Orion Crest, Series_1

Chapter 58: One wave subsided when another rises

          After breakfast in the cafeteria, Orion wandered slowly to the library. Finding a seat in the middle of nowhere, he sat down and prepared to start writing his thesis.

　　He had already chosen the topic long ago.

　　Maths was a research tool, and while it was useful to sharpen that tool, it would be a waste of his talent if he only stayed in the field of maths.

　　Having decided to move his feet into mathematical physics, Orion decided to choose 'Generalised Functional Analysis' as the direction of his thesis. Only this time, the target was no longer the 'Fourier Transform', but the unpredictable 'Hilbert Space'.

　　In quantum mechanics, there are infinitely many states, so the 'inner product space' has an infinite number of dimensions. Infinity involves the problem of convergence, when certain parameters are taken to infinity, in order not to let any physical state run out of space, so mathematically it is required that 'the limit of any sequence' is still in space, that is, it is required that 'complete metric space'.

　　The Hilbert space, on the other hand, satisfies this need of quantum mechanics.

　　A physical system can be represented by a 'complex Hilbert space', and the vectors in it are the wavefunctions that describe the possible states of the system.

　　Although the concepts related to Hilbert spaces have been introduced in general function analysis at the undergraduate level, they are at a somewhat elementary stage. At the forefront of mathematics, Hilbert space is one of the areas that can be discussed as a separate research direction, just like the Fourier transform.

　　For this graduation project, Orion would not follow the standards of an undergraduate student, but instead the criteria for a journal submission, just as a practice run!

　　Recalling those references that he had read some time ago, Orion put his hands on the keyboard and quickly typed out a line.

　　[Viscous Approximation Methods for Equilibrium Problems in Hilbert Spaces with Finite Non-Stretching Mappings]

　　This time, there is no need to redeem points for the paper, when he discussed the Hilbert space problem with Wynston earlier, Orion had just a little idea in his head, this is the right time to write it out.

　　Picking up a pen, Orion wrote on the draft paper.

　　[Let H be the complex Hilbert space endowed with inner products, and remember that L(H) is the entirety of the bounded linear operator and T ∈ L(H), then the numerical domain of the operator T is defined as the following set: W(T)={<Tx, x>|x∈H,||x||=1}......]

　　As the minutes passed, Orion just felt an unbelievable flow of ideas, and soon filled up the entire side of the draft, followed by a second sheet of draft paper.

　　It seemed that as he had previously guessed when he challenged the twin prime number conjecture, the benefits of the increased maths level didn't just include unlocking the access to the system's databases, but also an increase in his own ability to think in that area.

　　Brain development? Or something else?

　　In short, he could clearly feel the changes happening to him.

　　"Let's get the thesis done in the next few days, and then work on the system's tasks."

　　......

　　The paper on the proof of the twin prime conjecture finally arrived and published in the latest issue of the Annals of Mathematics.

　　After a few days of silence, he soon learnt the meaning of "The tree may wish to remain quiet, but the wind will not stop". The story was reported in the 'Youth Newspaper'.