Mathematical intuition is the instinctive impression regarding mathematical ideas which originate naturally without regard to formal mathematical proofs. It may or may not stem from a cognitive rational process.

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Intuition of Immersed versus Embedded Submanifolds

The definitions I read in Lee's Smooth Manifolds is: Embedded Submanifold: $S\subset M$ is an embedded submanifold if $S \to M$ is an embedding. Immersed Submanifold: $S\subset M$ is an immersed ...
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Geometric Intuition about the relation between Clifford Algebra and Exterior Algebra

It is common to see a relation being established between the Clifford Algebra and the Exterior Algebra of a vector space. Recently reading some texts written by Physicists I've seem applications of ...
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Geometric meaning of the contact condition?

I am trying to understand contact structures. The definition of a contact manifold is this: Let $M$ be a $2n + 1$-manifold and let $\omega$ be a differential $1$-form such that $\omega \wedge (d\...
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I don't understand why $\oint_\gamma f\, dz=0$ holds for holomorphic functions.

I've recently learned a proof of Cauchy's Integral Theorem, i.e, If $U\subseteq \Bbb C$ is open and simply connected, $f:U\to\Bbb C$ is holomorphic and $\gamma$ is a closed curve, $\gamma\...
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How do you visualize $\mathcal{P}(1)$ in constructive mathematics?

If I understand correctly, constructive mathematics doesn't prove that the powerset $\mathcal{P}(X)$ of a set $X$ is a Boolean algebra; in general, all we can say is that its a Heyting algebra. This ...
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Intuitive explanation of $(1-x)^{-a-1}=\sum_{j=0}^{\infty}{{a+j} \choose j}x^j$

Could anyone please explain me the reasoning behind this formula? $(1-x)^{-a-1}=\sum_{j=0}^{\infty}{{a+j} \choose j}x^j$ Thanks so much!
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36 views

What is an intuitive way to understand Cayley's formula?

Is there any intuition behind Cayley's formula $n^{n-2}$ for the number of spanning trees of a graph?
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What is the intuition behind covering spaces?

I've come to study this definition and become interested on the intuition behind it mainly because of the study of spinors, motivated by Quantum Mechanics. The definition of covering space is as ...
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Limit of $n-1$ measure of the boundary of a sphere

The measure of a sphere of radius $R$ centered in $0_{\mathbb{R}^n}$ in $\mathbb{R}^n$ is \begin{array}{l l}\int_{B_0(R)}dx_1\ldots dx_n & =\int_0^R\rho^{n-1}d\rho \int_{-\frac{\pi}{2}}^{\frac{\pi}...
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Why are second order linear PDEs classified as either elliptic, hyperbolic or parabolic?

Is there a geometric interpretation of second order linear partial differential equations which explains why they are classified as either elliptic, hyperbolic or parabolic, or is this just a naming ...
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42 views

Every fiber bundle with Cantor set fiber is the suspension of a homeomorphism of the Cantor set.

I've heard that every fiber bundle (over $\mathbb S^1$?) with Cantor set fiber is the suspension of a homeomorphism of the Cantor set. Can someone explain the intuition behind the fact? Is there a ...
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Intuition of a k-connected graph?

The formal definition of a k-connected graph $G$ is: $\nexists x\subseteq V(G)$ with $|x| \le k - 1$ such that $G-x$ is disconnected. What is the intuition behind this? What does it mean to be ...
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27 views

Intuition behind negative radians in an interval [duplicate]

Say a function's domain is $[−\pi,\pi]$. How should I interpret this interval? It starts from where? To where? In what direction?
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How to understand the notion of a differential of a function

In elementary calculus (and often in courses beyond) we are taught that a differential of a function, $df$ quantifies an infinitesimal change in that function. However, the notion of an infinitesimal ...
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How does the Fourier transform of a “zero avoiding” function look?

Let $n$ be a very large positive integer. Let $f \in\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function, satisfying $0\leq f\leq1$, and supported on $[-n,-\frac{1}{n}]\cup[\frac{1}{n},n]$ such ...
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71 views

Intuition about formal brances of a curve at a point

Consider an algebraic surface $X$ and a curve $Y\subset X$. Here $X$ is a $K$-scheme integral of finite type of dimension $2$ and $Y$ is a closed subscheme of dimension $1$. Fix a closed point $x\...
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43 views

How to describe a polynomial relation on $\mathbb{P}(\bigwedge^k V)$, and if the Zariski topology is canonical

I am working with the space $\mathbb{P}(\bigwedge^k V)$, where $V$ is some $n$ dimensional vector space over some field K. In here I want to define a variety, ie a solution to a set of polynomials. ...
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Visualizing the quotient of a torus and a circle

We were asked to compute the homology for the double torus, $X$, and a circle around one of the loops, $B$, of the torus (not a circle between the two halves of the torus) and were told that this ...
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Intuition probability of two pairs in poker dice

I need some intuition for an element of the following question: The answer starts with this: I would like to know how they get to 6 choose 2. If I write it out (1122, 1133, 1144, 1155, 1166, 2233,...
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Areas under the graphs of $\frac{1}{x}$ and $\frac{1}{x^2}$ from $1$ to $\infty$

A simple evaluation of the definite integral tells us that the area under the graph of $[\frac{1}{x}]^2$ from $1$ to $\infty$ is finite whereas that of $\frac{1}{x}$ for the same limits is infinite. ...
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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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Absolute value of a random variable

I have never encountered this concept before. Is this equation valid for $y>0$? $$\mathbb{P}(|X|>y) = \mathbb{P}(-|X|<y<|X|)$$ What about this? $$\mathbb{P}(|X|>y) = \mathbb{P}(X>...
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What comes after diagram chasing?

An early edition of Lang's algebra textbook gives the famous exercise to Take any book on homological algebra, and prove all the theorems without looking at the proofs given in that book. Here ...
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Matrix --> Scalar Valued Function: Differentiation

In class, we called a real-valued function from the space of matrices to the reals $f: \mathbb{R}^{m \times n} \rightarrow \mathbb{R}$ differentiable at $\mathbf{X}$ if: $$\lim_{\mathbf{H} \to \...
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Is there some geometric intuition for the quotient $G/Z(G)$, where $G=GL_n(\mathbb{R})$?

Let $G=GL_n(\mathbb{R})$ be the $n$th general linear group. Its center $Z(G)$ is given by all scalar matrices $aI$ with nonzero determinant. How can I get an intuitive picture of $G/Z(G)$? I know that ...
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Is differentiation with respect to a vector always defined componentwise?

When one takes the derivative of a function $f$ along the direction of some vector $\mathbf{v}$, i.e. the directional derivative of $f$ along $\mathbf{v}$ this operation is defined componentwise, i.e. ...
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47 views

Taking a derivative of a function with respect to another function

I read a set of notes recently (unfortunately I can't find the link) in which the author made a statement of the form "differentiation of a function with respect to a function doesn't make sense". By ...
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48 views

Intuition of weak solutions of elliptic equations in divergence form

Let $\Omega \subset \mathbb{R}^{n}$ a domain, and consider the following equation (1) $-D_{j}(a_{ij}D_{i}u) = 0$ (Einstein notation) The function $u \in H^{1}(\Omega)$ is a weak solution of (1) if $...
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Intuition about antisymmetrizing tensor equations

I was looking at the symmetries of the Riemann tensor, and tried to prove a couple of properties, namely If $\nabla$ is torsion-free, then: (i) $R^a_{\,[bcd]}=0$, and (ii) $R^a_{\,b[cd;e]}=0$. ...
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Intuitively, why does $\dfrac{a}{c} = \dfrac{1}{\dfrac{c}{a}}$?

For intuition, I reference objects. Imagine making a dessert with: $a$ as apples and $c$ as chestnuts. Question. How and why is $\dfrac{a}{c} \qquad (3) \quad = \quad\dfrac{\color{red}{1}}{\dfrac{c}{...
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Intuition for Adeles and Ideles

I'm currently studying some class field theory and read about the notion of adeles and ideles. However, the object seems a bit arbitrary to me; is there a natural way to think about the adele-ring? ...
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Intuitive proof for a Combinatorial Problem

Given a set $S$ such that $|S|=N$ and $S$ contains exactly $K$ $0$s $(K >0)$ and $N-K$ $1$s, then exactly half of the subsets of $S$ contain an $odd$ number of 1s, $indepedent$ of the value of $K$....
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Volume of a cone [duplicate]

I have a simple question about the formula for the volume of the cone. Let $C$ a cone, which base has radius $r$ and height equal to $h$. So its volume can be compute by the formula: $$\text{Vol}(C)=\...
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How to figure out the “idea behind” proofs in analysis?

I'm taking a course in Real Analysis, and for the most part I can follow the rote mechanics of a proof (e.g. manipulation to produce a chain of inequalities as desired, etc.), but I have difficulty ...
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Combinatorial Geometry explanation

I do not understand what is going on in $(4)$: for every flat $E \in \mathcal F$, $E \ne X$, the flats that cover $E$ in $\mathcal F$ partition the remaining parts. What is meant by "the flats ...
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Is this inverted ANOVA possible?

If my understanding is correct, in an ANOVA usually you start with a null hypothesis that all groups have the same mean. You then calculate the within group and between group variance, and do an F-...
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What is the appropriate ANOVA test for this situation?

So in this experiment I have have 7 devices. The response of each device (call it Y) are each measured ~20 at a 4 different levels of an independent variable (we'll call X). The Y response is known ...
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What are the intuitions about matrix algebra operations?

In my current data analysis problem I am using models with complicated penalty structure that is a result of operations on some matrix $Q$. I do know definitions of basic matrix operations: $Q^T$ $...
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What is the intuitive meaning of a determinant? [duplicate]

I know how to calculate a determinant, but I wanted to know what the meaning of a determinant is? So how could I explain to a child, what a determinant actually is. Could I think of it as a measure ...
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Why Taylor Series or any other approximation method give us approximation of function? Why not give exact equivalent of function?

Lately I am started studying approximation of functions by polynomials and the need for approximation of functions? But what I failed to understand and books did not explain me is that why finding ...
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Intuition about the second isomorphism theorem

In group theory we have the second isomorphism theorem which can be stated as follows: Let $G$ be a group and let $S$ be a subgroup of $G$ and $N$ a normal subgroup of $G$, then: The ...
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What's the intuition behind the identities $\cos(z)= \cosh(iz)$ and $\sin(z)=-i\sinh(iz)$?

I'm trying to understand in an intuitive manner the relationship between the circular and hyperbolic functions in the complex plane, i.e.: $$\cos(z)= \cosh(iz)$$ $$\sin(z)=-i\sinh(iz)$$ where $z$ is ...
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Practical intuition for standard deviation

Simply: What is the intuition you get about the data if you are given standard deviation? More detailed: It is easy to imagine some information about the data if you are being told for example mean ...
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Lagrange's theorem intuition

I cannot grasp the intuition behind |G|/|H|=[G:H]. Starting from the equivalence relation x~y if and only if x^(-1)*y is in H, I can see a sort of division, but in my mind, the equivalence relation ...
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Motivation behind Arithmetic Mean

I know that the arithmetic mean $(x_1+x_2+...+x_n)/n$ is the value that minimizes $f(x)=\sum_{k=1}^n (x_k-x)^2$; however, I'm looking for an intuitive relationship between the mean and $g(x)=\sum_{k=1}...
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Understanding limits and how to interpret the meaning of “arbitrarily close”

I have read several introductory notes on limits of functions, and in all of them they introduce the notion of a limit of a function $f(x)$ by discussing what happens to the value of $f$ as $x$ ...
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Why this intuition about natural transformations corresponds to its formal definition?

Almost everywhere people introduce the notion of natural transformations between two functors $ F$, $ G$ : $ \textbf C \Rightarrow \textbf D$ by examples like what follows: This is the intuition ...
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Covering maps as bundles

One geometric way to see a continuous map (or any set function really) is as a "fiber bundle" with the usual picture of a comb - the base space indexes the fibers of the map and there's a nice picture ...
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Pulling back along surjective étale maps vs being “locally in $\mathcal M$” vs being “locally in $\Sigma \mathcal M$”

(Closely related) This question centers around section 6.5 of Borceux and Janelidze's Galois Theories. Definition 1. Let $\mathcal M$ be a class of arrows in a category (in our case $\mathsf{Top}$). ...