Articles of soft question

Grothendieck's “Relative” Point of View

I have often read that Grothendieck’s insight was to put emphasis on studying the morphisms between schemes as opposed to just the schemes by themselves. What do we gain from this point of view? Why is it important that we study S-schemes, change of base, fibered products, and the like? Are there any specific concrete […]

Uniform distribution on $\mathbb Z$ or $\mathbb R$

I was assisting once the course in Probability Theory where students learnt quite quickly that there are ways to assign the uniform distribution to any finite set – or even subsets of $\mathbb R$ of a finite measure. They also clearly understtod the proof that there are no ways to put a uniform distribution (as […]

Stochastic Processes Solution manuals.

Does anyone have a link or a pdf stash of solution manuals for stochastic processes ebooks? I am doing a self-study on this course and I can’t seem to find any solution manual online to cross-check my solutions with. Any author or volume or version is ok with me. Thanks.

Is classifying one dimensional generalized quasicrystals worthwhile strategy to approach RH?

Works done: After fruitlessly poring over books on zeta functions, it seems Freeman Dyson’s sotto voce nudge to classify generalized one-dimensional quasicrystals is a way to go. As he writes: Question: Will this be a worthwhile strategy to pursue where a big picture akin to Atiyah-Singer index theorem needs to be made for symmetry?

Books like Grundlagen der Analysis in French

I am looking for some recommendations for a mathematics (text)book written in French. I am hoping to learn to read and write mathematics in French since I expect to take some mathematics courses that will be taught in French next year. Basically, I would like a book whose linguistic and mathematical level approaches that of […]

Dilemma for Studying Probability Theory while Waiting to Learn Measure Theory

I’m taking stochastic probability class but I’m now only taking analysis (with Rudin’s PMA) class. The stochastic probability class doesn’t depend heavily on the theoretic structures: rather, the professor wants to give the intution and that’s fine with me because I’ve taken set theory class and basic probability class before, so I can understand almost […]

Are there relations between elements of $L^p$ spaces?

I have read about dual spaces and the relation $1/p+1/q=1$ as mentioned in the Wikipedia page. Are there any more theorems or relations that connect elements between the $L^p$ spaces for different $p$’s?

Difference between Euclidean Space and $\mathbb{R}^n$ other than origin?

Often it is said that Euclidean $n$-dimensional space over $\mathbb{R}$ is different from $\mathbb{R}^n$, because in Euclidean space, all points are equivalent, in $\mathbb{R}^n$, there is origin. When I just heard this first time, I thought that the point $0$ is different from others in $\mathbb{R}^n$ when one concerns algebra: it is identity element (of […]

Multiplication, What is It?

What is multiplication? Upon review logarithms, and square roots, I realized that I have no intuitive grasp of multiplication-well no more so than I have for addition. Is it simply another thing we need to memorize? I understand that things like $\sqrt2$ could just be memorize as the thing, that when applied to itself, gives […]

Textbook Recommendation: Topological Dynamics

I need to take credits satisfying a topology requirement, and can structure it myself. My field of study is dynamical systems, can someone recommend a textbook that handles differential equations/dynamical systems from a topological point of view? Or is there another recommended field of study that would fill this?