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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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1answer
72 views

Meaningful interpretation for fixed point of a probability generating function

Suppose $f$ is the probability generating function for the Galton-Watson branching process. What intuition makes the fact that $f(s) = s$ is the probability of extinction obvious? Moreover, can one ...
2
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1answer
84 views

volume of parallelotope in $L^2(\mathbb R).$ [closed]

Let $L^2(\mathbb R)$ is complex Hilbert space with standard inner product. Does it make sense to talk of volume of parallelotope formed by following vectors in $L^2(\mathbb R):$ say, e.g., $$\{ f(...
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0answers
119 views

Best proof of Artin approximation?

I'm trying to learn deformation theory, where the algebraic Artin approximation theorem is crucial. However, the proofs I've seen* seem to go like: Keep reducing the theorem until one is in a ...
3
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2answers
144 views

Orthogonal Polynomials and Sturm Liouville operators

Classical Orthogonal polynomials (e.g., Hermite, Legendre) are eigenfunctions of Sturm Liouville operators. For example, define $L[u]=u''-xu'$, then the $n$-th order Hermite polynomial satisfies $...
3
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1answer
170 views

If $A$ is a (shifted) Poisson algebra, what does $A[\varepsilon]$ represent?

I have a question which is not really precise, unfortunately. Let $A$ be a Poisson $n$-algebra, i.e. a graded commutative algebra with a Lie bracket of degree $n-1$ s.t. the bracket is a biderivation ...
6
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0answers
189 views

“A typical pair of two-dimensional surfaces in four dimensions will intersect at finitely many points”

The title quotes C.H. Taubes in the Princeton Companion to Mathematics (p.404). I found this surprising despite the natural lower-dimensional analog (a typical pair of loops in $\mathbb{R}^2$ will ...
8
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0answers
240 views

Why are the open and closed adic discs defined the way that they are?

The closed adic disc is defined as $Spa(\mathbb{Q}_p\langle T\rangle,\mathbb{Z}_p\langle T\rangle)$, and the open adic disc is defined to be the fiber $Spa(\mathbb{Z}_p[[T]],\mathbb{Z}_p[[T]])_{\eta}$ ...
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1answer
150 views

Intuition for coercive functions

I have been working with $\Gamma$-convergence for some time now; it has lead me to wonder: What is the intuition behind coercive functions?
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1answer
569 views

How to visualize a Witt vector?

As the question title asks for, how do others "visualize" Witt vectors? I just think of them as algebraic creatures. Bonus points for pictures.
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2answers
199 views

Understanding reduced suspension of $S^1$ [closed]

I know this is just $S^2$. To see it, I use the CW structure of $S^1$ x $S^1$ , consisting of one 0-cell, two 1-cells and a 2-cell. Then since the reduced suspension is the cartesian product ...
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3answers
413 views

Intuition behind the Canonical Projective Resolution of a Quiver Representation

Let $Q$ be a finite acyclic quiver, and $X$ some representation of $Q$. For $i \in Q_0$ define the $kQ$-modules $P_i = kQe_i$, and $X(i) = e_i X$. The representation $X$ has a canonical projective ...
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0answers
148 views

Should the power series solution to $y' = y, y(0) = 1$ be obvious? [closed]

My Understanding: I would derive the Poisson random variable as follows: I consider an experiment which consists of a continuum of trials on an interval $[0,t)$. The result of the experiment takes ...
3
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1answer
185 views

Few questions regarding Heath-Brown's identity

Heath-Brown's identity states: Let $K \geq 1, z \geq 1.$ Then for any $n < 2 z^K$ we have $$ \Lambda(n) = - \sum_{1 \leq k \leq K} (-1)^k {{K}\choose{k}} \sum_{ \substack{ m_1 \cdots m_k n_1 \cdots ...
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4answers
542 views

What are Lie groupoids intuitively?

I am trying to understand about Lie groupoids but not able to get feeling for what it actually is. So, question here is, What are Lie groupoids? How similar are they to Lie groups, Groupoids and ...
6
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1answer
87 views

The Giraud-Benabou construction for splitting fibrations

I'm currently reading "Revisiting the categorical interpretation of dependent type theory" and they give a very terse description of the Giraud-Benabou construction: For a fibration $p : \mathbb E \...
18
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1answer
2k views

Reasons behind assuming the existence of Siegel zeros can be used to prove something stronger than assuming GRH?

There are few results that I am aware of where one can prove something stronger by assuming the existence of Siegel zeros than by assuming the GRH. For example Heath-Brown proved the existence of ...
18
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2answers
1k views

intuition for hochschild homology

According to this post Intuition for group homology, I wonder what is the intuition for Hochschild homology. The Hochschild homology is defined as the homology of this complex chain. Given a ...
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2answers
570 views

Is there a simple system that has $\text{SU}(3)$ symmetry?

The buckle at the end of a belt has $\text{SU}(2)$ symmetry, if the rotations around the three coordinate axes are taken as generators. See, for example, the paper by Hart, Francis and Kauffman, ...
16
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1answer
1k views

Geometric intuition for Fontaine-Wintenberger?

I asked my advisor the question in the title. He told me it was a stupid question and that I should focus on my research. Thus we're asking here. The statement of Fontaine-Winterberger, per their ...
2
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3answers
166 views

The independence of all sub-paths between adjacent returns to origin in random walks

In random walks, a path may return to origin for the $r$-th time in $n$-th step ($r$ is given but $n$ is not given), and under this condition, these $n$ steps can be split into $r$ sections where each ...
2
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0answers
123 views

Right adjoint completions

Forgive me if this question is not well thought out. I don't know how else to ask it. The nlab page on completion gives some examples of completions which are left adjoints. These completions are "...
5
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0answers
251 views

Grothendieck construction and coends

In category theory, both the Grothendieck construction and coends are represented by a sort of "integral sign", respectively: $$ \int F $$ for a functor $F:C\to\mathbf{Cat}$, and: $$ \int^x G(x,x) $$ ...
24
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2answers
881 views

Intuition for symplectic groups

My question essentially breaks down to How do you, a working mathematician, think about (real) symplectic groups? How do you visualize symplectic (linear) transformations? What intuition do you ...
10
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3answers
1k views

Why linearization leads to arithmetization?

Sorry for this question, but I think it is really important the intuition here. Motives can be seen as the 'best' way of linearizing the study of schemes, des-composing them into "cohomological atoms"...
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2answers
268 views

Non-trivial examples of taking the exponential of an integral

This question is inspired by a recent course I did on random matrix theory and also from common mistakes high-schoolers make in algebra :). In random matrix theory, one often encounters somewhat ...
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0answers
397 views

Mumford's intuition for flatness

In Mumford's book Algebraic Geometry II, he writes on page 179..."In order to get at what I consider the intuitive content of "flat" we need first a deeper fact..." After the deeper fact is proven he ...
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3answers
1k views

Understanding Vaughan's Identity

Vaughan's identity https://proofwiki.org/wiki/Vaughan%27s_Identity is a very useful identity in analytic number theory. The identity expresses the von-Mangoldt function $\Lambda(n)$ as a sum of ...
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1answer
545 views

Intuition behind the Dehn Invariant

EDIT: as pointed-out below, this has been posted on math.stackexchange. I'll leave it up to the community whether or not to delete this question, but I do think there is room for a more technical ...
3
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0answers
213 views

Intuition behind results in Mumford's “Lectures on curves on an algebraic surface”, II

NOTE: This is a followup to my question here. These are some questions concerning Mumford's "Lectures on curves on an algebraic surface". We concern ourselves with questions of the Picard variety $P$...
6
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1answer
637 views

Intuition behind results in Mumford's “Lectures on curves on an algebraic surface”, I

These are some questions concerning Mumford's "Lectures on curves on an algebraic surface". We concern ourselves with questions of the Picard variety $P$, and its dimension, of a complete nonsingular ...
3
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0answers
252 views

Intuition behind Artin/Grothendieck's Riemann existence theorem

The following result is due to Grothendieck and Artin and can be found in SGA 1. Let $X$ be a normal scheme of finite type over $\mathbb{C}$. Let $\mathfrak{X}'$ be a normal complex analytic space, ...
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0answers
97 views

Intuition for analysis of basic gradient descent variants

I'm currently learning the basic variants of gradient descent for minimizing convex functions under various assumptions, such as Lipschitz, smooth, strongly-convex, ... . I've found various sources ...
4
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1answer
411 views

Reason for putting log weight for exponential sums over primes?

In analytic number theory, for example as in ternary Goldbach problem via circle method, when one has to deal with exponential sums over primes often people use von Mangoldt function or log weight. In ...
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0answers
94 views

Posets as (0,1)-categories

I am reading on the nLab that a poset can be seen as a (0,1)-category. I was assuming all along that an ($n$,$r$)-category were a category where all morphisms of order larger than $n$ are trivial. ...
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1answer
230 views

Tarski-Grothendieck in the cumulative hierarchy

How can the Grothendieck-Tarski axiom seen to be true in the cumulative hierarchy of sets? What are intuitions that would convince us that this axiom is true?
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7answers
3k views

Intuition behind Harmonic Analysis in Analytic Number Theory

As far as I know, in analytic number theory, harmonic analysis appears often. The thing is that I would see the proof of some results where they use harmonic analysis, and I can follow the argument ...
5
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1answer
277 views

Decoding a Remark of Gödel on Complexity Theory

In Gödel's Collected Works (Vol 2), there is a discussion of von Neumann which was brought about by a query, made to Gödel, concerning the existence of a Turing machine which is so complex that its ...
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0answers
144 views

Intuition for Clifford Group

Clifford group $\Gamma$ of a Clifford algebra $C\ell (V,q)$ is defined to be the set of elements $g$ in $C\ell (V,q)$ for which there exists an inverse $g^{-1}$. This group can be represented by ...
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1answer
2k views

Mochizuki's Gaussian Integral Analogy

In his latest 115-page overview, Mochizuki spends some time explaining "alien copies" by the analogue of evaluating the Gaussian integral by squaring it and introducing a second variable/dimension. In ...
4
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0answers
274 views

Intuition: Smooth functions on Banach Spaces

On finite-dimensional vector spaces, we all have a reasonable idea of which functions are likely to be $C^1$ or smooth. When it comes to differentiation on Banach spaces, I find that my `intuition' ...
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0answers
65 views

How quickly can we mutliply Cayley-Dickson hypercomplexes?

Assuming that all of the coordinates of two Cayley-Dickson Hypercomplex numbers are non-negative integers less than a prime $p$, how quickly can we multiply these numbers? I'm also interested in what ...
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2answers
324 views

If mathematics is logic and intuition, then [closed]

I am just wondering why Mathematics is often defined as The study of Structures, Logic and Numbers which I can concur with but still retain various questions in mind. I am a postgraduate student of ...
5
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1answer
203 views

Norm vs A-norm in non-Archimedean Functional Analysis

Let $K =(K,| \cdot |)$ be a non-Archimedean valued field. Let $E$ be a $K$-vector space. A norm on $E$ is a map $||\cdot||:E\to[0,\infty)$ such that: $||x||=0$ if and only if $x=0$, $||\lambda x||=|...
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0answers
128 views

Is $n+\frac {1}{2}$ in Kendall-Mann numbers and quantum harmonic oscillator related?

It is known that quantum harmonic oscillator is connected to the symmetric group of infinite order which is isomorphic to the permutation group. According to Cayley's theorem any finite group is ...
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0answers
89 views

Intuition for universal quotient maps

The universal quotient maps are precisely the descent morphisms in the category of topological spaces. In some papers of Janelidze, Tholen, Sobral, and Reiterman (see for instance Reiterman-Tholen), ...
3
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1answer
324 views

Free operads and trees

I am currently working in my PhD thesis, and it became necessary to understand some facts about free symmetric operads. Hence I started to study this subject by myself, following Kapranov & ...
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0answers
48 views

A proper class for smooth chaotic function

This might be a little, soft, but I'll try Consider the interval $I=[-1,1]$. We will define a chaotic function $f:\mathbb{R}_+ \times I \to \mathbb{C}$ in the following traditional way: For every $...
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0answers
172 views

The application of recursive SVD [closed]

Given an n*m matrix A, the SVD decomposition of A is ${\rm SVD}(A)$= $USV^t$. The application of SVD to the product of U and S gives as a result the same matrices multiplied by the identity matrix, i....
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4answers
1k views

Expert, Intuitive, Organizing Analogies

In learning a new area it is very helpful to have high-level intuitive analogies that keep track of the various parts of an important argument or strategy. Experts have a store of such things, and ...
38
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2answers
3k views

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 ...