Mathematical intuition is the instinctive impression regarding mathematical ideas which originate naturally without regard to formal mathematical proofs. It may or may not stem from a cognitive rational process.

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What is the intuition behind / How can we interpret the eigenvalues and eigenvectors of Euclidean Distance Matrices ?

Given a set of points $x_1,x_2,...,x_m$ in the euclidean space $\mathbb{R}^n$, we can form a $m$ x $m$ Euclidean Distance Matrix $D$ where $D_{ij}={||x_i-x_j||}^2$. We know a little bit about these ...
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136 views

How strong is the analogy between spectra and abelian groups?

I am led to understand that spectra are some kind of $\infty$-analogue of (discrete) abelian groups, or perhaps more accurately, some kind of generalisation of chain complexes of abelian groups. How ...
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252 views

Behaviour at infinity of a function in terms of first and second derivatives

In a paper (dealing with spectra of certain Schrodinger operators) I found the following assumption for a function $f\in C^\infty(\mathbb R^n;\mathbb R)$: there exists a constant $C>0$ and a ...
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150 views

Why are injective $\mathscr{O}$-modules flasque?

Let $X$ be a topological space, and let $\mathscr{O}$ be a sheaf of rings on $X$. It is easy to verify that the functor $\Gamma (U, -) : \textbf{Mod}(\mathscr{O}) \to \textbf{Ab}$ is representable, ...
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58 views

Heuristics of the sum of squared naturals $(1^2 + 2^2 + 3^2 \cdots + n^2)$

I'm new and this is my first question (though I've been lurking). English is not my native language. Studying on my own. I'm really interested in deriving the formula $1^{2} + 2^{2} + 3^{2} + \cdots+ ...
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98 views

Apple game question

Player A and Player B play a game. On the middle of the table there is a pot full of $N$ apples of different weights. Player A starts first and chooses an apple and starts eating it. Losing no time ...
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144 views

Geometrical interpretation of a group action of $SU_2$ on $\mathbb S^3$

Background There're some nomenclatures from Michael Artin's Algebra to explain. 3-Sphere, or $\mathbb S^3$, is the locus of $x_0^2+x_1^2+x_2^2+x_3^2=1$, where $(x_0,x_1,x_2,x_3)\in\mathbb R^4$. ...
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176 views

Why is better to work with the spectrum of prime ideals than with the maximal one, for example in the definition of affine scheme.

When we have an algebraic variety we can identify the points of the variety with maximal ideals of the coordinate ring. I would like to know why it is more natural to define the main structure of ...
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176 views

Intuitive test of convergence

Are there any intuitive tests that might help one decide whether a sequence of functions converges / converges uniformly? For example, an intuitive test I have recently realized for uniform continuity ...
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57 views

Intuition behind variation of parameters method for solving differential equations

I have used the variation of parameters method (and have been taught it, although not hugely in depth) and I was wondering if I've understood the intuition behind it. In particular I've been thinking ...
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79 views

Distinction between nowhere monotone and nowhere differentiable

It is known that all functions that are continuous and nowhere differentiable are also nowhere monotone but that there is a function that is everywhere differentiable but nowhere monotone. I have ...
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123 views

What's the idea behind the covariant derivative?

I'm learning differential geometry from what I find on the Internet (to eventually find a grasp on General Relativity too). Right now I playing with a sphere. I have 3 functions ($x$, $y$, $z$) that ...
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234 views

Intuition behind the definition of a measurable set

This week I saw the definition of a measurable set for an outer measure. Let $\mu^*$ be an outer measure on a set $X$. We call $A \subseteq X$ measurable if $$\mu^*(E) = \mu^*(A\cap E) + ...
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210 views

Visualization of immersed submanifold

I am trying to visualize the difference between immersed submanifold and embedded submanifold. At first, I thought that, for example, if I can embed manifold $M$ in $\mathbb{R}^4$ and if my friend can ...
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225 views

Orthogonal Coordinate Systems Intuition

I'd really love it if you could give some intuition on how to derive the $x$, $y$ & $z$ coordinates from all/any of the orthogonal coordinate systems in this list, how you think about, say, ...
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200 views

What is the intuition of conjugacy classes?

How can I fully understand what are conjugacy classes are in groups? I know the definition, that $a$ and $b$ are conjugate if $gag^{-1}=b$ for some $g\in G$. But what is the intuition? Using a ...
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149 views

Intuition behind “Non-Archimedean” — two senses of “non-archimedean”.

There appear to be two senses of the qualifier "Archimedean" for fields. One is for ordered fields, and one is for "valued fields" (fields with an absolute value function defined). In the first case, ...
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279 views

Geometric intuition for Jordan normal forms (invariant subspaces, shearing, scaling, etc.)

I'm trying to visualize what a linear operator does to a vector space if that operator can be put into Jordan normal form. For concrete motivation, let's take $V = \mathbb{R}^3$, with some linear ...
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94 views

Is there any equivalence between the category of schemes over $\mathbb R$ and the category real manifolds

The equivalence of the category of smooth projective curves over $\mathbb C$ and the category of compact Riemann surfaces is, I believe, well documented. For example, it is mentioned on the wiki page: ...
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81 views

Quotient-lifting properties

I borrowed this terminology from K. Conrad's article on series of subgroups, in which he discusses solvability of groups. This property of certain groups satisfies Let $N\triangleleft G$. Then ...
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85 views

Ways to think about one-relator groups

What are some intuitive ways to think about one-relator groups? I am aware of the Freiheitsatz, and Bass-Serre theory. What I'm interested in are ways people who work extensively with one-relator ...
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84 views

Complex graphs in 3D

Does anyone have red-green 3D software for plotting 4D graphs in 3D with 3D glasses? I've seen a 4D hypercube done this way and it's very revealing...
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57 views

Does this integral variable change makes sense to you?

I was Reading a book about calculus when I've found this part about variable substitution in integrals: Consider $f$ defined in na interval $I$. Suppose that $x =\phi(u)$ is inversible, and its ...
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100 views

Intuition behind (statistical) completeness

I was wondering if any of the members of the MSE community would like to share his/her intuition about completeness in statistics. For the sake of "completeness", here's the definition, taken from ...
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85 views

Intuition behind a proof showing a square is homeomorphic to a quotient of an interval

There's a rather simple proof for the following theorem: There exists an equivalence relation $\sim$ on the unit interval $I=[0,1]$ such that the quotient $I/{\sim}$ is homeomorphic to the unit ...
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44 views

Henri Poincaré writings

I have heard that Poincaré writings were very intuitive in its approach and not very formal in the arguments. I'm searching for something like this to complement my study of dynamical systems. I ...
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84 views

A question on mathematical writing.

One of the problems I am grading this week is as follows: Given a simply connected bounded domain $\Omega$ on $\mathbb{R}^{2}$, prove that there exist a line that separates it into two parts of equal ...
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313 views

Taking a Fourier transform of Taylor series

My (naive) question is whether it is possible to take the Fourier transform of a Taylor series? Could one use multiplication with $\delta$ to get the function sampled at the point of expansion and ...
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107 views

A few questions about nonabelian cohomology of finite groups.

I apologize in advance if these questions are broad or basic. I tried to read about them at the Wikipedia, but everything is written in the language of category theory, in which I have had no formal ...
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126 views

Do the $p$-norms in $\mathbb R^2$ have a nice geometric intuition behind them?

For $p\geq 1,$ the $p$-norm of a vector $(x,y)\in\Bbb R^2$ is the number $\|(x,y)\|_p=(|x|^p+|y|^p)^{1/p}.$ I learned this definition some time ago, but I never really understood it. Is there a ...
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49 views

What is the variance of self-information (or surprisal)?

The self-information of an outcome $x_i$, or surprisal, is defined as: $$ I(x_i)=-\log P(x_i), $$ where $P$ means probability. This way, the Shannon entropy can be seen as the "average" or "expected" ...
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63 views

Intuitive Approach to Sheaf and Cech Cohomology

Sheaf and Cech cohomology $H^*(X,\mathcal{F})$ (which give the same result when applied to good enough topological spaces) are a useful generalisation of the concepts of de Rham and Dolbeault ...
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34 views

Protomodular categories

The axioms for abelian categories are nice and clear. The axioms for protomodular categories - and therefore semi-abelian categories - are beyond me entirely. I'm looking for a breakdown of the ...
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32 views

Derived Category in terms of Torsion Theory?

It is known that there's a bijection between hereditary torsion theories on, and localizations of, a fixed abelian category. Is this bijection natural? How/why not? How can I think of the derived ...
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57 views

Intuition for the Positive Real Number $\epsilon$ in Topology

Although this question might sound a little too simple, it is a problem that I must get addressed. In addition, there is no way for me to formally describe it. If you have something you can add, by ...
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40 views

Intuition for chain homotopy via tensor products

An approach to chain homotopies, alternative to the usual boundary relation, uses the monoidal (closed) structure of $\mathsf{Ch}_\bullet(R\mathsf{Mod})$ with $R$ a commutative ring. In particular, a ...
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37 views

Is there an intuitive reason why hippopede, the intersection curve of a sphere and a cylinder, is traced by composing two rotational motions?

The hippopede is historically famous because Eudoxus used its properties in the first mathematical model of planetary motion. He nested concentric spheres rotating at different inclinations to each ...
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74 views

Intuition behind the definition of Measurable Sets

I started studying "Measure Theory and Integration" and went through the first section which talks about Lebesgue Outer Measure of a set. All was well until I started with the second section which ...
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85 views

General formula for dependent probability distributions

Recently I encountered the following problem: What is the mean distance between two random points on a unit square? I understand pen and paper methods for solving this exist however I'm ...
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152 views

Visualize normal subgroup, normalizer, cosets.

A few important aspects of the relationship $H \lhd N_G(H) \le G$ are highlighted in Figure 7.31. First, the size of $N_G(H)$ is some multiple of |H|, and the size of G is some multiple of $N_G(H)$, ...
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329 views

How Would Arnold Explain the Jordan Normal Form to a 6 Year Old?

How would Vladimir Arnold explain the Jordan normal form, to a six year old, in full detail starting from nothing in a way that somehow explains everything in a deeper way, probably including topology ...
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186 views

Intuition behind continuity in topological spaces

I was approaching the following problem: "Let $f \colon X \to Y$ be continuous. Is it true that if $x$ is a limit point of $A \subset X$ then $f(x)$ is a limit point of $f(A)$?" The answer is that ...
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85 views

What is a Line Integral?

My question is very simple yet crucial to the understanding of many fields of mathematics. What is a line integral? If I choose some arbitrary line segment $\mathbb{A}$ to integrate a function ...
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43 views

What are some ways of showing that a structure is not minimal?

The question is really in the title. My background in model theory is very limited. Basically nothing past the definition of minimal structures and minimal subsets. I am interested in some ...
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145 views

Wilson's theorem intuition

Wilson's Theorem: $p$ is prime $\iff$ $(p-1)!\equiv -1\mod p$ I can use Wilson's theorem in questions, and I can follow the proof whereby factors of $(p-1)!$ are paired up with their (mod $p$) ...
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65 views

Intuition behind criterion for an irreducible Markov chain to be transient

I have been looking over my notes for Markov chains, and I have come across the following: Theorem: An irreducible Markov chain is transient iff for some state $i$ there exists a nonzero vector $y$ ...
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124 views

Intuition behind the proof for Wiener's theorem?

I am reading his proof for Wiener's theorem in Chp9 of Rudin's functional analysis. The theorems (9.4, 9.5 and 9.7) themselves are quite clear and Rudin did a good job explaining the intuition behind ...
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139 views

Is this calculus of variations intuition justifiable?

I'll preface this by saying that I haven't taken an in depth study of the calculus of variations and have only come across it recently in applications; in depth study is on my to do list. I'm ...
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209 views

Intuition in permutations for Laplace Determinant Expansion

Starting with the Leibniz formula for the determinant, I wish to derive the Laplace (Cofactor) Expansion. At the risk of being overly verbose, please see the proof here. Now I understand the idea of ...
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190 views

Ways to think about the binomial coefficient

Just to sharpen my intuition in combinatorics, I ask you of ways to think about interesting combinatorical quantities and expressions like the binomial coefficient, for example, for the binomial ...