On extension of solutions of a simultaneous system of iterative functional equations

Some sufficient conditions which allow to extend every local solution of a simultaneous system of equations in a single variable of the form $$\varphi(x) = h (x, \varphi[f_1(x)],\ldots,\varphi[f_m(x)]),$$ $$\varphi(x) = H (x, \varphi[F_1(x)],\ldots,\varphi[F_m(x)]),$$ to a global one are presented. Extensions of solutions of functional equations, both in single and in several variables, play important role (cf. for instance [M. Kuczma, Functional equations in a single variable, Monografie Mat. 46, Polish Scientific Publishers, Warsaw, 1968, M. Kuczma, B. Choczewski, R. Ger, Iterative functional equations, Encyclopedia of Mathematics and Its Applications v. 32, Cambridge, 1990, J. Matkowski, Iteration groups, commuting functions and simultaneous systems of linear functional equations, Opuscula Math. 28 (2008) 4, 531-541])

Student Group Dynamic Model Based on Understanding in Mathematics Subjects

This study discusses the interaction of students with a mathematical modeling point of view. This interaction involves students who understand and do not understand mathematics subject matter. The interaction process between groups is modeled in a two-dimensional system of differential equations. Variable A is the percentage of students who understand the material, and variable B is the percentage of students who do not understand the material. The dynamic analysis results obtained by one trivial equilibrium point and three non-trivial equilibrium points exist with several conditions. Based on the stability analysis of the non-trivial equilibrium point, it is found that the conditions without students do not understand mathematics subject matter. This condition is the goal of this study, which involves interaction between students; it can increase the learning process's success

Poincaré and complex function theory

Poincaré is still well known for the mathematical work that first made his name : his discovery in 1880--1881 of automorphic functions. New documents and insights were added in [Gray and Walter 1999], which can also be consulted for references to the well-known history of his work in this area. He is also remembered for one further theorem that grew out of that early work : the uniformisation theorem, which he sketched a proof of in 1883 and then proved rigorously in 1907, as did Koebe independently. The rest of his numerous contributions to complex function theory are more scattered and do not seem to have been the focus of much attention. In this paper I survey what he did and argue that they tell an eloquent story not only about the state of the subject in the years around 1900 but about Poincaré's place in the mathematical community of his day. To understand either of these it is necessary to give a quick summary of the prior development of complex function theory, which was growing rapidly into a central topic in all mathematics, and that will occupy the first half of this paper. The second half will consider Poincaré’s contributions. We will see that although he was actively involved in many aspects of the subject, his influence is scarcely to be noticed in the many books that were published, and I will investigate why that was and what it may tell us about relationship between research and teaching in the years around 1900

Generating and Solving Symbolic Parity Games

We present a new tool for verification of modal mu-calculus formulae for process specifications, based on symbolic parity games. It enhances an existing method, that first encodes the problem to a Parameterised Boolean Equation System (PBES) and then instantiates the PBES to a parity game. We improved the translation from specification to PBES to preserve the structure of the specification in the PBES, we extended LTSmin to instantiate PBESs to symbolic parity games, and implemented the recursive parity game solving algorithm by Zielonka for symbolic parity games. We use Multi-valued Decision Diagrams (MDDs) to represent sets and relations, thus enabling the tools to deal with very large systems. The transition relation is partitioned based on the structure of the specification, which allows for efficient manipulation of the MDDs. We performed two case studies on modular specifications, that demonstrate that the new method has better time and memory performance than existing PBES based tools and can be faster (but slightly less memory efficient) than the symbolic model checker NuSMV.Comment: In Proceedings GRAPHITE 2014, arXiv:1407.767