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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Do the equalities $\int_0^∞1dx·\int _0^∞1dx=2\int_0^∞xdx$ and $\int_0^∞e^xdx·\int_0^∞e^xdx=2\int_0^∞e^{2 x}dx-2\int_0^∞e^xdx$ make sense?

Previously I tried to define multiplication of divergent integrals, but my approach turned out to be umbral-like. Now, I decided to define multiplication of divergent integrals in a Hardy fields-like ...
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7 votes
1 answer
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Intuition/meaning behind/physical content of the concept of a smooth structure

Some mathematical structures are visualized very well. I imagine how a shapeless bunch of points (a set; the only property of which is quantity) is collected in one or another soft form (topological ...
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2 votes
1 answer
123 views

What do higher order diffusion terms do?

I have been trying to learn to work with the Python module FiPy, which is supposed to solve PDEs of the form $$ \frac {\partial(\rho \phi)} {\partial t} - [\nabla\cdot(\Gamma_i\nabla)]^n\phi - \nabla \...
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57 votes
4 answers
7k views

Is orientability a miracle?

$\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$This question is prompted by a recent highly-upvoted question, Conceptual reason why the sign of a permutation is well-defined? The responses made ...
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133 votes
31 answers
10k views

Conceptual reason why the sign of a permutation is well-defined?

Teaching group theory this semester, I found myself laboring through a proof that the sign of a permutation is a well-defined homomorphism $\operatorname{sgn} : \Sigma_n \to \Sigma_2$. An insightful ...
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30 votes
14 answers
3k views

Archiving mathematical correspondence

What are great examples of comprehensively archived mathematical correspondence (including both handwritten and electronic items)? Context: polished papers usually don't reveal the full process that ...
0 votes
0 answers
89 views

What is some algebraic intuition behind the fact that the (real part) of the logarithm of Bernoulli umbra plus $1$, is $-\gamma$?

Bernoulli umbra is defined in classical umbral calculus as in Taylor - Difference equations via the classical umbral calculus. Yu - Bernoulli Operator and Riemann's Zeta Function shows that $\...
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4 votes
2 answers
399 views

In what precise sense is quantum (i.e., non-commutative) probability not expressable in terms of classical probability?

The quantum set-up has many settings, so let's fix some definitions. I will be taking the Hilbert space approach with a minor modification that I will make explicit. We begin with a Hilbert space $\...
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4 votes
2 answers
305 views

Arzelà-Ascoli for $C_b(0,1)$? Or more generally, why is that continuous functions "live most naturally" on compact spaces?

I’m wondering if there is a version of Arzelà-Ascoli for continuous functions on not-necessarily compact metric/Hausdorff spaces $X$, i.e. a characterization of the compact subsets of $C_b(X)$ (under ...
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1 vote
1 answer
118 views

Intuition of the "work" done by random variables in Monte Carlo methods (incl. MCL)

(I've tried Math SE, but have so far come up empty handed, so I'm trying my luck here.) I would like to get a better intuitive understanding of why Monte Carlo works so well in approximating a ...
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-5 votes
1 answer
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Can we say that everywhere where it makes sense $\log_0 x=0^x$? Are they equal, the function is self-inverse? If so, what is deep intuition behind it? [closed]

It makes little reason to speak about $0^x$ and $\log_0 x$ on the set of real numbers, but in matrices, it seems, the expressions coincide, for instance, $0^ \left( \begin{array}{cc} \frac{1}{2} &...
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5 votes
0 answers
281 views

Geometric meaning of localization at $(1+I)$?

Let $I\vartriangleleft A$ be an ideal of a commutative ring. Consider the submonoid $1+I\subset A$. What is the geometric interpretation of localization at this submonoid? How does it relate to the ...
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1 answer
107 views

In what circumstances do we typically encounter expressions like $(c/2+1/2)^n \pm(c/2-1/2)^n$?

It attracted my attention that in many areas of mathematics we sometimes encounter expressions of the form $(c/2+1/2)^n \pm(c/2-1/2)^n$, where $c$ is some kind of a known constant. Split-complex ...
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2 votes
0 answers
171 views

Hypermodulus and what mathematical objects have it

When researching divergent integrals, I decided to introduce a concept of "modulus" or "determinant" of divergent integral (and series). Basically, it is the exponent of the real ...
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3 votes
2 answers
392 views

Why the sign in the definition of the discriminant?

Consider the split monic $f=\prod_{i=1}^n(x-x_i)\in \mathbb Z[x_1 ,\dots ,x_n,x]$. Its discriminant is usually defined as $$(-1)^{n(n-1)/2}\prod_{i=1}^nf^\prime(x_i)=\prod_{1\leq i<j\leq n}(x_i-x_j)...
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2 votes
0 answers
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Meaning of the coadjoint representation and its orbits

Given a Lie group $G$ there is a natural representation of $G$ on the dual of its Lie algebra $\mathfrak{g}^*$ given by the coadjoint representation. This representation is obtained by differentiating ...
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1 vote
0 answers
53 views

Topological intuition for the cancellation property of separated maps w.r.t a class of properties of continuous maps

Recall a continuous map is separated if its diagonal is closed. This is equivalent to the fibers being relatively Hausdorff in the total space. Proposition. Suppose $\mathrm P$ is a class of ...
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6 votes
1 answer
440 views

What is the intuition behind the Kantorovich potential in optimal transport?

From what I currently understand, under certain conditions one may turn the usual Kantorovich problem - a minimisation problem in terms of measures into a maximisation problem in terms of functions. ...
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1 vote
1 answer
319 views

Intuition behind formal neighborhood and local ring and formal power series

In The Geometry of Schemes by David Eisenbud and Joe Harris, on page 57, there is an explanation on "node" of a plane curve. The book says that, a curve $X\subseteq \mathbb A_{\mathbb C}^2$ ...
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5 votes
0 answers
395 views

Intuition behind exceptional inverse image?

The story is probably well-known: given a map $f:X\to Y$ of spaces (say schemes, but there are many other contexts), we have two classical operations between sheaves on $X$ and those on $Y$: the ...
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37 votes
3 answers
3k views

What is so geometric about symplectic geometry?

Symplectic geometry is often motivated by the Hamilton's equation which in turn are a reformulation of Newton's third law. But the subject itself is of independent mathematical interest. What I don't ...
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1 vote
0 answers
79 views

What intuitive meaning "determinant" of a divergency (divergent integral, series, germ, pole or a singularity) can have?

I am working on the algebra of "divergencies", that is, infinite integrals, series, and germs. So, I decided to construct something similar to the modulus or determinant of a matrix of these ...
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5 votes
2 answers
580 views

Intuition/elegant reason for why Langevin diffusion converges to $\exp(-U)$?

Given a potential function $U: \mathbb{R}^n \to \mathbb{R}$, Langevin diffusion is gradient descent plus a Brownian motion term: $X' = -\nabla U(X) + \sqrt{2} \text{ }dW$. It happens that the ...
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3 votes
1 answer
170 views

Is the solution to the Fredholm integral equation of the first kind a continuous analogue of Cramer's rule for matrix equations?

When we have a system of of $n$ linear equations represented by $$A \vec{x} = \vec{b} $$ with $\vec{x} = (x_{1}, x_{2}, \dots, x_{n})^{\intercal} $, we can solve for each component of this vector by ...
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7 votes
0 answers
491 views

Understanding the higher stack of perfect complexes

One of the most famous examples of higher Artin stacks is the stack of perfect complexes. I recall here the basic stuff: We fix a function $b: \mathbb{Z} \rightarrow \mathbb{N}$ which is zero ...
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6 votes
1 answer
128 views

Morphisms between compact quantum groups

Let $(A, \Delta_A)$ and $(B, \Delta_B)$ be two compact quantum groups (in the sense of Woronowicz). I would be tempted to define a morphism $(A, \Delta_A) \to (B, \Delta_B)$ to be a unital $*$-...
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31 votes
5 answers
5k views

How should you explain parallel transport to undergraduates?

The title is a bit deceiving, because what I really mean is the parallel transport that corresponds to the Levi–Civita connection. This is in the vein of many other questions on mathoverflow: What is ...
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23 votes
1 answer
2k views

Is an interpretation mathematics (fit for publication)?

Background I am a mathematician with two published papers. The first is based on my PhD thesis and generalised a tool to a more general setting. The thesis was cited a number of times by the time the ...
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3 votes
0 answers
394 views

What intuitive meaning "determinant" of a divergency (divergent integral or series) can have? [closed]

I am working on the algebra of "divergencies", that is, infinite integrals, series and germs. So, I decided to construct something similar to determinant of a matrix of these entities. $$\...
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2 votes
0 answers
258 views

Can be this "handwaving" idea about "counting" reals somehow put on solid ground?

We know that the Cantor's cardinality of a countable set is $\aleph_0$ and the cardinality of continuum is $2^{\aleph_0}=\aleph_0^{\aleph_0}$. Unfortunately, this measure is based on the idea of ...
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17 votes
2 answers
2k views

Intuitive explanation why "shadow operator" $\frac D{e^D-1}$ connects logarithms with trigonometric functions?

Consider the operator $\frac D{e^D-1}$ which we will call "shadow": $$\frac {D}{e^D-1}f(x)=\frac1{2 \pi }\int_{-\infty }^{+\infty } e^{-iwx}\frac{-iw}{e^{-i w}-1}\int_{-\infty }^{+\infty } e^...
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2 votes
0 answers
99 views

Interpretation of polar derivative and apolar polynomials

If $f(x)$ is a degree $n$ polynomial and $b \in \mathbb{C}$ is a complex number, then the polar derivative of $f$ with respect to $b$ is defined by $$D_b f(x) = nf(x) - (x - b)f'(x).$$ Two degree $n$ ...
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57 votes
9 answers
4k views

Examples of back of envelope calculations leading to good intuition?

Some time ago, I read about an "approximate approach" to the Stirling's formula in M.Sanjoy's Street Fighting Mathematics. In summary, the book used a integral estimation heuristic from ...
2 votes
2 answers
296 views

What concept does covariance formalise?

So for me the definition the independence of two random variables $X,Y$ is intuitivly very clear. But what I have never seen motivated is why the heck one would be interested in the covariance $$\...
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42 votes
5 answers
6k views

What is the Levi-Civita connection trying to describe?

I have seen similar questions, but none of the answers relate to my difficulty, which I will now proceed to convey. Let $(M,g)$ be a Riemannian manifolds. The Levi-Civita connection is the unique ...
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2 votes
0 answers
156 views

Dyadic models in number theory and "spillover"

In a classic blog post, Tao discusses the appearance of "dyadic models" in various guises in various areas of math. The number-theoretic version of the idea is to study polynomials over a ...
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2 votes
0 answers
251 views

Dimension of the moduli stack of vector bundles over a curve

Let $Vect_{n}(C)$ the moduli stack of vector bundles $V$ of rank $n$ over a smooth curve $C$ of genus $g$. It is well known that $Vect_{n}(C)$ is a smooth stack of dimension $\dim(H^{0}(C,End(V)))-\...
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14 votes
1 answer
1k views

Why should I look at the resolvent formalism and think it is a useful tool for spectral theory?

Wikipedia calls resolvent formalism a useful tool for relating complex analysis to studying the spectra of a linear operator on a Banach space. Sure, I believe you because I've seen results that use ...
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1 vote
0 answers
290 views

Why can one compute the sum of divisors of $n$ without factoring $n$?

Question links to paper which states: $$ \sigma(n)= \frac{6}{n^2(n-1)}\sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\sigma(n-k) \qquad (1) $$ where $\sigma(n)$ is the sum of divisors of $n$. Another similar ...
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10 votes
1 answer
350 views

Why are root data a natural candidate for classifying connected reductive groups?

For the purpose of this question, you may assume that we are working over the complex numbers. Given a connected reductive group $G$, one can choose a maximal torus $T$, and then let $T$ act on the ...
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11 votes
1 answer
537 views

Intuition behind choosing a specific test function

I am learning about elliptic PDEs using the book by Chen & Wu, especially on the maximum principle. The author uses the De Giorgi iteration technique to establish the weak maximum principle for ...
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9 votes
1 answer
454 views

Intuition for categorical fibrations?

I think I have a pretty good intuitive understanding of most types of fibrations of quasicategories: a (trivial) Kan fibration is a bundle of (contractible) spaces with equivalent fibers, a left/...
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4 votes
0 answers
346 views

Brouwer's fixed point theorem and the one-point topology [closed]

I posted this question last week on Math SE and got upvotes and helpful comments that allowed me to make the question more precise https://math.stackexchange.com/q/3765546/810513. As I did not get an ...
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2 votes
1 answer
403 views

Interpretation around conjugacy classes in group theory [closed]

this is Rajeev Srivastava and his colleague Ravinder Padmanabha, specializing in computational geometry algorithms for application in science and research. We would like to include methods from ...
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17 votes
3 answers
1k views

Intuition behind stability and instability in model theory

In A survey of homogeneous structures by Macpherson (Discrete Mathematics, vol. 311, 2011), a stable or unstable theory is defined as (Definition 3.3.1): A complete theory $T$ is unstable if there is ...
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11 votes
1 answer
642 views

Intuition about ordinal fixed points $\alpha = \aleph_\alpha$

I wanted to ask for your intuition about ordinal fixed points $\alpha = \aleph_\alpha$, where $\aleph_\alpha$ stands for the $\alpha$-th Aleph number in the Aleph sequence of cardinalities. For ...
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-2 votes
1 answer
59 views

Interpretation of the word Random [closed]

I have previous knowledge of what a random experiment is, but sometimes I get confused by the use of the word Random. I can express my doubts as the following questions: if something is random them it ...
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43 votes
3 answers
2k views

Why is the Vandermonde determinant harmonic?

It can be checked that the Vandermonde determinant defined as $$V(\alpha_1, \cdots, \alpha_n) = \prod_{1 \le i < j \le n}(\alpha_i-\alpha_j) $$ is a harmonic function, that is $\Delta V = 0$ where ...
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10 votes
0 answers
330 views

Grothendieck categories and their morphisms

I am not an algebraic geometer in the first place, and I am mainly familiar with topology and category theory. Recently I am studying Grothendieck categories and I am struggling with getting ...
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35 votes
8 answers
4k views

Interpretation of the action in classical mechanics

In classical mechanics the dynamics on a manifold $M$ are characterised by the minimisation of a functional $$ \min_{q \in C^\infty(\mathbb{R},M)} \int_{\mathbb{R}}L(q(t),\dot{q}(t))dt, $$ where $L:TM\...
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