Summary of the Project Project Title: Using algebraic combinatorics in evaluating discrete probabilistic measures Code number: Coordinator: Prof Dr Nguyen Nhuy (Chairman of the project), Dr Vu Hong Thanh, Dr Le Xuan Son, Master Nguyen Ngoc Quynh and Master Nguyen Duc Can Implementing Institutions: The College of Education, Vietnam National University of Hanoi Cooperating Institutions: Vinh University, Hanoi Institute of Ethnic Medicine Duration of performance: From April 2008 to July 2009 Objectives: Exploring local Fractal dimensions using Elementary Mathematics Main contains: - We present the basic definition of Golden ratio and its relationship to Fibonacci sequence of integers as well as the applications of Golden ratio in Art, Nature and Mathematics, particularly in defining the range of local projections in Fractal Geometry - The range of local projection of Fractal measure in combination with Problem (0, 1, 4) has explicitly been evaluated - The correctness of Shmerkin's hypothesis on the minimum value of the range of local projection of probabilistic measure of product of self similar measures on regular Cantor sets has successfully been proved - The results attained are up-to-date and provide a valuable scientific contribution to the problem of defining the range of local projection of Fractal measures in Fractal Geometry - In our project, this problem is solved only by using Elementary Math, though it has long been studied by advanced Mathematicians all over the World Results: - Under the financial support funded by the grant of the project, two papers have been finished and published on internationally known scientific journals - The scientific activities of the project provide a significant contribution to the results obtained in the dissertations of a PhD student graduating in 11/2008 and a Master student graduating in 12/2008 - Once more, the outcome of the project states that many open problems of advanced and modern Mathematics can be solved just by Elementary Math