Results

**1 - 2**of**2**### An Application of Reciprocally induced co-evolution: A computational metaphor in Mathematics

"... Natural phenomenon of coevolution is the reciprocally induced evolutionary change between two or more species or population. Though this biological occurrence is a natural fact, there are only few attempts to use this as a simile in computation. This chapter is an attempt to introduce reciprocally i ..."

Abstract
- Add to MetaCart

Natural phenomenon of coevolution is the reciprocally induced evolutionary change between two or more species or population. Though this biological occurrence is a natural fact, there are only few attempts to use this as a simile in computation. This chapter is an attempt to introduce reciprocally induced coevolution as a mechanism to counter problems faced by a typical genetic algorithm applied as an optimization technique. The domain selected for testing the efficacy of the procedure is the process of finding numerical solutions of Diophantine equations. Diophantine equations are polynomial equations in Mathematics where only integer solutions are sought. Such equations and its solutions are significant in three aspects-(i) historically they are important as Hilbert’s tenth problem with a background of more than twenty six centuries;(ii) there are many modern application areas of Diophantine equations like public key cryptography and data dependency in super computers (iii) it has been proved that there does not exist any general method to find solutions of such equations. The proposed procedure has been tested with Diophantine equations with varied powers and varied number of variables.

### A Connectionist Network Approach to Find Numerical Solutions of Diophantine Equations

"... Abstract. The paper introduces a connectionist network approach to find numerical solutions of Diophantine equations as an attempt to address the famous Hilbert‟s tenth problem. The proposed methodology uses a three layer feed forward neural network with back propagation as sequential learning proce ..."

Abstract
- Add to MetaCart

Abstract. The paper introduces a connectionist network approach to find numerical solutions of Diophantine equations as an attempt to address the famous Hilbert‟s tenth problem. The proposed methodology uses a three layer feed forward neural network with back propagation as sequential learning procedure to find numerical solutions of a class of Diophantine equations. It uses a dynamically constructed network architecture where number of nodes in the input layer is chosen based on the number of variables in the equation. The powers of the given Diophantine equation are taken as input to the input layer. The training of the network starts with initial random integral weights. The weights are updated based on the back propagation of the error values at the output layer. The optimization of weights is augmented by adding a momentum factor into the network. The optimized weights of the connection between the input layer and the hidden layer are taken as numerical solution of the given Diophantine equation. The procedure is validated using different Diophantine Equations of different number of variables and different powers.