Linear combinations of minor determinants
While thinking about properties of determinants, alternating multilinear maps, Grassmann algebras, I've stumbled upon some class of scalar- or vector-valued functions defined on spaces of matrices or...
View ArticleWant to learn more about cardinals beyond Aleph_n
I stumbled over this: Infinite Set is Disjoint Union of Two Infinite Setsbut from my current knowledge, this is far over my hat.At university, we have learned that $\aleph_0$ and $\aleph_1$ are...
View ArticleIs arithmetic with infinite numbers fictitious?
In 1933 Skolem constructed models for arithmetic containing infinite numbers. In a 1977 article Stillwell emphasized the constructive nature of Skolem's approach; see here. Is this at odds with...
View ArticleAn equivalent definition of isomorphism.
Let $G$ and $\overline G$ be groups of the same order, and $f\colon G\to\overline G$ a bijection. If $ f$ has theProperty. For all $g,h\in G$:$$f(gh)=f(g)f(1_G)^{-1}f(h)\tag1$$then $f^*:=f(1_G)^{-1}f$...
View ArticleModules over commutative absolutely flat rings vs. vector spaces
The theory of commutative absolutely flat rings (a.k.a. commutative von Neumann regular rings) is algebraic and furthermore it is the smallest variety containing all fields. Being a variety the...
View ArticleWhy is the solution to $(1+\frac{1}{x})^{x+1}=x$ close to pi?
$$\left(1+\frac{1}{x}\right)^{x+1}=x$$Solving for $x$ yields $x \approx 3.1410415254107885010$ which is close to $\pi$Why is this so? The only plausible reason I found is because $\ln{\pi} \approx...
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The beauty of calculus, looking for examples
My question is about the continuum between school math and research math.This started as a discussion at the Mittag-Leffler institute where I am visiting, and then I thought to post it here, to see if...
View ArticleScholarly work on the beauty of math
When reading mathematical books written for a general audience, or even searching questions on this site, the adjective beautiful is often used to describe mathematics. My question is whether there has...
View ArticleWhat are the current best GenAI tools (limits and sweet spot use cases) to...
My personal experience with GenAI is very mixed. It can be good on easy stuff, it can be miserable on more complex ones.I love the 'bad student' analogy in one of the referencesChatGPT is a good...
View ArticlePotential conceptual links between h-cobordism theorem (dimension 5+) and...
The h-cobordism theorem (in differential topology) is known to hold unconditionally for smooth manifolds of dimension ≥5, with the failure in lower dimensions closely tied to geometric intricacies...
View ArticleProblems that become easier in a more general form
When solving a problem, we often look at some special cases first, then try to work our way up to the general case.It would be interesting to see some counterexamples to this mental process, i.e....
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Motivation behind defining simplicial complexes (instead of just combinations...
I'm taking my first topology course this semester. We introduced the notion of an abstract simplicial complex, its topological realisation, and triangulation, which I think I'm fairly comfortable with....
View ArticleReference request: Open problems about finite free products of finite groups
I'm working with finite free products of finite groups, i.e. a group $G$ given by $$G = F_1 \ast \ldots \ast F_n$$ where each $F_i$ is finite.Do you know of any open problems as well as references...
View Articleancient principle of mathematics: figure = varying element
On page 83 in the book Conceptual mathematics by Lawvere et al. it says:An ancient principle of mathematics holds that a figure is the locus of a varying element.What does this quote mean? In...
View ArticleDouble assigment in a Definition for a math paper.
We are writing a number theory paper focused on integer tetration, the power tower $a^{a^ {\cdots ^a}}$$b$-times, where we have already stated that "$v_b(a)$" indicates the congruence speed of the base...
View ArticleSimplest 'proof of concept' that scheme theory can solve significant concrete...
In this MO answer, Felipe Voloch makes the claimActually, this is my only serious complaint about Hartshorne. He doesn't do any Number Theory and $\operatorname{Spec}\mathbb Z$ is where schemes really...
View ArticleRoadmap for self-learning pure mathematics from high school level onward....
Motivation:- I’m making this post because I know a lot of people who are genuinely interested in learning pure math but have no formal background or experience or help. Many of them are unsure how to...
View Article"So That" vs. "Such That"
In definitions and exercises, I notice that "so that" and "such that" are seemingly used interchangeably. Are they in fact interchangeable, or is one more appropriate for a specific context?Note:...
View ArticleA logical theory that puts meaning to (at first) nonsensical re-usage of a...
I noticed that things in everyday math such as $\forall x \in A, \forall y \in A \equiv \forall (x,y) \in A\times B$, a sort of "syntactic isomorphism" is formed. It's not easily written as...
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Why are the axes in coordinate geometry perperndicular?
In coordinate geometry, the $x$ and $y$ axis are perpendicular to each other. But is there any special reason for this (other than to make it simple)? Will coordinate geometry have contradictions if...
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