# 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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### 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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $*$-...
3k 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 ...
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 ...
367 views

49 views

### 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.&...
51 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 $n$ ...
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 ...