I felt like there is no actual difference between so-called applied mathematics and pure mathematics.
It seems that physicists and mathematicians are always joking around. But with all the jokes asides, there is no doubt that mathematics is widely applied to the area of physics, and these two subjects are connected in some sense. This fact becomes more realistic to me as I learn more in mathematics: In the analysis, we use Fourier analysis to solve the differential equations of the heat function; The area of quantum probability gives us the approach of some phenomenons in physics, like the two-slit experiment. Along with my study in ODE and SDE, I began to be fascinated by the world of physics. unfortunately, I haven’t learned physics for a long time, and mathematics physics may probably not be the future I would love to pursue.
“The difference between mathematicians and physicists is that after physicists prove a big result they think it is fantastic but after mathematicians prove a big result they think it is trivial.”
Lucien Szpiro (1941 – 18 April 2020 RIP)
Ordinary Differential Equations:
After a month, I was able to spend some time to read through sections 13-18 of the book Ordinary Differential Equations by Arnol’d. The goal is to learn the properties and computation of the Exponential of a linear operator, and how it applies to the Fundamental Theorem of the theory of linear equations with constant coefficients. Essentially, let be a linear operator, then the solution of differential equation for with initial condition is
Since the characteristic equation may have complex roots, it is important to study complex differential equations. The next topic is then to recall what is meant by the complexification of a real space and the realification of a complex space. I am sure that I have learned most of the concepts in Complex Analysis, but I may also learn some new things from this approach.
We recently are focusing on the theory of Ruler and Compass constructions, which illustrates how intriguing the connection between fields of mathematics can be. By proving that the polynomial is irreducible in , one can show that is not a constructible number from the triangle identity
It then implies that it is impossible to trisect an angle, in general, using only a ruler and compass. I also found a blog article by Professor Terence Tao very interesting, which gives a geometric proof of why angle trisection is impossible by ruler and compass.
By the way, I also wanna mention a game called Euclidea. It is a puzzle game that requires one to construct some shapes or graphs with limited steps using only a ruler and compass.
Stochastic Differential Equations:
Last week, we discuss how to solve linear Stochastic Differential Equations of the form
with the initial condition . Some concrete examples like Langevin’s equation and the Ornstein-Uhlenbeck process are presented in Evan’s book. I also learned how to solve SDE by changing variables, and how to prove the Existence and Uniqueness of solutions by a successive approximation with the help of Gronwall’s lemma and Borel-Cantelli’s lemma. This week’s discussion will recall the concept of stopping time in probability theory and consider stochastic integrals with stopping times.
Work hard to reward.