What is machine-assisted formalization of proofs good for? And when to do it?
I have been watching Terry Tao's lecture on machine-assisted proofs https://www.youtube.com/watch?v=AayZuuDDKP0&t=1460s.However in terms of the formalization of proofs via systems like Lean or Coq...
View ArticleWhen I finish an introductory abstract algebra book, where will I be? [closed]
In a few months, I think I'll be done with self-studying Introduction to Abstract Algebra by Nicholson. This book covers things like basic group theory, polynomials, factorization in integral domains,...
View ArticleReference request: Fractals from a very applied point of view [closed]
I would like a reference for a book about fractals from a very applied point of view.I have looked at other posts on SE, and have looked and spent time on Falconer's and Edgar's books.However these are...
View ArticleOptimization approaches to solving PDEs
In modern numerical methods, a PDE is often recast into the form of a variational problem, which is sometimes equivalent to a minimization problem.However in my courses on numerical analysis (say,...
View ArticleIs there a treatment/development of the Stokes' Theorem using differential...
I'm working through an analysis text independently to prepare for grad school, and the author has discussed the limitations of both the Riemann and Lebesgue integrals and only hinted at the power of...
View ArticleEquipping a set with an abstract notion of computability
I'm curious what the right abstraction is for equipping an arbitrary set with "something kind of like computability".Topologies don't seem to fit because the complement of an open set is not open in...
View ArticleMatrix presentation of cubic form and quartic form?
When I was studying my engineering degree, I learn from Linear Algebra about the quadratic form of ${x^T}Ax$. Therefore, I am just having simple academic curiosity regarding the presentation of order 3...
View ArticleComplexity of symbolic computation of matrix inverse?
I am an engineer who is working with some linear equation problems.In my application, I found out that in having the symbolic form of such matrix inverse actually speed things up (for example ${A^{ -...
View ArticleForm of ideal generated by subset of noncommutative nonunital ring
Given a noncommutative nonunital ring $R$ and a subset $S\subseteq R$, I know that the left and right ideals generated by $S$ in $R$ have the forms$$RS:=\{\sum^n_{i=1}...
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Chain Ladder Model: Unbiasedness of estimators for the variance parameters
I don't understand the equality of the middle term:from page 40 of "Wüthrich, M. V., & Merz, M. (2008). Stochastic claims reserving methods in insurance. John Wiley & Sons."I see the first...
View ArticleWhat other tricks and techniques can I use in integration?
So far, I know and can use a reasonable number of 'tricks' or techniques when I solve integrals. Below are the tricks/techniques that I know for indefinite and definite integrals separately.Indefinite...
View ArticleWhat exactly is the orbit-stabilizer theorem?
Obviously, being a professional group theorist, I know what the orbit-stabilizer theorem is. Or at least I thought I did.I thought that the orbit-stabilizer theorem was that if $G$ is a finite group...
View ArticleAre there any intuitive reasons for Goldbach conjecture to be true?
One thing puzzled me is that, despite its simple form, I have not seen any intuitive reasons for Goldbach conjecture to be true.Typical heuristic reason is based on probability arguments. Such...
View ArticleWhat is the "goal" of derived functors?
I've been learning about derived functors recently, and I had conceptualized them as fulfilling the following goal:Suppose that we had a left-exact functor $F:\mathcal{A} \to \mathcal{B}$ between two...
View ArticleStability of vector bundles as GIT quotient
I believe that the stability of vector bundles or coherent sheaves (defined as the inequality of 'slopes' of its subsheaves) comes naturally from GIT. However, in any literature I can find, it is...
View ArticleIs there a group theoretic proof that $(\mathbf Z/(p))^\times$ is cyclic?
Theorem: The group $(\mathbf Z/(p))^\times$ is cyclic for any prime $p$.Most proofs make use of the fact that for $r\geq 1$, there are at most $r$ solutions to the equation $x^r=1$ in $\mathbf Z/(p)$,...
View ArticleAlternative proof that base angles of an isosceles triangle are equal
The "classic textbook proof" of equality of base angles of an isosceles triangles which I studied in my school days is as follows:Let $\Delta ABC$ be a triangle with $AB = AC$ and let $D$ be the mid...
View ArticleHow can I (semi-formally) convince myself that Euclidean geometry comports...
I have always felt moderately "skeptical" about the notion that Euclidean geometry as presented in a mathematics course (whether in a synthetic or an analytic form) actually corresponds to our...
View ArticleHow to learn necessary concepts for research and avoiding a fake impression...
As an undergraduate engaged in applied mathematics research, specifically within mathematical programming, my work has quite some intersection with areas like algebraic geometry for polynomial...
View ArticleStory proofs in Combinatorics/Probability
I have been recently been going through Blitzstein in an attempt to put my probability on a stronger foundation then it currently is. There is a large emphasis on the use of "story" proofs/definitions...
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