The dynamical system theory started as an effort to describe and predict natural and social phenomena which evolve as time goes. The theory includes ergodic theory, topological dynamics, chaos and fractal theory as subfields. Moreover, it is related to various areas of mathematics, and has applications to natural sciences, engineering and social sciences. Recently it extends to data compression, storage and bioinformatics.
In Applied Mathematics Lab, mathematical modelling, analysis, and numerical solution of partial differential equations are studied. With the aid of computer programming and simulation, we attempt to explain and predict various phenomena of nature. Partial differential equations derived from engineering problems related to elasticity, fluid dynamics, and electro-magnetics, are typical subjects of research. Methods from harmonic analysis such as Fourier transform and wavelet transform are used as important tools. Solutions are approximated and simulated using finite element, boundary element, and spectral methods.
We deal with the uncertainty in natural or social phenomena. We find the information of probability distributions of the random quantity(random variables, vectors) we are interested in. We use this information to compute probability of important events related to the random quantity. Probability and Statistics are complementary each other.
The important information of a given distribution is included in several forms of deterministic quantities. For instance, the moments of a random variable contain most valuable probabilistic information; we obtain the mean and the variance from the first two moments. Given samplings, Statistics provide us with the methods of approximating these quantities and giving us the confidence on the size of samplings and the approximation. Based on the information of the distribution we get through Statistics, we use probability theory to approximate the probability of events of interest. Many probability theories of attacking complex and abstruse probabilistic event has been developed.
Algebra is one of the most fundamental fields in mathematics, whose objectives are the sets with binary operations. Algebraists consider the natural structures or the symmetries of algebraic objects, make descriptions of rules in nature using mathematical languages. The theories have wide applications in natural sciences, and cryptography as well as in engineering.
Our research interests in the `Algebra Lab’ include;