Archive for December, 2021

De Branges’s Proof of the Bieberbach Conjecture

It is a simple matter to post a paper on arXiv.org claiming to prove Goldbach’s Conjecture, the Twin Primes Conjecture or any of a large number of other interesting hypotheses that are still open. However, unless the person posting the article is well known, it is likely to be completely ignored.

Mathematicians establish their claims and convince their colleagues by submitting their work to peer-reviewed journals. The work is then critically scrutinized and evaluated by mathematicians familiar with the relevant field, and is either accepted for publication, sent back for correction or revision or flatly rejected.

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Number Partitions: Euler’s Astonishing Insight

In 1740, French mathematician Philippe Naudé wrote to Leonhard Euler asking in how many ways a positive integer can be written as a sum of distinct numbers. In his investigations of this, Euler established the theory of partitions, for which he used the term partitio numerorum.

Many of Euler’s results in number theory involved divergent series. He was courageous in manipulating these but had remarkable insight and, almost invariably, his findings, although not rigorously established, were valid.

Partitions

In number theory, a partition of a positive integer {n} is a way of writing {n} as a sum of positive integers. The order of the summands is ignored: two sums that differ only in their order are considered the same partition.

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Bernoulli’s Golden Theorem and the Law of Large Numbers

Swiss postage stamp, issued in 1994 for the International Congress of Mathematicians in Zurich, featuring Jakob Bernoulli and illustrating his “golden theorem”.

Jakob Bernoulli, head of a dynasty of brilliant scholars, was one of the world’s leading mathematicians. Bernoulli’s great work, Ars Conjectandi, published in 1713, included a profound result that he established “after having meditated on it for twenty years”. He called it his “golden theorem”. It is known today as the law of large numbers, and it was the first limit theorem in probability, and the first attempt to apply probability outside the realm of games of chance [TM225 or search for “thatsmaths” at irishtimes.com].

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Set Density: are even numbers more numerous than odd ones?

In pure set-theoretic terms, the set of even positive numbers is the same size, or cardinality, as the set of all natural numbers: both are infinite countable sets that can be put in one-to-one correspondence through the mapping {n \rightarrow 2n}. This was known to Galileo. However, with the usual ordering,

\displaystyle \mathbb{N} = \{ 1, 2, 3, 4, 5, 6, \dots \} \,,

every second number is even and, intuitively, we feel that there are half as many even numbers as natural numbers. In particular, our intuition tells us that if {B} is a proper subset of {A}, it must be smaller than {A}.
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Buffon’s Noodle and the Mathematics of Hillwalking  

In addition to some beautiful photos and maps and descriptions of upland challenges in Ireland and abroad, the November issue of The Summit, the Mountain Views Quarterly Newsletter for hikers and hillwalkers, describes a method to find the length of a walk based on ideas originating with the French naturalist and mathematician George-Louis Leclerc, Comte de Buffon [TM224 or search for “thatsmaths” at irishtimes.com].

[Image from November issue of The Summit, the Mountain Views Quarterly Newsletter.]

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