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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266 views

Why is better to work with the spectrum of prime ideals than with the maximal one, for example in the definition of affine scheme.

When we have an algebraic variety we can identify the points of the variety with maximal ideals of the coordinate ring. I would like to know why it is more natural to define the main structure of ...
22
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2answers
1k views

Can we think of a chain homotopy as a homotopy?

I'm taking a course in algebraic topology, which includes an introduction to (simplicial) homology, and I'm looking for a bit of intuition regarding chain homotopies. The definitions I'm using are: ...
4
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1answer
153 views

How can a subfield of an abelian extension fail to be cyclic when subjected to a norm-like condition. (How can I understand the supplied explanation)

I recently posted a question on MathOverflow (if you're interested it can be found here). While some answers were quickly produced there were a few points that I found confusing. I requested some ...
8
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4answers
1k views

Is there any motivation for Zorn's Lemma?

I have been reading Kreyszig's book on functional analysis, where it uses Zorn's lemma to prove the Hahn Banach theorem. However I don't quite get what Zorn's lemma is saying. I understand that it ...
3
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1answer
724 views

Combinatorial interpretation of Delannoy numbers formula

The Delannoy number $D(a,b)$ can be defined as the numbers of paths on $\mathbb Z^2$ from $(0,0)$ to $(a,b)$ using only steps $(0,1)$, $(1,0)$ and $(1,1)$. It is straightforward to see that they ...
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19answers
13k views

How do you explain the concept of logarithm to a five year old?

Okay I understand that it cannot be explained to a 5 year old. But, how do you explain the logarithm to primary school students?
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2answers
156 views

Identification of complex plane as $R^2$.

If we have following identification: $$(x,y)\to (z,\overline{z})$$ We will have $$\frac{\partial}{\partial x}= \frac{\partial}{\partial z}+\frac{\partial}{\partial \overline{z}}$$ and ...
8
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2answers
2k views

An explanation of spherical harmonics?

Could somebody please explain spherical harmonics in a simpler manner than it is demonstrated on various websites (like the Wikipedia page which simply overflows my buffer with symbols). I've tried ...
7
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0answers
170 views

Why are injective $\mathscr{O}$-modules flasque?

Let $X$ be a topological space, and let $\mathscr{O}$ be a sheaf of rings on $X$. It is easy to verify that the functor $\Gamma (U, -) : \textbf{Mod}(\mathscr{O}) \to \textbf{Ab}$ is representable, ...
8
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1answer
355 views

How can one visualize topological quotients or develop intuition for handling them?

This is a very open-ended question. I regret that -- I would like to be able to make it more precise, but I don't know how. I would appreciate comments on how to improve this question. I had my first ...
8
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1answer
633 views

Information captured by differential forms

My advanced calculus class is currently doing differential forms and I have a hard time really understanding what they are all about. I can read the proofs of the theorems given in Rudin's PMA chapter ...
3
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1answer
253 views

Intuition on the definition of “rational maps”

I'm studying some representation theory on $S_n$ and $GL(V)$ and tensor spaces, and have come across a lot of material involving rational representations. I'm not really an algebraic geometer by ...
3
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1answer
162 views

A Formal and Precise treatment of Simplification?

I am looking to gain a deeper understanding of, and increase my own skill in "Mathematical Simplification". But I've been finding the concept overly vague and haven't been able to find any good ...
11
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2answers
387 views

Is there a geometric interpretation of $F_p,\ F_{p^n}$ and $\overline{F_p}?$

I have been doing some exercises about finite fields lately and I think I've obtained some understanding of what they are. What seems to be missing though is some kind of picture. Learning to work ...
7
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1answer
535 views

Canonical example of a cosheaf

Sheaves can, like all modern mathematical constructions and abstractions, be counterintuitive beasts but, like all such constructions, a few examples can allow one to visualise them simply as a ...
13
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4answers
746 views

Why is Lebesgue integration better suited for convergence axioms?

I am trying to understand Lebesgue integration here. Here you basically make equal splits on the y-axis instead of the splits on the x-axis that Riemann does. I understand the proofs of the limit ...
5
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4answers
10k views

What is the purpose of the standard deviation?

I don't have any knowledge of statistics beyond high school common sense. Why is the standard deviation usually seen in combinatorics textbooks, and why is the standard deviation defined ...
8
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2answers
445 views

Intuition for limits

My basic intuition for limits/colimits was "limits suck up, colimits suck down". Now, having seen colimits used in presheaf categories, algebraic geometry, and topology, I have much clearer intuition ...
6
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2answers
1k views

The definition of independence is not intuitive

In the book "Introduction to Probability" by J. Charles M. Grinstead and Laurie Snell independent events are introduced in the following way: "It often happens that the knowledge that a certain event ...
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2answers
1k views

Why do we need (the abstract concept of) random variables (in discrete probability models)?

What we defined: Suppose we have a (discrete) probability model $\left(\Omega,P\right)$, where $P$ is the probability function (at least, that was the way it was introduced in a course I took; that ...
8
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8answers
2k views

Conceptual/Graphical understanding of the Fourier Series.

I've been reading about how the Fourier Series works, so like how the orthogonality cancels out all but the one that we're looking for. I've read derivations of the Fourier Series. What I would like ...
32
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4answers
910 views

Why do we look at morphisms?

I am reading some lecture notes and in one paragraph there is the following motivation: "The best way to study spaces with a structure is usually to look at the maps between them preserving structure ...
1
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1answer
143 views

Curvature and Radii

In my handout it is said that a circle $\Gamma(s)$ that is tangent to second order to a curve $\xi:[a,b]\to \mathbb R^2$ with unit speed and with curvature $\kappa$, then the radius of $\Gamma(s)$ is ...
9
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3answers
5k views

How to come up with the gamma function?

It always puzzles me, how the Gamma functions's inventor came up with it's definition $$\Gamma(x+1)=\int_0^1(-\ln t)^x\;\mathrm dt=\int_0^\infty t^xe^{-t}\;\mathrm dt$$ Is there a nice derivation of ...
4
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1answer
221 views

Intuition behind growth rate of some functions

This one really crushed my intuition. Let say a function $f$ grows faster than a function $g$ if $ \lim_{n \to \infty} \frac{f(n)}{g(n)} = \infty $ Which of the following functions grows the fastest ...
56
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4answers
7k views

Do you prove all theorems whilst studying?

When you come across a new theorem, do you always try to prove it first before reading the proof within the text? I'm a CS undergrad with a bit of an interest in maths. I've not gone very far in my ...
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2answers
1k views

Serge Lang´s remarks on the superiority of algebra. What it actually means? [closed]

I read two comments of Lang that basically places algebra over other math subjects. One of this comments is on his calculus book preface (see Remark 1 below); I am not finding his other comment, but ...
63
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9answers
6k views

What makes elementary functions elementary?

Is there a mathematical reason (or possibly a historical one) that the "elementary" functions are what they are? As I'm learning calculus, I seem to focus most of my attention on trigonometric, ...
11
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1answer
464 views

What is the intuitive meaning of “conspiracy” in number theory?

Assuming very little number-theoretic background from my part, could you please explain me what is the intuitive meaning behind "conspiracy" in number theory? There is no formal entry on Wikipedia and ...
20
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2answers
2k views

Geometric intuition of tensor product

Let $V$ and $W$ be two algebraic structures, $v\in V$, $w\in W$ be two arbitrary elements. Then, what is the geometric intuition of $v\otimes w$, and more complex $V\otimes W$ ? Please explain for me ...
5
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2answers
319 views

what does following matrix says geometrically

Let $M\subset \mathbb C^2$ be a hypersurface defined by $F(z,w)=0$. Then for some point $p\in M$, I've $$\text{ rank of }\left( \begin{array}{ccc} 0 &\frac{\partial F}{\partial z} ...
1
vote
1answer
132 views

Lorentz reflection

What is a Lorentz reflection of $\mathbb R^3$? Is there a way to visualize it? Suppose I have a plane, P, what would (Lorentz) reflecting in it differ from (Euclid) reflecting in it? I know that the ...
27
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5answers
4k views

Jacobi identity - intuitive explanation

I am really struggling with understanding the Jacobi Identity. I am not struggling with verifying it or calculating commutators.. I just can't see through it. I can't see the motivation behind it (as ...
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2answers
101 views

Geometric explanation of the product metric

Can someone describe to me the geometric intuition behind using a mapping $$ ((x_1,y_1),(x_2,y_2)) \mapsto \frac{d_1(x_1,y_1)}{1+d_1(x_1,y_1)} + \frac{1}{2} \frac{d_2(x_2,y_2)}{1+d_2(x_2,y_2)} $$ to ...
3
votes
1answer
350 views

Intuitive interpretation of these differential forms

Let $\pi: S^2-\{N\}\to \mathbb R^2$ be the stereographic projection map. Let $\sigma:\mathbb R^2\to S^2-\{N\}$ be its inverse. Let $p\in S^2-\{N\}$ and $x_1,x_2\in$ the tangent space of $S^2$ Would ...
7
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10answers
21k views

Explain for students: Why does 0 mod n equal 0 (zero)?

I told my students that the mod operator basically gives the remainder of division, so upon seeing: 0 mod 10 Some students (apparently) reasoned that, "10 goes ...
5
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2answers
258 views

The area problem!

We have to find area of the quadrilateral formed by joining the point of intersection of the four quarter circles that are drawn from each vertex in a unit square. $\hspace{4cm}$ The challenge is ...
6
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3answers
449 views

$\wedge,\cap$ and $\vee,\cup$ between Logic and Set Theory always interchangeable?

In "$\wedge,\cap,\times$ and $\vee,\cup,+$ are always interchangeable?" It has been shown that arithmetic shouldn't be included. So the new modified question is: The analogy of $\wedge,\cap$ and ...
4
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2answers
273 views

$\wedge,\cap,\times$ and $\vee,\cup,+$ are always interchangeable?

Update : Should have left the Arithmetic out of this question, the new modified question is posted here : $\wedge,\cap$ and $\vee,\cup$ between Logic and Set Theory always interchangeable? ...
15
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3answers
3k views

Line integration in complex analysis

In normal line integration, from what I understand, you are measuring the area underneath $f(x,y)$ along a curve in the $x\text{-}y$ plane from point $a$ to point $b$. But what is being measured with ...
6
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2answers
633 views

Organization of the Learning Process

Sorry for off topic. I'll delete this topic immediately when community decides it's useless, however if anyone finds it's interesting, share your opinion with us. I just want to know your opinion ...
15
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6answers
6k views

What is the best way to develop Mathematical intuition? [closed]

I want to develop my pure mathematics knowledge and would like to know what is the best way to develop mathematical intuition? I am going through exercises that ask for proofs and I don't have the ...
7
votes
5answers
2k views

What's an intuitive way of looking at quotient spaces?

I understand the concept of $\mathbb{Z}/n\mathbb{Z}$, but I am having a really hard time understanding how this concept of quotients applies to vector spaces. Suppose $V = \mathbb{F}[x]$ is a vector ...
3
votes
1answer
283 views

Avoiding algebraic integration by geometric arguments

Is there a geometric way of seeing why the integral $\int\limits_{-\infty}^\infty (x^2+y^2+z^2)^{-{3\over 2}}dz={2\over x^2+y^2}$? Otherwise what is a good way of evaluating it algebraically?
12
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2answers
535 views

Importance of 'smallness' in a category, and functor categories

I feel like, having spent a little time doing category theory now, this is probably a silly question, but I keep coming up to many things (definitions, examples etc.) where smallness is required. I ...
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1answer
1k views

Intuition behind homology with general coefficients

We just went over homology with general coefficients in topology and did some of the usual examples ($\mathbb{Z}_2$ for projective space and manifolds being the big examples) which led me to wonder ...
4
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0answers
88 views

Ways to think about one-relator groups

What are some intuitive ways to think about one-relator groups? I am aware of the Freiheitsatz, and Bass-Serre theory. What I'm interested in are ways people who work extensively with one-relator ...
21
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3answers
5k views

Differential equations and Fourier and Laplace transforms

Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and ...
2
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3answers
207 views

Games with human edge [closed]

Which are some two- or one-player games, where humans far outperforms the best computer programs? And how does the relative edge scale with time allowed to think? (In time frame 1 sec to 8 hours per ...
6
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1answer
388 views

Ways to visualize the real numbers?

I was just wondering if there are any diagrams for visualizing subsets of the real numbers, or totally 'radically' different ways of looking at them as a real line? The model of the line relies on ...