Most Popular
1500 questions
61
votes
1 answer
Are there simple examples of Riemannian manifolds with zero curvature and nonzero torsion
I am trying to grasp the Riemann curvature tensor, the torsion tensor and their relationship.
In particular, I'm interested in necessary and sufficient conditions for local isometry with Euclidean space (I'm talking about isometry of an open set -…
Selene Routley
- 2,655
61
votes
1 answer
A question about Sylow subgroups and $C_G(x)$
Let $G=PQ$ where $P$ and $Q$ are $p$- and $q$-Sylow subgroups of $G$ respectively. In addition, suppose that $P\unlhd G$, $Q\ntrianglelefteq G$, $C_G(P)=Z(G)$ and $C_G(Q)\neq Z(G)$, where $Z(G)$ is the center of $G$.
I want to prove there exist two…
Adeleh
- 1,357
61
votes
8 answers
"It looks straightforward, but actually it isn't"
In a previous topic, I asked about proof of statements which are simple but incorrect.
Here, I ask about statements which seems, at a first glance, straightforward, but if we try to write a proof, we can see it's much harder than it looked. So I…
Davide Giraudo
- 172,925
61
votes
17 answers
Intuitive understanding of the derivatives of $\sin x$ and $\cos x$
One of the first things ever taught in a differential calculus class:
The derivative of $\sin x$ is $\cos x$.
The derivative of $\cos x$ is $-\sin x$.
This leads to a rather neat (and convenient?) chain of…
Justin L.
- 14,532
61
votes
4 answers
What is the size of each side of the square?
The diagram shows 12 small circles of radius 1 and a large circle, inside a square.
Each side of the square is a tangent to the large circle and four of the small circles.
Each small circle touches two other circles.
What is the length of each side…
vamika2010
- 721
61
votes
16 answers
Is there such a thing as proof by example (not counter example)
Is there such a logical thing as proof by example?
I know many times when I am working with algebraic manipulations, I do quick tests to see if I remembered the formula right.
This works and is completely logical for counter examples. One specific…
SwimBikeRun
- 1,047
- 1
- 10
- 18
61
votes
7 answers
Could I be using proof by contradiction too much?
Lately, I've developed a habit of proving almost everything by contradiction. Even for theorems for which direct proofs are the clear choice, I'd just start by writing "Assume not" then prove it directly, thereby reaching a "contradiction." Is this…
user64844
- 593
61
votes
8 answers
The last digit of $2^{2006}$
My $13$ year old son was asked this question in a maths challenge. He correctly guessed $4$ on the assumption that the answer was likely to be the last digit of $2^6$. However is there a better explanation I can give him?
Keith Miller
- 743
61
votes
4 answers
Counterexample Math Books
I have been able to find several counterexample books in some math areas. For example:
$\bullet$ Counterexamples in Analysis, Bernard R. Gelbaum, John M. H. Olmsted
$\bullet$ Counterexamples in Topology, Lynn Arthur Steen, J. Arthur Seebach…
Amzoti
- 56,093
61
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:
$$ \mathrm{Base}(U)= \left\{ \left(1,1,0,-1\right), \left(0,1,3,1\right) \right\} $$
$$…
Cu7l4ss
- 963
61
votes
15 answers
Area of a square inside a square created by connecting point-opposite midpoint
Square $ABCD$ has area $1cm^2$ and sides of $1cm$ each.
$H, F, E, G$ are the midpoints of sides $AD, DC, CB, BA$ respectively.
What will the area of the square formed in the middle be?
I know that this problem can be solved by trigonometry by using…
Agile_Eagle
- 2,922
61
votes
3 answers
Proving $\left(1-\frac13+\frac15-\frac17+\cdots\right)^2=\frac38\left(\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\frac1{4^2}+\cdots\right)$
The equality$$\left(1-\frac13+\frac15-\frac17+\cdots\right)^2=\frac38\left(\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\frac1{4^2}+\cdots\right)\tag{1}$$follows from the fact that the sum of the first series is $\dfrac\pi4$, whereas the sum of the second…
José Carlos Santos
- 427,504
61
votes
3 answers
A very general method for proving inequalities. Too good to be true?
Update I 'repaired' this method, but it changed a lot and I have some different questions, so I posted it separately here.
As training for the olympiad, I have to solve a lot of inequalities. Recently, I found a very general method to solve…
Mastrem
- 8,331
61
votes
7 answers
Big O Notation "is element of" or "is equal"
People are always having trouble with "big $O$" notation when it comes to how to write it down in a mathematically correct way.
Example: you have two functions $n\mapsto f(n) = n^3$ and $n\mapsto g(n) = n^2$
Obviously $f$ is asymptotically faster…
Blnpwr
- 931
61
votes
10 answers
How to solve an $n$-th degree polynomial equation
The typical approach of solving
$$
f_2(x):=ax^2+bx+c=0
$$
is to solve for the roots
$$x_{1/2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}.$$
Here, the degree of $f$ is given to be $2$.
However, I was wondering on how to generalize this problem.
For example,…
Ayush Khemka
- 983