Motto: "**Few, but ripe.**" (Gauss)

**the best of math**gems and pearls

**classified**by stars,

**categorized**under the menu on the left side. You can

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## Notice

## Math Vault

**Math Vault** (mathvault.ca) is a online resource hub for people interesting in learning more about higher mathematics, and features a series of guides, articles, eBooks and modules with topics ranging from general math and college math to pure math and applied math.

## A Streamlined Proof of an Essential Calculus Fact *****

**GREAT NEW PROOF** of the Fundamental Theorem of Calculus

Stephen M. Walk: A Streamlined Proof of an Essential Calculus Fact, The American Mathematical Monthly, Vol. 117, No. 9 (November 2010), pp. 832-833.

(JSTOR abstract EN, idea online EN, CZ)

## A question: Is math well mixed?

One can build math starting from a simple original ideas. In terms of Kolgomorov complexity ( EN ) the "program for making math" is simple. The same appears when you put cocoa powder into milk. Just two ingredients in the beginning. When mixed a little it is an interesting fight between cocoa and milk. But quickly it will be well mixed and everything interesting disappears.

A question: **Is math well mixed**?

A nice discussion about the complexity and complexodynamics can be found in Scott Aaronson's *Shtetl-Optimized* blog ( EN , EN ).

## Square root of R^3

Asking cute questions is great.

Answering such questions is a nice challenge.

An article by Mark Dominus about this topics is here ( EN ).

## PRIMES is in P. Where in P?

.

.

In 2002-2003 Agrawal, Kayal, and Saxena in a great breakthrough "**PRIMES is in P**" ( Wiki EN ) proved that the PRIMES function (prime number test) is O(log ^8 n), in other words polynomial of degree 8 of the number of digits (log n) of a given number n.

Congratulations

BTW, where in P is its real position? Any idea?

## A long paused dialog cache *****

## FOUND !!! *****

In 2011 after 333 years was found a missing letter in the Newton's definition of the integral. The** DEFINITIVE definition of the integral** is due to Bendova & Maly (online EN ).

## Newton's 2011 Mystery Cache *****

Newton discovered the **Fundamental Theorem of Calculus**. To keep its mystery hidden he has sent a mail to Leibnitz with a **"mystery cache"**. For all cachers (reformulated to 2011 English setting) the message sounds

6a4c2d6e2g3i1k2l3n1p3r2s4t1u2v

This cache is a virtual one. Opening it you will read the answer to the Fundamental Question of Calculus.

While in those years Newton prepared the cache as "not to be found" (NTBF), we provide a hint here below.

## Cardinality theorem *****

Two lines proof of the Cantor-Schroeder-Bernstein theorem ( EN ).

## Swindle? *****

YES, you can swindle and prove theorems similar to this:

Swindle can be correctly used in many proofs, e.g.

**Irreducibility of the sphere**

**Cantor–Bernstein–Schröder theorem** ( Wiki EN )

...

Can you add the next one to the list?

## On rare integers between 0 and 1 *****

Sexy & clean proof that π is irrational on Dror Bar-Natan pages in 5 full stopped sentences written on a CD envelope ( EN ). The shortest & simplest one?

## Feynman: Is P=NP? No, next question? *****

Scott Aaronson talked on “TEDxCaltech / Feynman’s Vision: the Next 50 Years”. You can read about on his blog Shtetl-Optimized ( EN ).

The talk touches P=NP problem in a clever way.

The talk is presented online ( YouTube EN ) and is GREAT.

## Mathematics Genealogy Project *****

Mathematics Genealogy Project ( NDSU link EN, AMS link EN )

Tree of mathematicians and their students, e.g. Kurt Godel (no students known) was one of 4584 Bernard Bolzano's descendants.

## Turning a sphere inside-out *****

This possibility was discovered by Smale in 1958 ( wiki EN ) by observing homotopy groups.

An unbelievable animation (with intersections of course) is e.g. here ( YouTube EN ).

## Through one hole or two *****

A joke (or not?) breaking (or not?) the very basic topology fact that continuous movement cannot change topology invariants ( YouTube EN ).

## MathOverflow *****

MathOverflow - A EXCELLENT place for mathematicians to ask and answer **RESEARCH LEVEL** math questions ( EN ).

## Jordan curve theorem *****

A great collection of resources for the Jordan curve theorem ( Andrew Ranicki EN ).

In other words: **Once inside, once outside**