I just experienced the most intense week of this quarter, and now it is finally over.
When I researched the counterexamples for the measure theory, I accidentally found a book called Counterexamples in Analysis by Bernard R. Gelbaum and John M. H. Olmsted. It is now in my inventory and is very interesting and concrete to read and learn.
After the basics of Fourier series and expansion, we learned about the Sobolev space and the weak derivative. Also, the professor said that this topic would be discussed further in the next course of analysis. It annoys me sometimes because I always feel like that I can’t grasp the whole picture of these concepts. It seems that it has a lot of applications and can be elaborated to further extent, but now I only know how the weak derivative is defined and the Sobolev embedding from Sobolev spaces to space of continuous functions. It doesn’t feel good, but I don’t have extra time to get a source to learn. Anyway, life has to move on. The next chapter is about the bounded linear operators on a Hilbert space, and hope I can have a better understanding.
The terminology of chapter 8: bilinear form in the book Algebra by Michael Artin is the spectral theorem. The core is to diagonalize a linear operator or matrix and to achieve the goal of simple computations. It really sounds like a technique that can be used in machine learning, but it is just a quick thought and I am not sure about that. The next destination is the representation. This is an area full of unknowns for me, so I may need to spend more time on this in the future.
The course now moves on to the linear programming, which is what I am more familiar with. The definitions of halfspaces and polyhedrons were introduced, together with lemmas and theorems like Farkas’ lemma and Weyl-Minkowski’s representation theorem. Related text that I was reading is the Algebraic and Geometric Ideas in the theory of Discrete Optimization by professor De Loera himself and his collaborators Raymond Hemmecke and Matthias Köppe. Specifically, I was taught by professor Köppe before, and he is one of the knowledge, kind, and respected professors for me at UC Davis. Aside from class materials, my final project of this course has now begun, and some more progress has been made. My schedule will be more intense, and I hope that I can manage my time well.
The next Monday is the president day!! Praise the holiday to have some time to rest!!