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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Why are we interested in closed geodesics?

There's a lot of work about the existence, number and other properties of closed geodesics on a Riemannian manifold (belonging to some specific class of manifolds). In the case of geodesics ...
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5answers
2k views

Why is convexity more important than quasi-convexity in optimization?

In the mathematical optimization literature it is common to distinguish problems according to whether or not they are convex. The reason seems to be that convex problems are guaranteed to have ...
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1answer
129 views

Signs in binomial expansions

Edit the title as seems fit. $$\begin{align} (a^3+b^3) &= (a+b)(a^2 -ab+b^2) \\ &= (a+b)^3 -3ab(a+b) \end{align}$$ And so on and so forth. Right now, I only need these expansions in ...
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1answer
222 views

Explaining this particular version of the Implicit Function Theorem

I understand the general 'word' definitions of the Implicit Function Theorem and the simple examples such as the one on wikipedia but the version of the Implicit Function theorem given in our lecture ...
11
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2answers
723 views

Motivation behind the ingredients of First Cohomology group $H^1$

I started reading the Cohomology theory of groups. But I am not able to get any intuition or motivation behind the following : It is concerned with the formal definitions of crossed and principal ...
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1answer
268 views

Confusion about proof that first order logic without equality is not contradictory

I am having a problem understanding a proof from the field of mathematical logic. Seems like my brain cannot digest concepts from logic very well. I will quickly define some terminology and then ...
3
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2answers
422 views

$SO(3)$ Lie group

I'm a little stuck at the moment where to go next with this. I know that there is a fact that there is a curve in $SO(3)$, beginning and ending at the identity which cannot be deformed to the constant ...
2
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1answer
369 views

What is the intuition behind the Fermat-Euler's Theorem?

Can someone give me an intuition behind the working of Fermat-Euler's theorem? I am not looking for definition nor for proof (I know both of them). $$a^{\phi(p)} \equiv 1 \pmod p$$ This is what I ...
7
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1answer
821 views

Physical interpretation of the generating function for the Bessel functions.

It is well known that the generating function for the Bessel function is $$f(z) = \exp \left (\frac12 \left (z - \frac1z \right ) w \right ).$$ So, we have $$f(z) = \sum_{\nu = -\infty}^{\infty} ...
15
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8answers
17k views

Continuous versus differentiable

A function is "differentiable" if it has a derivative. A function is "continuous" if it has no sudden jumps in it. Until today, I thought these were merely two equivalent definitions of the same ...
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41answers
13k views

What are some examples of a mathematical result being counterintuitive?

As I procrastinate studying for my Maths Exams, I want to know what are some cool examples of where math counters intuition. My first and favorite experience of this is Gabriel's Horn that you see in ...
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1answer
837 views

Euler's sieve and wheel factorization

http://burntjet.co.uk/maths/primes/sieves.php#eq_sieve_div I have read that Sieve of Eratosthenes algorithm can be speeded up with wheel factorization. Can similar be done with Euler's sieve (while ...
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1answer
177 views

How to solve this by Galois theory? [closed]

Please focus on the concept to solve this problem, because I can't handle to research on difficult terminology. Thanks in advance. Find all real roots by Galois theory and find all other root to this ...
9
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2answers
3k views

How is Leibniz's rule for the derivative of a product related to the binomial formula? [duplicate]

Possible Duplicate: “Binomial theorem”-like identities The binomial formula describes the expansion of the $n$th power of the sum $(a+b)$: $$(a+b)^n = \sum_{k = 0}^n {n\choose ...
4
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2answers
2k views

Find all connected 2-sheeted covering spaces of $S^1 \lor S^1$

This is exercise 1.3.10 in Hatcher's book "Algebraic Topology". Find all the connected 2-sheeted and 3-sheeted covering spaces of $X=S^1 \lor S^1$, up to isomorphism of covering spaces without ...
20
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4answers
2k views

Perspectives on Riemann Surfaces

So, I have come to a somewhat impasse concerning my class selection for next term, and I have exhausted all the 'biased' sources. So, I was wondering if anyone in this fantastic mathematical community ...
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1answer
139 views

Problems with infinite $\Omega$, when trying to define product spaces of discrete probability spaces

Definitions In our course we defined a discrete probability space as a tuple $\left(\Omega,P\right)$, where $P:\mathcal{P}(\Omega)\rightarrow\left[0,1\right]$ and $\Omega$ is at most countable, such ...
5
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4answers
288 views

Theories and models

I apologize if my question is not well formed. The reason for it is that I don't understand the concepts enough to be able to ask a completely meaningful question. In the classes we said that a ...
25
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2answers
2k views

Geometric intuition behind the Lie bracket of vector fields

I understand the definition of the Lie bracket and I know how to compute it in local coordinates. But is there a way to "guess" what is the Lie bracket of two vector fields ? What is the geometric ...
6
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0answers
271 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 ...
23
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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 ...
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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
748 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
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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 ...
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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
171 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
358 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
637 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
391 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
547 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
759 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 ...
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448 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 ...
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2answers
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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
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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
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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
917 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 ...
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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 ...
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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
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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 ...
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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, ...
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1answer
467 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 ...
22
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3answers
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 ...
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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} ...