these are notes from introductory survey lectures given at the institute for studies in theoretical physics and mathematics (ipm), teheran, in 2008 and 2010. we present the definition and the fundamental properties of fomin-zelevinsky’s cluster algebras. then, we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generators of ...

These are notes from introductory survey lectures given at the Institute for Studies in Theoretical Physics and Mathematics (IPM), Teheran, in 2008 and 2010. We present the definition and the fundamental properties of Fomin-Zelevinsky’s cluster algebras. Then, we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generator...

In this paper, an interesting and easy method to solve the multi-dimensional  knapsack problem is presented. Although it belongs to the combinatorial optimization, but the proposed method belongs to the decision making field in mathematics. In order to, initially efficiency values for every item is calculated then items are ranked by using Multiple Criteria Decision Making (MCDA).  Finally, ite...

A classical system of mathematics is homogenous without contradictions. But it is a little ambiguous for modern mathematics, for instance, the Smarandache geometry. Let F be a family of things such as those of particles or organizations. Then, how to hold its global behaviors or true face? Generally, F is not a mathematical system in usual unless a set, i.e., a system with contradictions. There...

Consider the experiment of tossing a 6-sided die. Suppose the outcome is 2. Does the event that the outcome is even happen? The answer is yes. Now assume that the toss produces an outcome 4, does the event that the outcome is even happen? The answer is yes. Actually observe that when the outcome is from the set {2, 4, 6}, the event happens. Thus an event in a probabilistic experiment is actuall...

The aim is a description of discrete mathematics used in a project 1 devoted to the implementation of a software package for the simulation of combinatorial chemistry. A. Hage-dorn contributes to this \crossing the interface" by making helpful remarks on applications of quantum mechanical resonances, which play an important role in chemical reactions. The report of the National Academy is very ...