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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143 views

How does the internal language of a topos come to be?

There are several books and articles on topos theory which mention the internal language, but I can't manage to see the big picture from any of them. I would like a soft explanation of how the ...
9
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187 views

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 ...
7
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62 views

What's the intuition behind the direct integral of a family of Hilbert spaces?

In order to understand better the mathematically rigorous version of Dirac's formalism in Quantum Mechanics I've been reading about direct integrals of Hilbert spaces, projector-valued measures and so ...
7
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169 views

How to intuitively understand prolongations

This question is concerned with the algebraic side of the theory of prolongations as explained in this paper by V. Guillemin and S. Sternberg. Let me first introduce my notation. We're working with a ...
7
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145 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 ...
7
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166 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, ...
6
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59 views

Sheaf cohomology intuition

I am working on understanding specifically what the $n^{th}$ Cech cohomology group $H^n(\mathcal{U}, \mathcal{F})$ measures, where $\mathcal{U}$ is a locally finite open cover on a topological space ...
6
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148 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 ...
6
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111 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 ...
6
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165 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 $(x_0,x_1,x_2,x_3)\in\mathbb R^4$. ...
6
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268 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 ...
6
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252 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 ...
5
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64 views

Geometric intuition for coherent rings, modules, and sheaves

Throughout, all rings are commutative. Definition 1. A ring $R$ is coherent if the solutions $\mathbf x=(x_1,\dots,x_n)$ to a linear equation $\mathbf{rx}=0$ are a finitely generated $R$-submodule of ...
5
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509 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 ...
5
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183 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 ...
4
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46 views

How to intuitively understand that an open subset of the reals can contain the rationals and have finite measure?

A question that one could ask is the following: if $U \subset \mathbb{R}$ is an open subset such that $\mathbb{Q} \subset U$, then is the measure of $U$ infinite? The answer is no, as the ...
4
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77 views

Mathematics Wallpapers

I know that this sounds very silly. But I don't know where else to ask. Is there a good free site for mathematics wallpapers , pictures etc ? Most of the time it is very difficult to find exact ...
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96 views

Learning Galois theory geometrically?

Recently I started poking at algebraic geometry and commutative algebra. My background is basic category theory and basic algebraic topology. I don't know a lot of other mathematics. I noticed Galois ...
4
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86 views

Connecting a vector space to its dual - why?

Can someone explain to me - intuitively - why embedding a vector space into its dual should naturally fix its geometry? I mean, I can run the usual statements through my mind - "The injection into the ...
4
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35 views

Intuition for homotopy (co)limits in triangulated categories

The following definition is taken from Daniel Murfet's Triangulated Categories Part I notes. Let $\mathcal T$ be a triangulated category with countable coproducts. Suppose we are given a ...
4
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68 views

Intuition behind generic point of a scheme?

I've been reading a little about algebraic geometry and how there seems to have existed this notion of "generic point" on a variety which wasn't carefully defined at first. But often times, ...
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51 views

How to show $k$-algebras are isomorphic in practice

I am working through some problems which require me to show when some $k$-algebra ($k$ a field) maps are isomorphisms. Unfortunately, I've got myself a bit confused with definitions and the like, and ...
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140 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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87 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 ...
4
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242 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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281 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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163 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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329 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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100 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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95 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 ...
4
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88 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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90 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...
3
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22 views

Moving from sheaves over spaces to sheaves over sites

The first example of a sheaf that I have consciously come across is the sheaf of continuous (real) functions on some topological space. The fact it is a sheaf is equivalent to the pasting lemma, which ...
3
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49 views

What was the genesis of Hua's identity?

Many resources I have read prove Hua's identity more-or-less mechanically. I have seen there is more than one raison d'être for Hua's identity: e.g. its connection to the fundamental theorem of ...
3
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63 views

The Intution Behind Real Symmetric Matrices and Their Real Eigenvectors

I am wondering about the geometric intuition behind real symmetric matrices and their corresponding linear transformations. Is it possible to understand geometrically why real symmetric matrices ...
3
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108 views

Is there a way to visualize a group?

Is there a way to picture a group in ones head? I want to "see" the difference between abelian and non-abelian group. And if f is a group homomorphism, is there a way to see that Ker(f)=1<=>f ...
3
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37 views

Need some help understanding this exercise about injective plane curve

Let $\gamma (t) = (x(t), y(t))$ be a smooth regular plane curve $\gamma: I \to \mathbb R^2$ where $I$ is some open interval. Now consider the following exercise: Let $\varphi (u,v) = (x(u), v + ...
3
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78 views

How to think/see point-set topology abstractly?

I've started learning point-set topology this semester. I've learned basic material about: topology on a set topological space open sets closed sets clopen sets closure neighborhoods interior point ...
3
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62 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 ...
3
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147 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 ...
3
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137 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 ...
3
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221 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 $N_G(H)$, ...
3
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355 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 ...
3
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51 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 ...
3
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0answers
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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430 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 ...
3
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110 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 ...
3
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166 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 ...
3
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132 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 ...
2
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38 views

Intuition for AB5 and Grothendieck categories

I'm trying to get some intuition for AB5 categories and Grothendieck categories by asking primitive questions. First of all, why ask for exact filtered colimits? Are they there simply to have some ...