Questions tagged [intuition]
Questions asking for the intuition behind some definition, conjecture, proof etc. In other words, questions designed to improve or to acquire understanding on a conceptual or intuitive level, as opposed to on a technical or formal level. When asking such a question it can be helpful to include a rough description of ones understanding of the subject at hand (on a technical level).
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Étale morphisms, a priori
I don't know what the etiquette is for asking old questions from MSE, but I would very much like to re-ask this one here at MO:
Background: The notion of an étale morphism has proved itself to be ...
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Equivalence of Functor Categories holding for $\infty$-Categories but failing in Homotopy Categories
When one starts with $C$ an ordinary co- & complete $1$-category with choosen class $W \subset \text{Mor}(C)$ of weak equivalences, classically there are two ways to obtain a new category from $C$,...
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What is a closed condition? [migrated]
I have had one or two professors in various classes who have proved that a set is closed because it is defined by a closed condition. For example: given a group $G$, define the set of invariant means $...
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Why do these two lines have the same probability of intersecting the circle?
Consider three tangent circles with radii in a geometric sequence, as shown.
Random point $A$ is chosen on the red circle. Random points $B$ and $C$ are chosen on the green circle. (All random points ...
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Hochschild cohomology is to associativity, as Hochschild homology is to _?
I apologize if this question is a bit of an ill-defined one and not research level. However, I think this community would be a better place to ask this question for a deeper understanding.
This ...
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Random triangle on a ring of three tangent circles: Show that the probability that the triangle contains the centre is 1/2
A triangle's vertices are random (uniform and independent) points on a ring of three mutually tangent congruent circles, with one vertex on each circle.
Show that the probability that the triangle ...
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Semmes' construction of a rapid filter
Originally posted on MSE. There are several expositions of Raisonnier's simplification of Shelah's proof that "all subsets of $\mathbb{R}$ are measurable" implies "$\aleph_1$ is ...
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Understanding the proof of Allard's integral compactness theorem
Lately, I have spent some time trying to understand the proof of Allard's compactness theorem for integral varifolds. The sources I have been looking at are
Section 6 of Allard's original paper (&...
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Geometric interpretation of residue functional on $A[x]/\langle f\rangle$?
Let $A$ be a commutative ring and $f\in A[x]$ monic. The residue functional on the quotient $A[x]/\langle f\rangle$ is defined by taking the $\deg f-1$ coefficient of minimal degree representatives.
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Mathias vs Cohen
This question is more about intuition than about working out the nitty-gritty of forcing constructions.
In Mathias forcing, conditions are $(s, A)$ where $s, A \subseteq \omega$, $|s| < \omega$, $|...
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Intuitive/combinatorial proof for Boppana entropy inequality $H(x^2)\ge\phi xH(x)$, i.e. $\binom{\phi p n}{\phi p^2 n} \leq \binom{n}{p^2n}$
Let $H(x)$ on $[0,1]$ denote the binary entropy function (the base of the logarithm does not matter for this whole discussion). Let $\phi:= \frac{1+\sqrt 5}{2}$ denote the golden ratio $\approx 1.618$....
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Paradox involving probability that a triangle is acute in circles and spheres
For a set of points $s$, let $P_s=$ probability that three undependent uniformly random points chosen from $s$ are the vertices of an acute triangle.
We have:
$P_\text{circle}=\frac14$
$P_\text{disk}=...
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Calculi of pseudodifferential operators and K-theory
I am reading the thesis of Chris Kottke (https://dspace.mit.edu/bitstream/handle/1721.1/60193/681923895-MIT.pdf) and I would need some help to try to understand intuitively why he makes the choice of ...
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What intuitive notion is formalized by condensed mathematics?
Preface: I ask this question from the position of a curious layperson who is excited about new conceptual advances in mathematics.
It is often said that the notion of "topology" formalises ...
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Intuition for proximal point method using L2 regularization
To minimize a function $f$, the proximal point method is defined as
$$x_{k+1} := \operatorname*{argmin}_x f(x) + \frac{1}{2\eta}\|x - x_k\|^2.$$
What's the intuition for why we want to use L2 ...
