Diversions in Mathematics 3b: The Cauchy-Schwarz inequality (part 2)

In the previous post, we introduced the Cauchy-Schwarz inequality purely as a statement about real numbers:

\displaystyle \sum_{k=1}^n x_k y_k \le \left( \displaystyle \sum_{k=1}^n x_k^2 \right)^{1/2} \left( \displaystyle \sum_{k=1}^n y_k^2 \right)^{1/2}.

We proved this inequality using mathematical induction. However, one could argue that the method did not yield much insight. As our proof was entirely algebraic, it is not easy to see why the sums that appear in the inequality are reasonable quantities to consider, and there is no hint as to how they could arise naturally in mathematical problems. Therefore, in this post, we will look at the Cauchy-Schwarz inequality from a geometric perspective, which reveals a much richer structure. Moreover, we will see how the inequality arises naturally from a basic optimisation problem.

Note: I continue with the convention from the previous post that a quantity x is called positive if x \ge 0. We say that x is strictly positive if x > 0.

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Diversions in Mathematics 3a: The Cauchy-Schwarz inequality (part 1)


After a long time, I have decided to resurrect the mathematics part of this blog! I started Diversions in Mathematics as a way for me to try to explain mathematics to the general public. This continues to be the main goal of this series of blogposts — for a more detailed introduction, please read the introductory remarks. I wrote two posts in this series, but then abruptly stopped. Part of the reason was that my studies got in the way, but I was also unsure exactly what material to present, and how to present it. I wrote down some of my thoughts on this matter in a previous update post.

One of the main issues as a writer is to consider the readers’ background in mathematics. For posts targeting the general public (like the previous two Diversions), I have tried to assume as little as possible while maintaining the discussion at an intelligent level, i.e. without “dumbing down” anything) However, this already assumes familiarity with many mathematical concepts taught in high school, or at least, some level of maturity in regard to abstract reasoning. Consequently I have decided to relaunch Diversions in Mathematics with high school mathematics as a foundation.

In this blogpost, I will introduce the Cauchy-Schwarz inequality, one of the most fundamental results in mathematical analysis, with the aim of connecting various topics that are typically studied in the Year 12 HSC maths curriculum in NSW. The main article is below.

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Conversation with Katie Bergløf

Recently (1 June in Australia) I spoke with hornist Katie Bergløf over Zoom about focal dystonia. Check out the conversation here:

There are many more conversations with other musicians who are dealing with various forms of focal dystonia. I highly recommend checking out the Youtube playlist: https://www.youtube.com/user/HeldenCors/playlists

Katie’s website: https://www.focalembouchuredystonia.com/

On the value of recording your mistakes (occasionally)

Let me tell you about a mistake I made recently. Some months ago, I made a few calculations which seemed promising, and I thought they could lead to some new results. When I showed the calculations to my supervisor, he was skeptical, and rightly so. Our project involved some fairly abstract concepts in the theory of operator semigroups, so his immediate advice was to check the calculations against some simple, concrete examples . So of course I did that, and quickly realised that the calculations were totally wrong! Moreover, they were wrong for a rather silly, elementary reason — a pitfall that I thought I should be capable of detecting with my current mathematical experience.

To clarify what I mean by “elementary”: no, it wasn’t a silly error in the sense of a missing minus sign or multiplying two numbers incorrectly. Mathematicians say that something in their field of study is “elementary” if it is a basic fact in that particular field. However, to paraphrase my supervisor, “basic does not mean easy”, so if I were to describe the mistake to a non-mathematical audience, it would still be extremely difficult. Another interesting point is the following. When you do research in pure mathematics, quite often you deal with abstractions upon abstractions, and it is easy to forget the humble origins of these abstract concepts. For example, the concept of a linear operator is fundamental to many areas of maths, and as cool as it is to be working with linear operators on infinite dimensional Banach spaces or whatever, it is also worth checking a statement about abstract linear operators on, let’s say, a 2-by-2 matrix as encountered in 1st year university maths (or even high school, depending on where you were educated). This comic sums up the sentiment quite well (excuse the pun).

Source: https://mathwithbaddrawings.com/2017/01/11/why-are-mathematicians-so-bad-at-arithmetic/

Now this brings us to the main point of this blogpost. The wrong calculation I described above is not the mistake I want to write about; rather, I want to focus on what happened next. I assume that most people do not enjoy looking at their failures. I am not sure what I did with those few pages of calculations: either I threw them out, or I shelved them away somewhere I had forgotten. In any case, recently I wanted to return to the same problem with a different perspective, but now I have lost my point of reference. Because I had misplaced or thrown out the incorrect calculations, I cannot really remember what I had got wrong the first time! To be fair to myself, I have a vague idea of the kind of error it was, and it is very unlikely I will make exactly the same error again, but nevertheless it would have been more efficient to keep the original calculations, wrong as they are, so that I can remember exactly what the lapse of judgment was. As I have learned by now, mathematical work is not only about putting one true statement after another. It is equally important and even illuminating to know what kind of result cannot be true. In hindsight, throwing away those calculations was a bigger mistake than the calculations themselves.

There are certain similarities with the performing arts, although I have to admit that the comparison gets fuzzy on closer inspection. For a start, mistakes in mathematics are objective errors: you are objectively wrong if you claim that 1 + 1 = 3, for instance. On the other hand, speaking from a background in Western classical music, even though there are certain objective errors (e.g. playing a wrong note), usually we are unhappy with a performance because we feel that we could have done certain things better. One could say this is a “subjective” error. Despite this fundamental difference though, both mathematicians and performing artists undeniably improve by learning from their mistakes, objective or subjective. Indeed, I would say that learning from mistakes is an essential part of development in the two disciplines.

As a violin student, I used to dislike making recordings — what a way to highlight one’s faults and weaknesses! However, in hindsight I should have taken more opportunities to record my playing, and to embrace whatever mistakes (objective or subjective) occurred during a performance.  One should not take this viewpoint too pessimistically. The idea is not to focus only on failures; rather it is about using those failures strategically to improve one’s understanding or problem solving technique in the case of mathematics, or one’s expression capabilities in the case of the performing arts.

I end this post with a nice little comic, which comes with some wise words from the great mathematician Paul Halmos:


Update post (July 2019)

It seems that every time I return to write something on this blog, it’s about how infrequently I write (and that I’m awfully sorry, and I promise to write more, but it never happens). I see three main reasons for this:

  • I am now in a masters program in the School of Mathematics at the University of Sydney! My honours year went quite well overall, and my supervisor was happy to take me as a postgraduate research student.
  • Since March, I have also been working casually for Matrix Education, a private tutoring company that is well-known in particular for its HSC preparation courses.

From the two reasons listed above, you can appreciate that most of my time is spent doing mathematics in some way — reading articles and learning new theory for my research project, working on exercises from textbooks, or preparing tutorials and classes. The third reason is really a blend of two:

  • My own laziness coupled with a lack of direction about this blog.

This update post will explain this last point further.


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On the importance of chamber music

On 19 and 20 January, I had the pleasure of teaching at the Zhang Violin Summer School for young violinists (AMEB grade 3 to A.Mus equivalent). This was a 4-day intensive course, where students received training in preparing solo repertoire and chamber music, as well as Dalcroze eurhythmics. I was invited to take chamber music lessons, a task that I gladly accepted. There was quite a wide range of ages: the youngest students were in years 1 and 2, while the eldest ones were senior school students. As a result, there was a similar range of experience in chamber music. Teaching at the summer school has reinforced my belief of the high importance of chamber music in music education, and in this blogpost I would like to share some of my thoughts on the subject.

First of all, I remark that it is a happy coincidence that the summer school was held at Barker College, where I was a former student. It was at Barker that I first experienced playing chamber music — specifically, the string quartet. As it turned out, I played string quartets throughout my entire education at Barker College, and it was one of the most important factors in my development as a musician — not only as a performer, but also as a composer! The main point that I will emphasise is this: playing chamber music makes great demands on the musicians involved, in technical aspects (i.e. the physical part of instrumental playing) as well as conceptual (i.e. intellectual understanding of the music). These demands are quite different from those encountered in solo playing, and furthermore, I will try to argue that chamber music should play a significant role in a student’s musical development.

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Composer’s Notebook #7: Variations for wind quintet

This blogpost is a short commentary on my new quintet for winds. Click here for the Youtube video.

The Variations were not my first attempt to write for wind instruments, but it is my first finished work for winds. (I had previously attempted solo pieces for oboe and bassoon, and “sonata”-like pieces for oboe and piano. I did not think any of these could have been successful, and so I have discarded all such fragments). The piece bears the subtitle Small Steps and Giant Leaps, primarily because the foundation of the whole piece is the main theme of John Coltrane’s famous piece Giant Steps. The subtitle also plays on Neil Armstrong’s famous words “One small step for man, one giant leap for mankind.” One could interpret this as representative of the fact that the quintet is my first completed work for winds, and hence it represents an important new step in my compositional development. At the risk of disappointing music analysts, the real reason is much more mundane. I think it is simply a rather cool title. If one has to interpret it, then I can offer the following suggestion: I tend to write melodic lines with large jumps, while the Giant Steps theme heavily features steps of major and minor thirds. It is the combination of these steps and leaps that characterise much of the material in this piece.

On the concept of ‘variation’

If you take a look at my Youtube channel, you will find other attempts at variation form. Perhaps the best one preceding the wind quintet is the Intermezzo festivo for string quartet. However, that piece follows the classical variation form more closely than the wind quintet, at least initially. (The piece transitions into a freer form halfway through). In the quintet, the Giant Steps theme does not appear until the very end of the work. This is in opposition to the classical form, where one hears the theme at the beginning, and then follow the variations. In this sense, even Arnold Schoenberg’s Variations for Orchestra (Op. 31) is classical, although the ways in which he varies the theme are of course more intricate and abstract than what is generally encountered in pre-20th century works. How, then, is my quintet a set of variations? Although the score is divided into sections which may indeed be identified with particular variations, it is not obvious (or at least it is not supposed to be obvious) how they are variations of a theme, which is not explicitly stated until the end anyway. Here, the word ‘variation’ conveys a much more general principle, which I think is similar to the term developing variation, often used in connection with Brahms. Another term I like is thematische Arbeit, or ‘thematic working’, which is often attributed to Joseph Haydn. I think the second term is more flexible, and hence easier to appropriate into a modern context. The general recipe is to start with a basic idea (the simpler the better), and see how much can be generated from it. Then introduce some embellishments, pertubations, variations — this produces a new but related idea. Now consider variations of this second idea, and so on. Of course, this process can happen in a nonlinear way. Moreover, the basic idea need not be a melodic fragment (although this is often a natural choice), but it can be something rather abstract. In my quintet, the basic idea, or Ursatz (a gross misuse of a term from Schenkerian analysis), consists of the following pair of elements:


The chord is comprised of the opening bass notes, while the second element is first five notes of the melody in Giant Steps. Observe that both elements coincidentally contain five notes, which is perfect for a quintet. An important secondary idea is the following voice-leading pattern, also featuring extensively in the melodic line of Giant Steps:


These two ideas comprise the essence of my wind quintet. I noted above that my use of the term Ursatz is a gross misrepresentation. In Schenkerian analysis, the Ursatz is supposed to be the fundamental structure of the entire piece — to put it facetiously, this means that “all of classical music is essentially the chord progression I-V-I.” However, in my current compositional process, the basic idea only needs to affect the ‘surface’ of the piece, and it does not necessarily determine the large-scale structure. (Controlling large-scale structure remains one of my greatest challenges — you will notice that all the pieces I have written so far are quite short). The fact that the elements presented above do affect the large-scale structure of the wind quintet is the reason for the title Variations. These three elements are collectively the ‘theme’ of the composition.

Remarks on the structure

I will offer some comments on the structure of the quintet that may be helpful for both players and listeners. It is easily observed that each variation features one of the instruments of the quintet. The order of appearance is: flute, clarinet, oboe, horn, bassoon. After these five variations, there is a fugal variation, which leads into the coda, where the Giant Steps theme is finally present. The way the variations are organised suggests an embedded multi-movement structure. One possible partition of the piece into movements is as follows:

  • 1st movement: introduction, flute variation, clarinet variation (“Scherzo”)
  • 2nd movement: oboe variation (“Adagio”), horn variation, bassoon variation (“Cadenza”)
  • 3rd movement (Finale): fugue, Giant Steps coda

Notice also that the bassoon “cadenza” recalls material from the introduction (namely, the staggered chord-building entries).