Educational websites of Taketechease.com are focused on providing readers with a clear and a relatively simple explanation to some mathematical, numerical and physical problems. Do not be mislead. This task is not easy at all. To find a professional answer to difficult and complex questions that is as clear as it is only possible becomes a real adventure. The educational part is divided into two main branches: the advanced and the intermediate. Within the advanced courses one can find topics related to mathematical and theoretical physics while the intermediate courses are aimed at developing programming skills with help of basic math and at preparing strong analytic background of elementary mathematics. At the beginning one can make use of tutorial to integration techniques, finding derivatives or limits of sequences.
At first have a look at the following topics:
The Finite Element Method (FEM)
You can learn how to deal with differential problems with help of the finite element method (fem). It is the most effective tool to solve numerically differential equations being either boundary – value or initial – value problems. One and two dimensional problems are discussed. You can enjoy the Adobe Flash Player presentation explaining one – dimensional case and/or read articles about: two Solver 2D and three Solver 3D dimensional differential equations.
Genetic algorithms are one of the methods for search and optimization. The algorithms are based on the main evolutionary rule: gradual change that takes place over many generations, during which organisms slowly change some of their physical features. Genetic algorithms are a method of search in a space of solutions to find the 'best' fitted artificial organism. The method of genetic search makes use of random exchange of information. Then a 'better fitted' organisms are more likely to survive and reproduce. In that way, genetic algorithms resemble evolutionary mechanisms that govern life of biological creatures. This method enables searches in many spaces of coding sequences that is why it can be used as an alternative way of solving mathematical or economical problems.
The genetic-based Machine Learning is one of methods of adaptation. This method is based on analysis of environmental conditions applied to an artificial organism. This organism can be also influenced by evolutionary rules described in the previous paragraph. Generally, the genetic-based machine learning method is based on a set of rules called 'classifiers' and on genetic algorithms. Each rule consists of a pair of condition and corresponding organism answer when this condition is satisfied. The learning system teaches simple rules that enables an interaction between the organism and its environment. 'Classifiers' define an organism response to stimuli from surroundings. Any piece of information coming from environment is analyzed and answered by finding rules corresponding to obtained signals. On that basis organism generates an answer to the outer impact. The feedback given by the organism is verified by the environment. After that, the organism can be granted or punished. Since organism is granted then the rules involved in generating the 'good' answer have stronger position than those that has been punished. A small random perturbations are added to this set of rules by applying genetic algorithms. It gives an evolution of the set of rules towards optimization of organism behavior.
The following topics are considered:
Stochastic processes are dynamical processes of random sources. Its evolution in time is not deterministic. Equations describing evolution of stochastic systems have explicitly written random term that presence is the source of unpredictable behavior of such systems. Time-dependent random force is given by a sequence of random values taken from an assumed probability distribution in subsequent moments in time. When a dynamical system is perturbed by such a random term then one can generate an ensemble of system evolutions in time (called phase trajectories) starting from the same initial value. Each phase trajectory has different shape. The whole ensemble of phase trajectories gives a stochastic process when the number of generated trajectories tends to infinity. Stochastic processes are used each time when a system is perturbed in a very complicated way and it cannot be described by deterministic equations from classical physics or such classical approach rises enormously difficulty of finding the solution of this problem.
Chaotic Systems - Nonlinear Dynamics
Chaotic systems are dynamical systems that fulfill deterministic equations (i. e. given by classical equations), however, they act quite similar to stochastic systems (they are randomly perturbed - see above). The reason for this is that chaotic systems are very sensitive to initial conditions. This behavior appears only for nonlinear dynamics. If a system is sensitive to initial conditions it means that even very small inaccuracy in defining an initial condition leads to a large discrepancy in a final result and in consequence it means lack of prediction. It is very symptomatic of random events, however, the source of this unpredictable behavior in both stochastic and chaotic cases is quite different.