Category Archives: Technical

Probably Banach spaces or algebras

Jozef Schreier, Schreier sets and the Fibonacci sequence

This post has some more technical bits, as it’s aimed mainly at people with an undergraduate-level background in mathematics. I’ve included a historical note on Schreier, because it’s easy to forget that mathematical objects are named after real people.

These five sets (each of which have their elements written in numerical order) are Schreier sets:

\{1\},\{2,3\},\{2,6\},\{3,4,5\},\{3,6,17\},\{4,5\},\{5,6,7,8,9\}.

The following six sets are not Schreier sets:

\{1,2\},\{2,6,17\},\{3,4,5,6\},\{3,5,7,9,10\},

\{4,5,7,13,25\},\{5,6,7,8,9,10\}.

Can you spot the simple rule that defines whether a set of numbers is a Schreier set or not?

Historical Background

Jozef Schreier was born 1908/1909 in Lwów (then part of the Austro-Hungarian empire, now in Ukraine) and later based at the university there (whilst it was part of Poland). Schreier was killed in April 1943 during the Second World War in his home town of Drohobycz. In his book “Adventures of a Mathematician“, Stanislaw Ulam wrote about Schreier, his friend and collaborator:

We would meet almost every day, occasionally at the coffee house but more often at my house. His home was in Drohobycz, a little town and petroleum center south of Lwów. What a variety of problems and methods we discussed together! Our work, while still inspired by the methods then current in Lwów, branched into new fields: groups of topological transformations, groups of permutations, pure set theory, general algebra. I believe that some of our papers were among the first to show applications to a wider class of mathematical objects of modern set theoretical methods combined with a more algebraic point of view. We started work on the theory of groupoids, as we called them, or semi-groups, as they are called now.

Schreier was also a major contributor (ten questions) to the now famous Scottish book (an important book of questions, famous at least amongst Banach space theorists, and which you can read in English); in fact, he and Ulam pair were the only undergraduate student participants in the Scottish Cafe, a coffee shop where mathematicians from Lwów met, and scribbled maths directly onto the marble tabletops. The Scottish book was a notebook to save the work from being cleaned away at the end of the day.

The Baire-Schreier-Ulam and Schreier-Ulam theorems both bear Jozef Schreier’s name. He has on occasion been referred to as “Julian Schreier” (possibly by conflating him with Juliusz Schauder?): whereas Ulam’s account, the Portugese Wikipedia article, which seems to have been written by his family, and this journal from the time all name him Jozef (or a variant: Josef/Józef/Joseph). He definitely shouldn’t be confused with the contemporary mathematician Otto Schreier, who specialised in groups.

One of Jozef Schreier’s conjectures is now referenced in computer science literature: inspired by an article of Lewis Carroll, in the 1932 paper “On tournament elimination systems“, he conjectured that to find the second largest number in an unordered list requires at least n + \lceil \log_2 n \rceil -2 comparisons, which was later confirmed in the 1960s.

I’ve included a list of some of Schreier’s publications at the end of this article.

Schreier sets

In 1930, Schreier constructed a counterexample to a question of Banach and Saks using the following idea:

A (non-empty) subset X of the natural numbers is a Schreier set if its size is not larger than its least element.

|X| \leq \min X

The collection of all such subsets of \mathbb{N} is known as the (first) Schreier family. For example: \{1\}, \{2\} and \{3,5,42\} are all Schreier sets.  This definition is the first my PhD supervisor showed me in my very first supervision.

You can think of building Schreier sets by picking the smallest element, n say, and then filling the set up with at most (n-1) strictly greater distinct numbers.

The Schreier family has the following simple properties (that make it useful when defining norms on Banach spaces, especially the Schreier space):

  • The singleton \{n\} is Schreier set for all n \in \mathbb{N}
  • The family is hereditary: each (non-empty) subset of a Schreier set is itself a Schreier set.
  • The Schreier family is spreading: if \{m_1, m_2, \ldots m_k\} is a Schreier set and n_j \geq m_j for all 1 \leq j \leq k, then \{n_1, n_2,\ldots, n_k\} is also a Schreier set. (These don’t necessarily need to be in ascending order).

Schreier and Fibonacci

Many of areas of maths have links to the Fibonacci sequence: here’s one, involving the Schreier sets, that I haven’t seen before.

Define M_n to be all those Schreier sets whose greatest element is n. For instance M_5 = \{ \{5\}, \{5,2\}, \{5,3\}, \{5,4\}, \{5,4,3\} \}. Then the number of elements |M_n| = F_n the n-th Fibonacci number.

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Jenga blocks and Gowers’ hyperplane space

This post is an attempt to communicate some of the feel of Banach space theory to those who aren’t familiar with it. I once tried to explain my research to a six year old using Jenga blocks, but fortunately only got as far as the triangle inequality. Near the end of my Phd, at my supervisor’s suggestion, I started to explore the complicated Banach space that is Timothy Gowers’ solution to Banach’s hyperplane problem. These experiences inspired the following explanation of one relatively simple observation (that I included as an example in my thesis) through the delightful medium of building blocks.

Our object of study are towers of good old-fashioned building blocks. Each block has a number written on its side, so each tower built from these blocks gives a sequences of numbers (x_1, x_2, x_3, \ldots). These don’t have to be positive natural numbers, but you won’t lose much by pretending, in this post, that they are. There are innumerably many different brands of towers, but we’ll concentrate on one particular brand: the ‘Gowers Towers’. Let’s say the number written on each block represents how heavy the block is, and is inversely proportional to the length of the block. So we’d represent the sequence (0, 1, 5, 0, 3, 5, 5, 1, 10) with the Gowers Tower pictured.

It’s worth mentioning that the Gowers Towers include every individual tower of finite height that you can build with your unlimited set of Gowers branded building blocks (and lots of infinite height, but you don’t really need to worry about those here).

Let’s pretend we’ve got a measure of the instability of a tower (the norm of the sequence), and whenever we increase the instability beyond a certain threshold, K, the tower collapses.

Blocks with higher numbers are heavier, as well as narrower and perhaps inherently more unstable. How the blocks of different weights at different heights affect the stability of the Towers of Gowers is extremely complicated. However, the towers do have some nice, intuitive properties.

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Getting into Norms: a technical postscript

For mathematicians’ eyes only. The post doesn’t require much theoretical knowledge to understand, but I haven’t given many definitions.

In the last two posts I’ve been talking about an example I made with four-dimensional vectors a, b, c such that \|a\|_1 > \|b\|_1 > \|c\|_1, \|b\|_{2} > \|c\|_{2} > \|a\|_{2} and \|c\|_{\infty}>\|a\|_{\infty}>\|b\|_{\infty}. Finding it was more difficult than I at first expected, so I thought I would write the investigation up, which happily gives me an excuse to introduce a useful inequality.

My first thoughts were to choose something like c=(10,0,0,0) and a=(6,6,0,0) or a=(4,4,4,0), and then I’d got stuck choosing b. So I decided to try to prove the opposite.

First of all, it’s not possible to create such an example in two dimensions. Continue reading

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