When is it sufficient to use logic as proof for an intuitive answer

Say I have the following limit limx→∞3xex−1 In this case it’s simple enough to write it as \lim_{x\rightarrow\infty}{3^x}{e^{1-x}} and then show it approaches infinity, but if there wasn’t an easy algebraic solution would it be sufficient to state something along the lines of: 3^x approaches infinity more quickly e^{x-1} because 3 is larger than e … Read more

Relation between topological denseness and denseness over poset

In the theory of forcing, the notion of dense set is important. Formally, a subset $D$ of a poset $P$ is dense if, for any $p\in P$ we can find some $q\in D$ with $q\le p$. Intuitively, denseness of $D$ suggests the “property $D$” is easy to get via extension of condition. I wonder why … Read more

Improper integral involving trigonometric function

I was wondering what happens when evaluating an improper integral involving a trigonometric function where the denominator is a rational function with a zero at x=0. The example I have in mind is \int_{-\infty}^{\infty}\frac{sin(ax)}{x(x-i)(x+i)} dx If I rewrite the sine in terms of the exponential and then evaluate two integrals, one for the upper half-plane … Read more

Understanding orientability of surfaces

I’m trying to understand the intuitive meaning behind the definition of orientability of 2-dimensional surfaces. I understand the formalities, but I just can’t seem to get what exactly it means for a surface to be orientable. Specifically, I’d like an intuition that was intrinsic: if I were a little ant living inside a surface S, … Read more

Smallest subgroup and ⟨a⟩\langle a \rangle.

Gallian says in Chapter 3 of Contemporary Abstract Algebra that For any element a of a group G, it is useful to think of ⟨a⟩ as the smallest subgroup of G containing a. But wouldn’t this mean the set ⟨a⟩ is simply {a,a−1,e}? Answer The set {a,a−1,e} has an identity and is closed under inverses, … Read more

How can one intuitively understand the notion of quantile preferences in stochastic cooperative game theory?

Background For my thesis, I’m trying to wrap my head around a relatively small field within mathematics called “Stochastic Cooperative Game Theory”. To that end, I’ve read a few papers on the subject, including Cooperative games with stochastic payoffs (Suijs et al., 1999, link) and Convexity in stochastic cooperative situations (Timmer et al., 2005, link). … Read more

Intuition behind Eigenvalue solution matrix

I’ve been watching the excellent course by 3Blue1Brown on Linear Algebra which is oriented towards giving students intuition into Linear Algebra concepts. I am trying to find an intuitive way to understand the matrix we use to calculate eigenvalues. Specifically, I am trying to get an intuition for the matrix shown below (screen snapshot from … Read more

Intuition about Euler’s Theorem on homogeneous equations

I wonder, what would be the intuition or motivation to studying Euler Formula for homogeneous function $f:\mathbb{R}^k \to \mathbb{R}$ such that $f(tx) = t^n f$, for all $t>0$ . $\sum x_i \frac{\partial f}{\partial x_i} = n f$ I understand its proof and can do some problem but it feels really artificial or rather just manipulation … Read more