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0

I will employ here again the same web resource by Eric Moorhouse I linked yesterday in this answer to a related question. This finite group and this other have both order $27$: since there's no $n$ such that $n!=27$, they are not symmetric; since there's no $m$ such that $2m=27$, they are not dihedral. Just a glance at the bottom right corner of the two ...


-1

I have another one, but I'm quite unwilling to post this here because it's MINE, I haven't found it anywhere, so don't steal this. Let us take the four most important mathematical constants: The Euler number $e$, the Aurea Golden Ratio $\phi$, the Euler-Mascheroni constant $\gamma$ and finally $\pi$. Well we can see easily that ...


0

An awesome pattern. 1 x 1 = 1 11 x 11 = 121 111 x 111 = 12321 1111 x 1111 = 1234321 11111 x 11111 = 123454321


0

The first thing for me is the working of an equation. it is, to me, like a stanza of a poem that tells us many things in minimum words. No one would have ever thought of describing a geometrical figure. Every one used to draw it before math's entry in the real world. It's awesome for a mathematician to say that write me a circle, ellipse etc. In order to ...


0

I think one of the most surprising results of this type is the following (and I could just be very naive): Kervaire and Milnor showed that diffeomorphism classes of oriented exotic spheres form the non-trivial elements of a finite abelian group under the connected sum for dimension not equal to $4$.


0

A set of permutations closed under composition. These were historically the first thing called groups, every group algebra is instantiated by some set of permutations closed under composition and most of the theorems you see in group theory books have been developed to understand permutations closed under composition better and their relationship with ...


0

Since associativity is not immediate to be verified for finite structures whose Cayley table is given, it is probably not obvious that magmas like this one or this other are groups. It is easier instead to catch at a first glance that both have identity or that none of these two is commutative.


0

$\bullet$ Patience! $\bullet\bullet$ Persistence. $\bullet\bullet$ Work hard. $\bullet\bullet\bullet$ Learn things very well. (in detail) $\bullet\bullet\bullet\bullet$ Ask yourself lots of questions, even stupid ones! (when does this lemma work? when it doesn't? is there a generalization of it? is there a similar lemma about ...) ...


0

Check out the portal Applied Topology, which arose from Gunnar Carlsson's research group in Stanford. There are many applications areas mentioned with relations to statistics, data-mining, biology etc.


3

Sandpile groups have the curious feature that the identity element is very complicated. See http://www.ams.org/notices/201008/rtx100800976p.pdf.


0

Topology is at least partially built into human intuition because it talks about invariants - general properties and classification independent of fine details - exactly what humans are best at! There are many real-life examples, for instance, the hairy ball theorem is encountered every time you wonder how come we can't have a map of the earth without ...


0

Topology also has applications within computer science. Directed algebraic topology is a branch of algebraic topology that has applications in concurrency theory when trying to avoid and resolve deadlocks and starvation. See for example here. Topological data analysis is an alternative to standard data mining, which allows one to infer global and ...


0

Kuratowski's Theorem characterizing planar graphs (http://en.wikipedia.org/wiki/Kuratowski%27s_theorem) has application to circuit boards where certain nodes must be connected but without any edge crossings.


0

Irving Adler in 'a new look at geometry' gives a circular chain of proofs of alternate versions of the fifth postulate (that if a line striking each of two lines at the same angle, then the two lines do not cross at any distance). The idea of such a proof shows that all of these propositions are identical in function.


1

The first example that comes to my mind is the Cauchy-Schwarz inequality. Here there is a paper I found some time ago in which the authors claim (I write claim, because I did not check each proof) to be able to show twelve different proofs of the result. Actually, there is an entire book on inequalities that starts from this very basic one, with various ...


2

Here I found that 20 different proofs for the Euler Formula$$V-E+F=2.$$


8

The absolute least value you can get is a rectangle topped by a half circle (the circle has the best area to arc length ratio of any shape) with a total arc length of $2 \big(1 - \frac{\pi}{8}\big) + \frac{\pi}{2} \approx 2.78539$. If you use Fourier approximation, you can come arbitrarily close to this limit. (I assume the fun of this challenge is to find ...


-1

$$f(x)=\begin{cases} cx & \text{ if } 0 \le x \le \frac{1}{c} \\ -cx+c & \text{ if } 1-c < x \le 1 \\ 1 & \text{ otherwise } \\ \end{cases}$$ If we take $c\to\infty$ we get that it is an arc length of 3.


1

I'm not sure if this is a real-world application, but topological methods can be used for a formal proof that a system of real (possibly nonlinear) equations has solution. Assume that $f: K\to\mathbb{R}^n$ is continuous and you want to prove that $f^{-1}(0)$ is nonempty. If, for example, $K$ is an $n$-manifold, then a nonzero degree of $f/|f|: \partial ...


3

The fundamental theorem of algebra Every nonconstant polynomial with coefficients in $\mathbb{C}$ has a root in $\mathbb{C}$ can be shown using algebraic topology. See for example Hatcher (http://www.math.cornell.edu/~hatcher/AT/AT.pdf) Page 31. Theorem 1.8. Sketch of the proof: Take a polynomial: $p(z) = z^n +a_1 z^{n−1} +\cdots+a_n$ . Then ...


1

A proof that needs five seconds rather than five minutes, but I find instructive nonetheless: Suppose you have a line in $\mathbb{R}^2$ together with two points $A$ and $B$ on the same side of the line. Determine the point where the distance travelled by an object moving from A to B being reflected at the line is minimal. Too lazy to draw up a diagram. Show ...


2

Hindman's Finite Sums Theorem: If you partition $\mathbb N$ into finitely many classes, there is an infinite sequence $a_1\lt a_2\lt a_3\lt\cdots$ in $\mathbb N$ such that all of the finite sums $a_{i_1}+a_{i_2}+\cdots+a_{i_k}$, where $i_1\lt i_2\lt\cdots\lt i_k$ and $k\in\mathbb N$, belong to the same partition class. The proof uses some kind of topological ...


1

I was really surprised about the following theorem. (General Question, sharp result) Theorem: If a compact Lie Group $G$ acts freely on a sphere, then $G$ is either finite or isomorphic to $S^1$, $S^3$ or the Normalizer of $S^1$ in $S^3$. (As usual i write $S^1=SO(2)$ and $S^3=SU(2)$. The proof of this results is an easy (If you know a tiny bit of Lie ...


4

The Necklace Splitting problem in combinatorics has been beautifully solved by Alon and West using the Borsuk-Ulam theorem. http://en.wikipedia.org/wiki/Necklace_splitting_problem


9

The Nielsen–Schreier theorem (a subgroup of a free group is itself free) can be proven using methods of elementary algebraic topology. My personal favourite is the pretty deep result that every finite dimensional divison algebra over $\mathbf{R}$ has dimension $1,2,4$, or $8$. This result seems to be due Kervaire and Milnor; the proof uses methods of ...


5

The proof of the Cayley–Hamilton theorem in the case of different eigenvalues is very easy. The extension to general case in any field is possible using the Zariski topology.


6

Francis Su described in 1999 ("Rental Harmony: Sperner's Lemma in Fair Division", Amer. Math. Monthly, 106, 1999, 930-42) how to apply Sperner's Lemma---which says that every so-called Sperner coloring of a triangulation of an $n$-simplex contains a cell colored with a complete set of colors---to produce a list of variously sized rents for rooms in a shared ...


2

The Borsuk-Ulam Theorem states that for any continuous map $f : \mathbb{S}^n \to \mathbb{R}^n$, there are two antipodal (opposite) points of $\mathbb{S}^n$ that $f$ maps to the same value. (This follows from the fact that every antipodes-preserving map from the sphere to itself has odd degree.) For example, in the case $n = 2$, we can conclude that there ...


9

How about Furstenberg's proof of the infinitude of prime numbers?


4

The Brouwer fixed-point theorem. The Wikipedia lists the following methods: Homology Stokes's theorem Combinatorial (Sperner's lemma) Reducing to the smooth case and using Sard's theorem Reducing to the smooth case and using the COV theorem Lefschetz fixed-point theorem Using Hex


0

In a finite semigroup $S$, every element has an idempotent power, i.e. for every $s \in S$ there exists some $k$ such that $s^k$ is idempotent, i.e. $(s^{k})^2 = s^k$. For the proof consider the sequence $s, s^2, s^3, \ldots$ which has to repeat somewhere, let $s^n = s^{n+p}$, then inductively $s^{(n + u) + vp} = s^{n + u}$ for all $u,v \in \mathbb N_0$, so ...


1

Why not some properties of powers of (natural) numbers: $$\forall\,a,b,m,n\in\mathbb{N}\setminus\{0\},\quad a^m\cdot a^n=a^{m+n},\quad (a^m)^n=a^{mn},\quad a^n\cdot b^n=(ab)^n.$$ Another one, which is not totally elementary as you meant it but I think may be very formative in secondary school level is the following: If $p$ is a polynomial in variable ...


-2

My favourite 'immediately proven' theorem is that all natural numbers are interesting. Suppose the contrary, that some natural numbers are not interesting. Then by natural ordering there would exist the smallest uninteresting natural number ...and there would be only one such number, which would be very interesting! So, there is no uninteresting natural ...


1

Classic Set Theory: for Guided Independent Study by Derek Goldrei is supposed to be a "guided discovery" book, but I think it reads pretty much like any standard textbook. Might be worth a try though for those interested in set theory.


5

Problem: A red ribbon is tied tightly around the earth at the equator (assume the earth is a perfect sphere). How much more ribbon would you need if you raised the ribbon 1 ft above the equator everywhere? Answer: Only a tad bit more than 6 ft! Solution: Let $r$ be the radius of the earth in feet. Then the circumference (length of the ribbon) is $2\pi r$. ...


1

How about Marilyn vos Savant's explanation of which door to choose in a game show? http://marilynvossavant.com/game-show-problem/ The problem asked of her was: Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the ...


3

My answer is quite philosophical and the books concerned with guided discovery, which I read, are less concrete than proposed by the others: George Polya, “Mathematical discovery: on understanding, learning and teaching” Henri Poincaré, “Mathematical Creation”. Imre Lakatos, “Proofs and refutations. The Logic of Matematical Discovery” Jacques Hadamard, ...


4

There's also a well known overpowered proof of the infinitude of primes: If $P$ is the set of primes, the Euler product formula tells us $$\prod\limits_{p\in P}\frac{1}{1-\frac{1}{p^2}}=\zeta(2)=\frac{\pi^2}{6}$$ However, $\pi$ is transendental, so $|P|$ cannot be finite (otherwise the product would be rational).


0

One of the first proofs I was shown was in a discrete structures class. I don't think this would survive the rigors of formal proof, but I really enjoyed it. Definitions: $N$ is the set of numbers starting with 0, 1, 2, $\dots $ A number in our set $N$ is odd if and only if there exists another number in our set $N$ that when multiplied by two and added ...


0

I have spent many decades studying why so many highly intelligent people are so mystified by mathematics. Lockhart's view is very serious and cannot be negated by the personal experiences of mathematically inclined people. My study has clearly shown that the best advice is to be simple and sensible. For example, our place number system is an ingenious ...


1

If Euclid knew that the series of reciprocals of prime numbers diverges then we would not have his proof that there are infinitely many primes. :-) (see more here).


4

I really like proofs using the pigeonhole principle I give two examples I think most people should know should know. Example 1: In a party with $n$ persons there are always two persons who have shaken hands with the same number of people. Proof: clearly in parties people don't shake hands with the same person twice (for sufficiently low alcohol levels). ...


4

In my old paper I used Four Colors Theorem to prove a two dimensional case of the following Proposition. Every open subset of the space $\Bbb R^m$ can be partitioned into $n$ homeomorphic parts if $n\ge 2^{m+1}-1$ or $m=2$ and $n\ge 4$. Proof. For every positive integer $k$ let $A_k$ be the set of all points $x\in\Bbb R^m$ such that all coordinates of the ...


3

While possibly a bit silly, I find that the (utterly trivial) proof of the uniqueness of identity elements very nicely illustrates how "abstract" mathematical proofs "work" and how, at least not totally trivial, questions can get very simple answers if posed correctly. While the proof itself obviously does not require 5 minutes to present one would probably ...


3

There's no way to tune a piano in perfect harmony. There are twelve half-steps in the chromatic scale, twelve notes in each octave of the keyboard. Start at middle "C", and ascend a perfect fifth to "G". That's seven half steps up, with a frequency ratio of 3/2. Drop an octave to the lower "g" -- that's twelve half steps down, and a frequency ratio of ...


1

Jones' index rigidity formula comes to mind. In fact, there was no way to "observe" it in the special cases. It was widely believed that the index could take all values between $1$ and $\infty$- so the theorem came as a surprise. (The theorem states that the index takes values in a discrete series between $1$ and $4$).


1

Soap Film Problem - this is actually another term for "minimal surface problems", since soap bubles or other similar soap forms tend to minimize their surface. Links here and here.


1

Proof that $\sqrt 2$ is irrational: Any non-integer fraction multiplied by itself cannot be an integer. (So a full length proof along these lines would first have to show that the prime factorization of integers is unique, and this turns out to be rather hard. But any young kid who has learned about prime factorization will accept this without proof.)


1

I always loved the explanation of the Hilton-Eckmann argument, given by J. Baez in This week's finds in mathematical physics 100. The Hilton-Eckmann argument itself is a rather easy result (which one could certainly present in an undergrad abstract algebra class as an exercise), and the "visual proof" (using "higher-dimensional reasoning") which is hinted ...



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