Mathematics

Most Popular

more tags
1500 questions
116
votes
7 answers

Variance of sample variance?

What is the variance of the sample variance? In other words I am looking for $\mathrm{Var}(S^2)$. I have started by expanding out $\mathrm{Var}(S^2)$ into $E(S^4) - [E(S^2)]^2$ I know that $[E(S^2)]^2$ is $\sigma$ to the power of 4. And that is as…
MathMan
  • 1,369
116
votes
16 answers

If $(a_n)\subset[0,\infty)$ is non-increasing and $\sum_{n=1}^\infty a_n<\infty$, then $\lim\limits_{n\to\infty}{n a_n} = 0$

I'm studying for qualifying exams and ran into this problem. Show that if $\{a_n\}$ is a nonincreasing sequence of positive real numbers such that $\sum_n a_n$ converges, then $\lim\limits_{n \rightarrow \infty} n a_n = 0$ . Using the definition of…
dls
  • 4,796
116
votes
3 answers

$\int_{-\infty}^{+\infty} e^{-x^2} dx$ with complex analysis

Inspired by this recently closed question, I'm curious whether there's a way to do the Gaussian integral using techniques in complex analysis such as contour integrals. I am aware of the calculation using polar coordinates and have seen other…
JasonMond
  • 4,114
116
votes
9 answers

If $S$ is an infinite $\sigma$ algebra on $X$ then $S$ is not countable

I am going over a tutorial in my real analysis course. There is an proof in which I don't understand some parts of it. The proof relates to the following proposition: ( $S$ - infinite $\sigma$ -algebra on $X$ ) $\implies $ $S$ is uncountable. Proof…
Belgi
  • 23,722
116
votes
2 answers

Does a four-variable analog of the Hall-Witt identity exist?

Lately I have been thinking about commutator formulas, sparked by rereading the following paragraph in Isaacs (p.125): An amazing commutator formula is the Hall-Witt identity: $$[x,y^{-1},z]^y[y,z^{-1},x]^z[z,x^{-1},y]^x=1,$$ which holds for any…
Alexander Gruber
  • 28,325
116
votes
25 answers

Can an infinite sum of irrational numbers be rational?

Let $S = \sum_ {k=1}^\infty a_k $ where each $a_k$ is positive and irrational. Is it possible for $S$ to be rational, considering the additional restriction that none of the $a_k$'s is a linear combination of the other ? By linear combination, we…
User1
  • 1,943
116
votes
5 answers

Difference between gradient and Jacobian

Could anyone explain in simple words (and maybe with an example) what the difference between the gradient and the Jacobian is? The gradient is a vector with the partial derivatives, right?
Math_reald
  • 1,417
  • 2
  • 12
  • 12
115
votes
8 answers

Is there any difference between mapping and function?

I wonder if there is any difference between mapping and a function. Somebody told me that the only difference is that mapping can be from any set to any set, but function must be from $\mathbb R$ to $\mathbb R$. But I am not ok with this answer. I…
Hassan Muhammad
  • 4,570
115
votes
5 answers

Fibonacci number that ends with 2014 zeros?

This problem is giving me the hardest time: Prove or disprove that there is a Fibonacci number that ends with 2014 zeros. I tried mathematical induction (for stronger statement that claims that there is a Fibonacci number that ends in any number of…
VividD
  • 16,266
  • 10
  • 66
  • 120
115
votes
1 answer

Gross-Zagier formulae outside of number theory

(Edit: I have asked this question on MO. ) The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer conjecture. It relates the value at $1$ of the derivative…
Bruno Joyal
  • 56,355
115
votes
5 answers

An integral involving Airy functions $\int_0^\infty\frac{x^p}{\operatorname{Ai}^2 x + \operatorname{Bi}^2 x}\mathrm dx$

I need your help with this integral: $$\mathcal{K}(p)=\int_0^\infty\frac{x^p}{\operatorname{Ai}^2 x + \operatorname{Bi}^2 x}\mathrm dx,$$ where $\operatorname{Ai}$, $\operatorname{Bi}$ are Airy functions :…
Cleo
  • 1
115
votes
7 answers

Is black hole pattern possible in Conway's Game of Life that eats/clears everything?

Is black hole pattern possible in Conway's Game of Life that eats/clears the infinite universe plane ? Formally, is there a pattern that satifies following requirements? The pattern has finite size. The universe plane has infinite size. Each cell…
VainMan
  • 1,745
114
votes
24 answers

Why do we still do symbolic math?

I just read that most practical problems (algebraic equations, differential equations) do not have a symbolic solution, but only a numerical one. Numerical computations, to my understanding, never deal with irrational numbers, but only rational…
totalnoob
  • 1,263
114
votes
1 answer

In categorical terms, why is there no canonical isomorphism from a finite dimensional vector space to its dual?

I've read in several places that one motivation for category theory was to be able to give precise meaning to statements like, "finite dimensional vector spaces are canonically isomorphic to their double duals; they are isomorphic to their duals as…
Ben Blum-Smith
  • 21,660
114
votes
4 answers

Least prime of the form $38^n+31$

I search the least n such that $$38^n+31$$ is prime. I checked the $n$ upto $3000$ and found none, so the least prime of that form must have more than $4000$ digits. I am content with a probable prime, it need not be a proven prime.
Peter
  • 87,158
1 2 3
…
99 100