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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List of Local to Global principles

What are some of the local to global principles in different areas of mathematics?
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The Degree of Zero Polynomial.

I wonder why the degree of the zero polynomial is $-\infty$ ? I heard that, it is $-\infty$ to make the formula $\deg(fg)=\deg(f)+\deg(g)$ hold when one of these polynomials is zero. However, if that ...
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What is characteristic time?

What is characteristic time? Where is it useful? From this answer by Joriki: The characteristic time is usually defined to be the time in which a quantity decreases by $1/e$. Why is ...
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1answer
29 views

Formalizing continuously indexed spaces in fiber bundles?

This MSE question asks for clarification of the local triviality condition imposed in the definition of a fiber bundle. As mentioned there, the point of local triviality seems to somehow ensure a ...
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1answer
67 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 ...
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35 views

How much can we compress a sequence of bits?

Suppose that we have a sequence of bits defined for each natural less than $n$. For example, if we have 3 bits (either 0 or 1), we can represent the sequence as a function of $x < 4$ in the ...
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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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1answer
213 views

Intuition behind variation of parameters method for solving differential equations

I have used the variation of parameters method (and have been taught it, although not hugely in depth) and I was wondering if I've understood the intuition behind it. In particular I've been thinking ...
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273 views

Intuition behind functional dependence

What is the intuition behind functional independence ? (This is defined in the following way: Let $k\leq n$. The $C^1$ functions $F_1,\ldots,F_k:\mathbb{R}^n\rightarrow \mathbb{R}$ are functionally ...
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129 views

Connecting a vector space to its dual - why?

Can someone explain to me - intuitively - why embedding a vector space into its dual should naturally fix its geometry? I mean, I can run the usual statements through my mind - "The injection into the ...
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4answers
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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 = ...
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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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Chain Rule Intuition

We know that the chain rule is used to differentiate a composite function ,say $$f(x) = h(g(x))$$ It's defined as the derivative of the outside function times the derivative of the inner function or ...
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1answer
92 views

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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2answers
156 views

Law of Excluded Middle Controversy

I was reading an introductory book on logic and it mentioned in passing that the Law of Excluded Middle is somewhat controversial. I looked into this and what I got was the intuistionists did not ...
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3answers
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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, ...
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279 views

Explicit formula for floor(x)?

In number theory we have so-called explicit formula's in terms of the Riemann zeta zero's. For instance to count the sum of the logarithms of the primes below some given integer. (second Chebyshev ...
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1answer
23 views

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 ...
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3answers
51 views

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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What are Some Tricks to Remember Fatou's Lemma?

For a sequence of non-negative measurable functions $f_n$, Fatou's lemma is a statement about the inequality $$\int \liminf_{n\rightarrow \infty} f_n \mathrm{d}\mu \leq \liminf_{n\rightarrow ...
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4answers
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The standard role of intuitive numbers in the foundations of mathematics

In my career I've been formed mostly in the formal side of mathematics, that is, standard set theory and every classical branch of mathematics that uses set theory. However, I am not quite sure about ...
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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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1answer
34 views

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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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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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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1answer
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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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1answer
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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 ...
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1answer
28 views

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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What are Aleph numbers intuitively?

I cannot get my head around the concept of the `types' of Aleph infinity. What is an easy intuitive way to see when you are given the integer numbers $\aleph_0$ the $\aleph_1$ will follow?
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1answer
96 views

Does a retraction really “retracts” something?

I wonder what is the intuition behind the definitions of a section and retraction in Category Theory. From Awodey's book: Definition 2.7. A split mono (epi) is an arrow with a left (right) ...
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1answer
64 views

Synthetic differential geometry and formally étale morphisms?

Upon looking throug Kostcki's synthetic differential geometry notes, I stumbled upon the following definition. (Here $R$ is the geometric line, $W$ is a Weil algebra, and $\operatorname{Spec}_RW$ is ...
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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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Puzzle: Players $A,B,C,D$ are in a line

Players $A,B,C,D$ stands in a line. Players $A, D$ do not move. round $1:$ player $B$ moves one distance closer to the midpoint of $A$ and $C$ round $2:$ player $C$ moves one distance closer to ...
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1answer
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Sheaf cohomology intuition

I am working on understanding specifically what the $n^{th}$ Cech cohomology group $H^n(\mathcal{U}, \mathcal{F})$ measures, where $\mathcal{U}$ is a locally finite open cover on a topological space ...
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0answers
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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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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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1answer
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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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2answers
451 views

Characteristic function of union of two sets formula and intuition

From http://topologicalmusings.wordpress.com/2008/03/20/inclusion-exclusion-principle-counting-all-the-objects-outside-the-oval-regions-2/ Is there an easier proof or way to calculate $1[A \cup ...
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2answers
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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, ...
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2answers
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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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Trying to understand why circle area is not $2 \pi r^2$

I understand the reasoning behind $\pi r^2$ for a circle area however I'd like to know what is wrong with the reasoning below: The area of a square is like a line, the height (one dimension, length) ...
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1answer
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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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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 ...
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1answer
40 views

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) = ...
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Intuitive way to understand covariance and contravariance in Tensor Algebra

I'm trying to understand basic tensor analysis. I understand the basic concept that the valency of the tensor determines how it is transformed, but I am having trouble visualizing the difference ...
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What is the intuition behind / How can we interpret the eigenvalues and eigenvectors of Euclidean Distance Matrices?

Given a set of points $x_1,x_2,\dots,x_m$ in the euclidean space $\mathbb{R}^n$, we can form a $m\times m$ Euclidean Distance Matrix $D$ where $D_{ij}={\|x_i-x_j\|}^2$. We know a little bit about ...
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Geometric Interpretation: Parallel forms are harmonic

Let $(M,g)$ be a Riemannian manifold. The canonical volume form $\mu=\sqrt{\det g_{ij}}\mathrm{d}x^1\wedge\dots\wedge\mathrm{d}x^m$ is parallel w.r.t. the induced Levi-Civita conection $\nabla$ ...
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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 ...