Top new questions this week:

Why do people lose in chess?

Zermelo's Theorem, when applied to chess, states: "either white can force a win, or black can force a win, or both sides can force at least a draw [1]" I do not get this. How can it be proven? …

(logic) (game-theory)

asked by Brika 37 votes answered by Hagen von Eitzen 68 votes

Finding every $n$ such that $a_n$ is an integer

Let us define $\{a_n\}$ as $a_1=a_2=1$,$$a_{n+2}=a_{n+1}+\frac{a_n}{2}\ \ (n=1,2,\cdots).$$ Then, is the following true? If $a_n$ is an integer, then $n\le 8$. I conjectured this by using …

(sequences-and-series) (recurrence-relations)

asked by mathlove 24 votes answered by J. J. 9 votes

What is Abstract Algebra essentially?

In the most basic sense, what is abstract algebra about? Wolfram Mathworld has the following definition: "Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic …

(abstract-algebra) (group-theory) (ring-theory) (field-theory) (definition)

asked by Colin 23 votes answered by Arkamis 22 votes

Lamport claims there is an error in Kelley's proof of the Schroeder-Bernstein theorem. What is it?

In section 4.1 of his note How to write a proof, Leslie Lamport mentions an error in Kelley's exposition of the Schroeder-Bernstein theorem: Some twenty years ago, I decided to write a proof of …

(elementary-set-theory) (proof-verification) (fake-proofs)

asked by Marion 22 votes answered by Henning Makholm 13 votes

How does this discontinuity occur in evaluating a nested square root?

This question is based on a comment I made on a question likely to be closed. Let $$y=\sqrt {x+ \sqrt {x+ \sqrt {x+ \sqrt {x+ \sqrt {x+ \dots}}}}}$$ be the classic nested square root which has …

(real-analysis) (limits) (continuity)

asked by Mark Bennet 20 votes answered by jef 9 votes

Who named "Quotient groups"?

Who decided to call quotient groups quotient groups, and why did they choose that name? A lot of identities such as $$\frac{G/A}{B/A}\cong \frac{G}{B}$$ suggest that whoever invented the notation …

(abstract-algebra) (group-theory) (notation) (terminology) (math-history)

asked by NotNotLogical 19 votes answered by KCd 19 votes

Gap year to study math

This is a plan in its earliest and thus least concise stage, so either bear with me or don't read the following babble (I bolded some of the important stuff): I am a high school graduate who is about …

(soft-question) (advice) (undergraduate-research)

asked by user154154 16 votes answered by gnometorule 17 votes

Greatest hits from previous weeks:

How many squares actually ARE in this picture? Is this a trick question with no right answer?

This is one of those popular pictures on sites like Facebook. I always see a huge variation of answers such as 8, 9, 16, 17, 24, 28, 30, 40, 41, 52 etc, yet I've never seen a definitive answer on any …

(puzzle)

asked by user1092719 22 votes answered by A.L 98 votes

Fourier transform for dummies

A vague question of Kevin Lin which didn't quite fit at Mathoverflow: So ... what is the Fourier transform? What does it do? Why is it useful (both in math and in engineering, physics, etc)? …

(fourier-analysis)

asked by user218 122 votes answered by Mark Eichenlaub 136 votes

Can you answer these?

A proper local diffeomorphism between manifolds is a covering map.

The following is an exercise taken from "Manifolds and Differenial Geometry" by Jeffrey M. Lee. Let $\widetilde M$ and M be (connected) $C^r$ manifolds. Let $f: \widetilde M \to M$ be a proper map …

(manifolds) (covering-spaces)

asked by Saal Hardali 4 votes

$(b-a)^2-2ab$ is a perfect square.

I'm in need of some help if possible, about a formula, theorems, old works, ideas, or even an existing solution are welcome. The problem is that i have two distinct natural numbers as $b > a > …

(elementary-number-theory) (diophantine-equations) (square-numbers)

asked by ixenzire 5 votes

Show that $x$ is rational.

Sincerely, I don't have the slightest idea for this one : Suppose that $a,b$ are distinct positive integers and that the numbers $\lfloor a^n x\rfloor $ with $n\in\Bbb{N}$ and $x$ a fixed real …

(elementary-number-theory)

asked by Edwin 6 votes