Articles of soft question

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?

Beginner material for mathematical logic

I am looking for study and beginner material to study mathematical logic. I understand that it is a very broad topic but I would like to know what the best path there is to learning mathematical logic. Where should one start? What are the best resources? If someone could paint a time line of events […]

Is the infinitesimal generator for Lie groups the same as the infinitesimal generator of a Markov semigroup?

Is the infinitesimal generator for Lie groups related to the infinitesimal generator of a Markov semigroup? Or are they totally different concepts? https://en.wikipedia.org/wiki/Lie_group#The_exponential_map https://en.wikipedia.org/wiki/Infinitesimal_generator_(stochastic_processes) Both seem to be related to the exponential map. The connection would also explain why so many people, when discussing infinitesimal generators of a Markov process, seem to have such a […]

Can I skip the first chapter in Rudin's Principles of Mathematical Analysis?

I am a statistician who wishes to learn real analysis in order to better understand the foundations of statistics. With that aim in mind I plan to go through Rudin’s classic on “Principles of Mathematical Analysis”. Given the above context can I skip chapter 1? It seems to me that the material in chapter 1 […]

Lie algebra-like structure corresponding to noncrystallographic root systems

In the classification of Coxeter groups, or equivalently root systems: $$A_n, B_n/C_n, D_n, E_6, E_7, E_8, F_4, G_2, H_2, H_3, H_4, I_2(p)$$ with $p \geq 7$, the last four fail to generate any simple finite dimensional Lie algebras over fields of characteristic zero because of the crystallographic restriction theorem. However, I know that there exist […]

Why are Lie Groups so “rigid”?

This is probably a naive question, but here goes. To motivate my question, I’ll consider a unit circle in $\mathbb C$ or $\mathbb R^2$. This is a compact Lie group equipped with the usual exponential map. However, any deformation no matter how smooth of the unit circle makes it lose the group closure property (say […]

Learning differential calculus through infinitesimals

In class, we’ve studied differential calculus and integral calculus through limits. We reconstructed the concepts from scratch beginning by the definition of limits, licit operations, derivatives and then integrals. But the teacher really did everything to avoid talking about infinitessimals. For instance when we talked about variable changes we had to swallow that for a […]