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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15 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 ...
3
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2answers
163 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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1answer
108 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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447 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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2answers
220 views

Line graphs called “graph derivatives”: any intuition?

Short version: in several papers, line graphs (and closely related graphs) are called graph derivatives or derived graphs; is there any intuition for such terminologies, in connection with the ...
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67 views

Intuition behind the geometry of n-ball and n-cube in non-Euclidean space

The paradox in the volume of $n$-dimensional balls and cubes has been discussed many times (e.g., History of the high-dimensional volume paradox). It is also quite clear how to derive the ratio of ...
6
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1answer
102 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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4answers
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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 ...
24
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1answer
1k 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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367 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. $$\...
2
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0answers
227 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 ...
12
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2answers
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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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Intuition on constrained optimisation

I am trying to understand and develop some intuition about Problem Defition (Section 3) in the following paper: Agarwal, Deepak, et al. "Online parameter selection for web-based ranking problems.&...
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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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9answers
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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 ...
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63 views

Geometric meaning of second tangent bundle, or of microsquares in SDG

In synthetic differential geometry (SDG), we have $D\subset R$ comprised of the second order nilpotents. The Kock-Lawvere axiom (KL axiom) implies that a function $D\times D\to R^n$ is of the form $...
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2answers
260 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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5answers
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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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0answers
143 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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0answers
204 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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1answer
457 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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273 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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1answer
316 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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1answer
447 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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1answer
317 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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0answers
328 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 ...
2
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1answer
270 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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3answers
990 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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1answer
507 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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1answer
58 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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2answers
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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0answers
155 views

Geometric meaning of Todd classes for non-tangent bundles

While Todd classes are defined for arbitrary vector bundles, the only cases I actually see in practical use are Todd classes of tangent bundles (as in Hirzebruch-Riemann-Roch or Grothendieck-Riemann-...
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0answers
309 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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8answers
3k 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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0answers
194 views

Intuition for the McGerty-Nevins compactification of quiver varieties

In Section 4 of the paper Kirwan surjectivity for quiver varieties (Inventiones Math. 2018) McGerty and Nevins define a compactification of the moduli space of representations of the preprojective ...
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371 views

Seeing what gets Harvey Friedman's “tangible incompleteness” principles into large cardinal territory

I'm trying to wrap my head around some of Harvey Friedman's recent, unpublished work on his tangible incompleteness project, and I'm trying to see the link between his "tangible statements" (...
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89 views

Intuition behind the local limit theorem in hyperbolic groups

Let $\Gamma$ be a finitely generated group and let $\mu$ be a probability measure on $\Gamma$. Denote by $X_n$ the induced random walk. Finally, let $p_n=\mu^{*n}(e)=P_e(X_n=e)$. The local limit ...
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1answer
693 views

Geometric intuition behind this chain homotopy

My question has to do with the chain homotopy that appears in Lee's Introduction to Topological Manifols and Rotman's Introduction to Algebraic Topology proofs that the inclusion $$C_\bullet^\mathcal{...
4
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1answer
306 views

Geometric intuition for Mather's cube theorem

Mather's cube theorem for the category of topological spaces says that given a homotopy-commutative cube: If one pair of opposite faces are homotopy pushouts and the two remaining faces adjecent ...
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1answer
354 views

Geometric interpretation of sections $H^0(\Theta_X, X)$ of the Tangent sheaf over curve

I'm reading Mumford's & Oda's Algebraic Geometry II and I'm confused about explanations on geometric intuition of sections $H^0(\Theta_X, X)$ of the tangent sheaf on page 287: Let $X$ a ...
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38 views

Geometric interpretation for uniformly elliptic pde of 2 second order

Let $\Omega \subset \mathbb{R}^{2}$ a domain,let $u \in C^{2}(\Omega)$, the operator $Lu= tr(A.D^{2}u) + <\nabla u,b> +cu$ where $A$ is a symmetric matrix, $b$ is a vector field continuous ...
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1answer
283 views

Building intuition in algebraic number theory [closed]

How do you build your intuition in algebraic number theory? Generally my intuition in elementary number theory came from just numerically fiddling with python (and also elementary number theory is ...
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1answer
1k views

Is there any deep philosophy or intuition behind the similarity between $\pi/4$ and $e^{-\gamma}$?

Here is a couple of examples of the similarity from Wikipedia, in which the expressions differ only in signs. I encountered other analogies as well. $${\begin{aligned}\gamma &=\int _{0}^{1}\int _{...
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2answers
244 views

How should I think about concrete functors and in particular about concrete isomorphism?

All the definitions that follow is taken from The Joy of Cats. Definition 1. Let $\bf{X}$ be a category. A concrete category over $\bf{X}$ is a pair $({\bf{A}},U)$, where $\bf{A}$ is a category ...
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0answers
429 views

Deligne's Mixed Hodge Theory

Deligne constructs Mixed Hodge Structures (MHS) on the cohomology, $H^{*}(X)$, of an algebraic variety $X$, in his papers Hodge II and Hodge III. Really the question below is rather vague, and is ...
5
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2answers
435 views

Tricks for getting a creative idea [closed]

Caveat: I fear that people will criticize me for asking this potentially inappropriate question here, but I guess that the community here is quite unique in the ability of potentially answering my ...
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3answers
1k views

What does the torsion-free condition for a connection mean in terms of its horizontal bundle?

I must have read and re-read introductory differential geometry texts ten times over the past few years, but the "torsion free" condition remains completely unintuitive to me. The aim of this ...
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0answers
73 views

Imagining linear maps between finite fields

I can't imagine a right picture of a linear transformation $\mathbb{F}_{p} \mapsto \mathbb{F}_p$ or $\mathbb{F}_{p^2} \rightarrow \mathbb{F}_{p^2}$ etc (over the field $\mathbb{F}_p$) although they ...
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364 views

Additive and multiplicative convolution deeply related in modular forms

From the fact spaces of modular forms are finite dimensional, from the decomposition in Hecke eigenforms, and from the duplication formula for $\Gamma(s)$ there are a lot of identities mixing additive ...
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4answers
3k views

How to visualize the Riemann-Roch theorem from complex analysis or geometric topology considerations?

As the question title asks for, how do others visualize the Riemann-Roch theorem with complex analysis or geometric topology considerations? That is all Riemann would have had back in the day, and he ...

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