1

I know... reading lot of proofs and comments about them and working hard by myself on proving theorems are probably the only good solutions. But in the same time, it is not a solution at all because If I'm stock or if I only manipulate formulas without understanding what I am doing, I can spend my whole life working and reading without being able to prove anything...

But fortunately, I'm not so dumb. I only have the fault of not being a professional mathematician. In my physical research activity as PhD student, I often start with an intuition based on vague conceptual view of physical process. Then I imagine a "solution" based on that intuition (in the physicist sense, it only means to propose something that increase our knowledge of the physical process at hand). I have to end up with an analytical expression from which one can extract physical insights.

Unfortunately, physicists do not know exact solutions for most of the problems (anyway... it would not be a problem anymore!). Thus, our strategy is to use approximations (sorry about that, I did not invented that trick). So the rules of the game is to make only approximation that have as less impact as possible.

In the meanwhile, we have a lot of flexibility, but our ability is constrained by the mathematical tools we have at hand. I am generally able to figure out intuitive solutions, then I try to derive them from first principle (+ 1 or 2 approximations). But I often need relations that I think are correct but that are not formally proven or that I cannot find existing proof.

For example, I would like to use this relation:

$$\bigg( \sum_{i=1}^N a_i x_i \bigg) \bigg( \sum_{j=1}^N a_j x_j \bigg) = \sum_{\{i_n,j_n\}\in S}^{N_S} \bigg( a_{i_n} x_{i_n} +a_{j_n} x_{j_n} \bigg)^2$$

where $S$ is the set of all the distinct doublets of $i_n\in (1,N)$ and $j_n\in (1,N)$.

I think it is true, because we can trivially construct.

$$ (1,2,3,...)\otimes(1,2,3,...) = (1,2)\otimes(1,2) + (1,3)\otimes(1,3)+ (2,3)\otimes(2,3) + ...$$ Where $(1,2)\otimes(1,2) = \big\{(1,1),(1,2),(2,1),(2,2)\big\}$.

It looks like it is basic tensor algebra...

So, my questions are:

  • I am totally wrong about this example?
  • Which proof technic could I use to prove or refute this relation?
  • In general, which are the most fundamental proof technics?
  • Which kind of proof is more suited to which kind of problem? (and as a corollary, Is there some type of theorems that cannot be solved by some type of proofs).
  • Which knowledge would you recommend my to learn to maximise my "proof skill" while minimizing the time I spend on learning math? (sorry, I do not have enough time this week to learn thousand of years of knowledge accumulation).
  • Which knowledge would you recommend my to learn to improve my ability to search (and find!) existing proofs?
jvtrudel
  • 131
  • One rule of thumb is to try simple examples. What happens in the case where $ N =1$? – littleO Apr 19 '15 at 16:39
  • Yeah, interesting comment. But your example is a bit too trivial... It would give $(a_1 x_1)^2$ which is not really instructive (in my opinion). I gave a simple example that is the tensor product of the tuples. This is an abstraction of my problem combined with substitution of variables and functions by numbers. It allows me to see that I am right and I could continue with that. But I want to improve my skills in math (especially for proof) and, up to now, I do not know how to achieve it. – jvtrudel Apr 19 '15 at 17:14
  • But I am worried that the tensor product idea is actually too fancy / complicated. When $N = 1$, isn't the right hand side equal to $(2 a_1 x_1)^2$? It's possible that I'm misunderstanding what you mean by $S$. When $N = 1$, I think $S$ has a single element, which is the ordered pair $(1,1)$. – littleO Apr 19 '15 at 19:27
  • Yeah, you are right... I am not really experienced with tensor algebra. I'm doing things sloppily here, because I do not know how to rigorously state my idea. If in the definition of $\otimes$ I just keep distinct element, it works. But it was just a way to think faster (which is probably not a good idea). In fact, I have $(x_1+x_2+x_3+...) (x_1+x_2+x_3+...)=(x_1+x_2)^2 +(x_1+x_3)^2+(x_2+x_3)^2+... $. $S$ is the set of all distinct doublets ${i,j}$ in $(x_i+x_j)^2$ ... how should I say that? – jvtrudel Apr 19 '15 at 20:26
  • Ok. I was correct. $S$ do not contain doublets of ${i,i}$. As you can see in the second equation of my question. So it seems to be true except for $N=1$ ! And maybe when $N =\infty$ – jvtrudel Apr 19 '15 at 20:32
  • So when $N = 3$, and $a_i = 1$ for all $i$, would your formula say that $(x_1 + x_2 + x_3)^2 = (x_1 + x_2)^2 + (x_1 + x_3)^2 + (x_2 + x_3)^2$? But that formula isn't true. – littleO Apr 20 '15 at 06:47
  • Ok, you are right. Your comments are really useful and I begin to understand the problem I have. I do not do pure mathematics: it is more something like mathematic modelling. I want to understand interferences phenomena. It arise from measurement on wave functions that I simplified as $\sum x_i$. But I did not mention that it is a complex function (and it appear that the algebraic structure is important for my model to work!) A mesure is an mapping of the wave function to a real number, so it has the mathematical form of $(\sum_i x_i^\dagger) (\sum_j x_j)$. $^\dagger$is the complex conjugate. – jvtrudel Apr 20 '15 at 08:46
  • So it solve the problem of counting the mixed term, because $x_i^\dagger x_j \ne x_j x_i^\dagger$. In fact, the interferences comes from there. So, these are the most important terms for the physical analysis of my problem. I just reproduce what I learned up to now: it is common to see physicists dropping terms of a formula that are not of interest for them to focus on others. But I agree with you that It do not make sense from the mathematical point of view! So I still do not have the right counting of pure ($|x_i|^2$) terms in the right hand side... I work on that and I come back. – jvtrudel Apr 20 '15 at 09:10
  • Does this answer your question? Basic book about mathematical proofs – user1110341 Oct 07 '23 at 09:17

1 Answers1

1

One book that people often recommend (though I haven't read it myself) is How to Prove It by Velleman.

littleO
  • 51,938