ""subsequences"" and "" rearrangements"" properties, it was shown that through one of the proposed properties it is possible to characterize reflexive Banach spaces. The strong law of large numbers was derived under sufficiently general conditions. The unconditional convergence of weakly subGaussian series in the Banach space was also studied.  1. Vakhania N., Kvaratskhelia V. Weakly SubGaussian Random Elements and Banach Spaces with Finite Cotype. Bull. Georgian Acad. Sci., 171 (2005),No.2, 221224 2. Vakhania N., Kvaratskhelia V., Tarieladze V. Weakly subgaussian random elements in Banach spaces. Ukr. Math. J. 57, 9, 1187 – 1208. 3. B. Mamporia, A. Shangua, V. Tarieladze. Permutations and convergence in probability. Bull. Georgian Acad. Sci., 172, 2, 2005, 2325. 4. S.A. Chobanyan. Strong law of large numbers under a general moment conditions. Elec¬tro¬nic communications in probability, 10, 2005, 218222, (coauthors: S. Levental, H. Sa¬le¬hi) 5. A. Shangua, V. Tarieladze. Two Permutational Versions of the BanachSaks Property. Bull. Georgian Acad. Sci., 2006, 229232. 
1.18.04  Development of Numerical Methods for Solving Problems of Certain Classes of Operations Research and Mathematical Economics  Grant of The Georgian National Academy of Sciences  Georgian National Academy of Sciences    2004  2005  J. Giorgobiani  Nachkebia M., Nikoleishvili M.  The modification of the mixed gradients method for the convex programming (nonquadratic) problem is discussed; A problem of optimal inventory control theory for multichannel random flows. For the records of plant species, the unified format of the status fields has been studied and established, the dendroflora species conservation base of Tbilisi streets and squares has been created and processed. The issue of solving two different classes of continuous games was studied. A general model of coordination of economic sectors of the states has been elaborated. A mathematical model has been created for the joint optimal functioning of energy systems of neighboring states.  A modification of the conjugated gradient method for the convex programming (nonquadratic) problem was considered. In the developed method, the initial step is not necessarily equal to the antigradient, and in the case of a variable conditioning operator, the process converges at the rate of geometric progression. A problem of optimal inventory control theory for multichannel random flows when demands are predictable is formulated. A practical analogy of the model is a regional energy system (it can be a single country or a union of neighboring countries). The model belongs to the task of mathematical programming. In comparison with the previously developed model, the issue of energy accumulation and the differentiation of base and peak energy are introduced here. The criterion of optimality is the maximum of the total profit. At the expense of a slight violation of the adequacy of the model to reality, we approach the problem of linear programming. A unified format of status fields has been studied and determined for the records of plant species in the database. The research was carried out on the basis of information reflecting the status and applied values of plant species in the characteristic information. The format includes information on the species' relictness, extinction, rarity, endodomy, medical use, ornamentality, human consumption, technicality, agricultural use and economics. The issue of solving two different classes of continuous (socalled duel type and generalized market) games was studied. The methods of their solution are elaborated, which are based on the principle of dynamic programming (in the first case) and the calculation of components of the Shapley vector (in the case of the second game). A general model of coordination of the economic sectors of the states has been elaborated, taking into account the prospects of development and expected economic conflicts. This last nuance leads to the inclusion of elements of game theory in the model. In cooperation with the Institute of Energy, a mathematical model was created for the joint optimal functioning of energy systems of neighboring states. This model is practically calculated for the energy system of Georgia. The dendroflora species conservation database of Tbilisi streets and squares was created and processed, which is a prerequisite for the creation of an extensive information system that reflects the variety of plants in Tbilisi streets and squares for the purpose of assessing and maintaining (conservation) their current condition. The employees of Muskhelishvili Institute of Computational Mathematics, Dekanosidze, D. Chaligava, E. Chakhunashvili, took part in the implementation of the mentioned work along with the main performers of the grant topic and the representatives of the Tbilisi Botanical Garden.  1. Giorgobiani D.A., Nachkebia M.D. Mathematical model for optimizing the longterm operation of a complex energy system. Proceedings of int. scientific conference "Problems of Management and Energy", Reports, 8, 2004, Tbilisi, 532535. 2. M. Nachkebia. Optimization of the underground metro route. Georgian Engineering News, 3, 2005, 137141. 3. D. Giorgobiani, M. Nachkebia, M. Gegechkori. Optimal management of water and fuel reserves in the energy system. Proceedings of the Institute of the control systems, 9, 2005, 132137. 4. M. Nikoleishvili. Determining and evaluating the characteristics of the level of specialization of agricultural enterprises. Economica, 34, 2005, Tbilisi, 114121. (in Georgian). 
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Manana MirianashviliDoctor of Science / Principal Researcher 

Duglas UgulavaDoctor of Science / Principal Researcher 

Zaza TabagariAcademic Doctor of Science / Researcher 

Marina MenteshashviliDoctor of Science / Principal Researcher 

Valeri BerikashviliAcademic Doctor of Science / Researcher 

mziana nachkebiaDoctor of Science / Senior Researcher 

Mikheil NikoleishviliDoctor of Science / Researcher 

Hamlet MeladzeDoctor of Science / Principal Researcher 

Kartlos KachiashviliDoctor of Science / Principal Researcher 

Paata TsereteliDoctor of Science / Principal Researcher 

Giorgi BaghaturiaDoctor of Science / Senior Researcher 

Vakhtang KvaratskheliaDoctor of Science / Principal Researcher 

George GiorgobianiAcademic Doctor of Science / Principal Researcher 

Merab PkhovelishviliAcademic Doctor of Science / Principal Researcher 

Mamuli ZakradzeAcademic Doctor of Science / Principal Researcher 

George ChelidzeAcademic Doctor of Science / Senior Researcher 

Joseb KachiashviliAcademic Doctor of Science / Researcher 

Zviad KalichavaAcademic Doctor of Science / Researcher 

Zaza SanikidzeAcademic Doctor of Science / Senior Researcher 

Nikoloz VakhaniaDoctor of Science / Principal Researcher 

Badri MamporiaAcademic Doctor of Science / Principal Researcher 

Jemal SanikidzeDoctor of Science / Principal Researcher 

Aleksandre ShanguaAcademic Doctor of Science / Senior Researcher 

Zaur KhukhunashviliAcademic Doctor of Science / Senior Researcher 

Dimitri KurdgelaidzeDoctor of Science / Principal Researcher 

Aleksandre ChaduneliDoctor of Science / Principal Researcher 

Gaioz KhatiashviliDoctor of Science / Principal Researcher 

Guram KutateladzeAcademic Doctor of Science / Senior Researcher 

Konstantine NinidzeAcademic Doctor of Science / Senior Researcher 

Kote KupatadzeAcademic Doctor of Science / Researcher 

Nodar KandelakiDoctor of Science / Principal Researcher 

Tamaz ChantladzeAcademic Doctor of Science / Senior Researcher 

Jimsher GiorgobianiDoctor of Science / Principal Researcher 

Givi SilagadzeDoctor of Science / Principal Researcher 

Zurab ErgemlidzeDoctor of Science / Principal Researcher 

Abram JuJunashviliAcademic Doctor of Science / Senior Researcher 

Amiran ToronjadzeDoctor of Science / Principal Researcher 

Alexander LashkhiDoctor of Science / Principal Researcher 

Edison AbramidzeAcademic Doctor of Science / Researcher 

Mikheil TutberidzeAcademic Doctor of Science / Researcher 

Dimitri GoguadzeDoctor of Science / Principal Researcher 

Zurab TsintsadzeDoctor of Science / Senior Researcher 

Gela BendelianiDoctor of Science / Principal Researcher 