Mathematics

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Cover of "Gödel, Escher, Bach"

Consider the cover image of the book "Gödel, Escher, Bach", depicted below. The interesting feature is that it shows the existence of a subset of $\mathbb{R}^3$ which projects onto $\mathbb{R}^2$ in three different ways to form the letters of the…
user900250
100
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12 answers

Why do bell curves appear everywhere?

Why do most probability graphs show a bell curve? I've been wondering why... Is it just something natural, like the fibonacci sequence?
Shreyas Shridharan
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Given this transformation matrix, how do I decompose it into translation, rotation and scale matrices?

I have this problem from my Graphics course. Given this transformation matrix: $$\begin{pmatrix} -2 &-1& 2\\ -2 &1& -1\\ 0 &0& 1\\ \end{pmatrix}$$ I need to extract translation, rotation and scale matrices. I've also have the answer (which is…
metavers
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Mathematical precise definition of a PDE being elliptic, parabolic or hyperbolic

what is the general definition for some partial differential equation being called elliptic, parabolic or hyperbolic - in particular, if the PDE is nonlinear and above second-order. So far, I have not found any precise definition in literature.
shuhalo
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0 answers

Why is a PDE a submanifold (and not just a subset)?

I struggle a bit with understanding the idea behind the definition of a PDE on a fibred manifold. Let $\pi: E \to M$ be a smooth locally trivial fibre bundle. In Gromovs words a partial differential relation of order $k$ is a subset of the $k$th jet…
hase_olaf
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10 answers

Explain "homotopy" to me

I have been struggling with general topology and now, algebraic topology is simply murder. Some people seem to get on alright, but I am not one of them unfortunately. Please, the answer I need is ideally something very elaborate and extensive,…
John Trail
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100
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12 answers

What is the best book for studying discrete mathematics?

As a programmer, mathematics is important basic knowledge to study some topics, especially Algorithms. Many websites, and my fellows suggest me to study Discrete Mathematics before going to Algorithms, so I want to know which Discrete Mathematics…
idonno
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18 answers

Proof of $\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.$

I am looking for a short proof that $$\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.$$ What do you think? It is kind of amazing that $$\int_0^\infty \frac{\sin x}{x} \mathrm dx$$ is also $\frac{\pi}{2}.$ Many proofs of this…
TCL
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100
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5 answers

A matrix and its transpose have the same set of eigenvalues/other version: $A$ and $A^T$ have the same spectrum

Let $ \sigma(A)$ be the set of all eigenvalues of $A$ . Show that $ \sigma(A) = \sigma\left(A^T\right)$ where $A^T$ is the transpose matrix of $A$ .
Zizo
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7 answers

How can adding an infinite number of rationals yield an irrational number?

For example how come $\zeta(2)=\sum_{n=1}^{\infty}n^{-2}=\frac{\pi^2}{6}$. It seems counter intuitive that you can add numbers in $\mathbb{Q}$ and get an irrational number.
user41414
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99
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4 answers

If $G/Z(G)$ is cyclic, then $G$ is abelian

Continuing my work through Dummit & Foote's "Abstract Algebra", 3.1.36 asks the following (which is exactly the same as exercise 5 in this related MSE answer ): Prove that if $G/Z(G)$ is cyclic, then $G$ is abelian. [If $G/Z(G)$ is cyclic with…
Altar Ego
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8 answers

Zero probability and impossibility

I read a comment under this question : There are plenty of events that can occur that have zero probability. This reminds me that I have seen similar saying before elsewhere, and have never been able to make sense out of it. So I was wondering if…
Tim
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14 answers

Formal proof for $(-1) \times (-1) = 1$

Is there a formal proof for $(-1) \times (-1) = 1$? It's a fundamental formula not only in arithmetic but also in the whole of math. Is there a proof for it or is it just assumed?
tvamsisai
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6 answers

What is a covector and what is it used for?

From what I understand, a covector is an object that takes a vector and returns a number. So given a vector $v \in V$ and a covector $\phi \in V^*$, you can act on $v$ with $\phi$ to get a real number $\phi(v)$. Is that "it" or is there more to it?…
Mike Flynn
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6 answers

Not every metric is induced from a norm

I have studied that every normed space $(V, \lVert\cdot \lVert)$ is a metric space with respect to distance function $d(u,v) = \lVert u - v \rVert$, $u,v \in V$. My question is whether every metric on a linear space can be induced by norm? I know…
Srijan
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