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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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 ...
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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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Geometric interpretation of trace
This afternoon I was speaking with some graduate students in the department and we came to the following quandary;
Is there a geometric interpretation of the trace of a matrix?
This question ...
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What is a twisted D-Module intuitively?
When I think about $\mathcal{D}$-Modules, I find it very often useful to envison them as vectorbundles endowed with a rule that decides whether a given section is flat. Or alternatively a notion of ...
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What is (co)homology, and how does a beginner gain intuition about it?
This question comes along with a lot of associated sub-questions, most of which would probably be answered by a sufficiently good introductory text. So a perfectly acceptable answer to this question ...
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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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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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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
$$
\frac {\partial(\rho \phi)} {\partial t} - [\nabla\cdot(\Gamma_i\nabla)]^n\phi - \nabla \...
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1
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Intuition for Picard-Lefschetz formula
I'm trying to develop some intuition for the (local) Picard-Lefschetz formula (which I'm encountering for the first time in Deligne's paper "La Conjecture de Weil, I").
To summarize the ...
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answer
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Grothendieck - sheaves as meter sticks
I'm trying to read parts of McLarty's Grothendieck on Simplicity and Generality. In the article, I read Grothendieck thought of sheaves over some topological space as meter sticks measuring it.
...
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Intuitive explanation why "shadow operator" $\frac D{e^D-1}$ connects logarithms with trigonometric functions?
Consider the operator $\frac D{e^D-1}$ which we will call "shadow":
$$\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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Zagier's one-sentence proof of a theorem of Fermat
Zagier has a very short proof (MR1041893, JSTOR) for the fact that every prime number $p$ of the form $4k+1$ is the sum of two squares.
The proof defines an involution of the set $S= \lbrace (x,y,z) \...
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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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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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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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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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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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9
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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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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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What is the "intuition" behind "brave new algebra"?
Y.I. Manin mentions in a recent interview
the need for a “codification of efficient new intuitive tools, such as … the “brave new algebra” of homotopy theorists”. This makes me puzzle, because I ...
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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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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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Why do Littlewood-Richardson coefficients describe the cohomology of the Grassmannian?
I'm looking for a "conceptual" explanation to the question in the title. The standard proofs that I've seen go as follows: use the Schubert cell decomposition to get a basis for cohomology and show ...
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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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Philosophy behind Mochizuki's work on the ABC conjecture
Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy ...
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Tips on cohomology for number theory
I am curious about what is a good approach to the machinery of cohomology, especially in number-theoretic settings, but also in algebraic-geometric settings.
Do people just remember all the rules and ...
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Why is a topology made up of 'open' sets? [closed]
I'm ashamed to admit it, but I don't think I've ever been able to genuinely motivate the definition of a topological space in an undergraduate course. Clearly, the definition distills the essence of ...
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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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What is convolution intuitively?
If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability ...
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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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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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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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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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Why are lacunary series so badly behaved?
Hi!
I just came across the Ostroski-Hadamard gap theorem, and while I can understand the proofs as well as the principle that the series $\sum_{n=0}^\infty z^{2^n}$ ought to have a singularity at ...
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What makes a set random?
There are many results in number theory, where the existence
of some $B \subseteq \mathbb{N}$ with certain properties is proved by
a probabilistic argument employing "random sets". One such ...
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Does anyone know an intuitive proof of the Birkhoff ergodic theorem?
For many standard, well-understood theorems the proofs have been streamlined to the point where you just need to understand the proof once and you remember the general idea forever. At this point I ...
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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 ...
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Vector bundle $L$ admits connection if and only if degree of every direct summand of $L$ divisible by $\text{char}\,k$, intuition
Consider the following theorem of Atiyah.
Let $X$ be a connected smooth projective curve over an algebraically closed field $k$. Then a vector bundle $L$ on $X$ admits a connection if and only if the ...
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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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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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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} $$ with $\vec{x} = (x_{1}, x_{2}, \dots, x_{n})^{\intercal} $, we can solve for each component of this vector by ...
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Morphisms between compact quantum groups
Let $(A, \Delta_A)$ and $(B, \Delta_B)$ be two compact quantum groups (in the sense of Woronowicz). I would be tempted to define a morphism $(A, \Delta_A) \to (B, \Delta_B)$ to be a unital $*$-...
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Flips in the Minimal Model Program
In order get a minimal model for a given a variety $X$, we can carry out a sequence of contractions $X\rightarrow X_1\ldots \rightarrow X_n$ in such a way that that every map contracts some curves on ...
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Is an interpretation mathematics (fit for publication)?
Background
I am a mathematician with two published papers. The first is based on my PhD thesis and generalised a tool to a more general setting. The thesis was cited a number of times by the time the ...
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Geometric imagination of differential forms
In order to explain to non-experts what a vector field is, one usually describes an assignment of an arrow to each point of space. And this works quite well also when moving to manifolds, where a ...
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What intuitive meaning "determinant" of a divergency (divergent integral or series) can have? [closed]
I am working on the algebra of "divergencies", that is, infinite integrals, series and germs.
So, I decided to construct something similar to determinant of a matrix of these entities.
$$\...
- 9,312
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votes
2
answers
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Grothendieck says: points are not mere points, but carry Galois group actions
Apologies in advance if this question is too elementary for MO. I didn't find an explanation of these ideas in any algebraic geometry books (I don't know French).
The following is an excerpt from ...
CommunityBot
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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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How to explain the concentration-of-measure phenomenon intuitively?
One way to phrase the
"concentration-of-measure"
phenomenon is that,
for a Euclidean sphere $S^d$ in $d$ dimensions, for large $d$,
"most of the mass is close to the equator, for any equator."1
Q. ...
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0
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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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