Mathematics

Most Popular

more tags
1500 questions
66
votes
7 answers

How to find a basis for the intersection of two vector spaces in $\mathbb{R}^n$?

What is the general way of finding the basis for intersection of two vector spaces in $\mathbb{R}^n$? Suppose I'm given the bases of two vector spaces U and W: $$ \mathrm{Base}(U)= \left\{ \left(1,1,0,-1\right), \left(0,1,3,1\right) \right\} $$ $$…
Cu7l4ss
  • 1,013
66
votes
2 answers

Conway's "Murder Weapon"

The following quote is an excerpt from an interview with John Conway: Coxeter came to Cambridge and he gave a lecture, then he had this problem for which he gave proofs for selected examples, and he asked for a unified proof. I left the lecture room…
Oiler
  • 3,511
66
votes
26 answers

What are some mathematically interesting computations involving matrices?

I am helping designing a course module that teaches basic python programming to applied math undergraduates. As a result, I'm looking for examples of mathematically interesting computations involving matrices. Preferably these examples would be easy…
providence
  • 4,538
  • 2
  • 35
  • 46
66
votes
4 answers

Why can we use induction when studying metamathematics?

In fact I don't understand the meaning of the word "metamathematics". I just want to know, for example, why can we use mathematical induction in the proof of logical theorems, like The Deduction Theorem, or even some more fundamental proposition…
183orbco3
  • 2,097
66
votes
7 answers

$\int_{0}^{\frac{\pi}{4}}\frac{\tan^2 x}{1+x^2}\text{d}x$ on 2015 MIT Integration Bee

So one of the question on the MIT Integration Bee has baffled me all day today $$\int_{0}^{\frac{\pi}{4}}\frac{\tan^2 x}{1+x^2}\text{d}x$$ I have tried a variety of things to do this, starting with Integration By Parts Part 1 $$\frac{\tan…
Teh Rod
  • 3,176
66
votes
6 answers

What is the required background for Robin Hartshorne's Algebraic Geometry book?

It seems that Robin Hartshorne's Algebraic Geometry is the place where a whole generation of fresh minds have successfully learned about modern algebraic geometry. But is it possible for someone who is out of academia and has not much background,…
Hooman
  • 3,827
66
votes
18 answers

Is there a simple function that generates the series; $1,1,2,1,1,2,1,1,2...$ or $-1,-1,1,-1,-1,1...$

I'm thinking about this question in the sense that we often have a term $(-1)^n$ for an integer $n$, so that we get a sequence $1,-1,1,-1...$ but I'm trying to find an expression that only gives every 3rd term as positive, thus it would read;…
Kristaps John Balodis
  • 1,526
66
votes
1 answer

Generating correlated random numbers: Why does Cholesky decomposition work?

Let's say I want to generate correlated random variables. I understand that I can use Cholesky decomposition of the correlation matrix to obtain the correlated values. If $C$ is the correlation matrix, then we can do the cholesky decomposition:…
Flux Capacitor
  • 823
66
votes
10 answers

Symbol for "probably equal to" (barring pathology)?

I am writing lecture notes for an applied statistical mechanics course and often need to express the notion that something is very probably true for functional forms found in the wild, without launching into a full digression for pathological…
mewahl
  • 719
66
votes
3 answers

Is "A New Kind of Science" a new kind of science?

A couple of years ago I was reading "New Kind of Science" (NKS) by S. Wolfram, and it presented lot of interesting ideas for a young Physics undergraduate. Now that I am studying Mathematics however, I realise that many ideas of NKS seem to be not…
Andrea
  • 2,503
66
votes
5 answers

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 an axiom for defining a Lie algebra). Could…
aelguindy
  • 2,718
66
votes
3 answers

"Every linear mapping on a finite dimensional space is continuous"

From Wiki Every linear function on a finite-dimensional space is continuous. I was wondering what the domain and codomain of such linear function are? Are they any two topological vector spaces (not necessarily the same), as along as the domain is…
Tim
  • 49,412
65
votes
1 answer

Why is the absolute value function not differentiable at $x=0$?

They say that the right and left limits do not approach the same value hence it does not satisfy the definition of derivative. But what does it mean verbally in terms of rate of change?
user187397
  • 651
65
votes
2 answers

Why is $L^{\infty}$ not separable?

$l^p (1≤p<{\infty})$ and $L^p (1≤p<∞)$ are separable spaces. What on earth has changed when the value of $p$ turns from a finite number to ${\infty}$? Our teacher gave us some hints that there exists an uncountable subset such that the distance of…
Andylang
  • 1,751
65
votes
2 answers

Computation with a memory wiped computer

Here is another result from Scott Aaronson's blog : If every second or so your computer’s memory were wiped completely clean, except for the input data; the clock; a static, unchanging program; and a counter that could only be set to 1, 2, 3, 4, or…
Casebash
  • 9,497
1 2 3
…
99 100