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The big argument: Is the Einstein summation convention worth it?

The Einstein summation convention is a way to write and manipulate vector equations in many dimensions. Simply put, when you see repeated indices, you sum over them, so $\sum_{i=1}^N a_i b_i$ is written $a_i b_i$ for example.

Yes: worth it, argues Ellen Jolley

This debate boils down to just one question: how much of your life do you spend doing tensor algebra? Those of us who undertake a positive amount of tensor algebra or vector calculus know that the goal is to be done with it as fast as possible! Try tensor algebra even five minutes without using the summation convention—I promise you will tire of constantly explaining “yes, the sum still starts from $1$, and yes, it still goes to $N$.”

You’ll scream, “All of them! I am summing over all indices! Obviously! Why’d I ever skip some??” If you’re confused how many you’ve got, use this simple guide: physicists use four; fluid dynamicists use three; and Italian plumbers use two. Wouldn’t it be nice to avoid saying this in every equation?

You may cry that it’s easier to make mistakes with the convention; but for applied mathematicians, the joy comes in speeding ahead to the answer by any means—time spent on accuracy and proof is time wasted. And as the great mathematician Bob Ross said: there are no mistakes, just happy little accidents!

No: not worth it, argues Sophie Maclean

Before writing this argument, I had to Google ‘summation convention’ which is all the evidence I need for why it’s just not worth it. I’ve learnt how to use the convention—multiple times! In fact, I’d say it’s something I’m able to use, yet I’m still not sure I know exactly what it is.

Some of our readers won’t have ever heard of it (which is one strike against it). Some have heard of it but won’t know much about it (another strike). But I guarantee none would be confident saying they can use it without making any errors (if you think you would be, you’re in denial).

We don’t even have need for the convention! We already have a suitable way to notate summation:
\[\sum\]
It’s taught to schoolkids. There is no ambiguity. And it’s so much less pretentious. Yes, the summation convention is fractionally faster to write out, but mathematicians are famed for being lazy and aloof—maybe dispensing with it is all we need to break that stereotype!

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Top ten vote issue 13

What is the best wave?

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Dear Dirichlet, Issue 13

Moonlighting agony uncle Professor Dirichlet answers your personal problems. Want the prof’s help? Send your problems to deardirichlet@chalkdustmagazine.com.

Dear Dirichlet,

As a successful author on spies who are also fish, I’m looking to branch out a little. What with the number of streaming platforms, I’m hoping I can get a TV company to make my series of novels into a ten-episode drama. But it feels like a buyers’ market—how can I hook a producer? Let minnow!

— Micholas Herron, Oxford

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Prize crossnumber, Issue 13

Our original prize crossnumber is featured on pages 58 and 59 of Issue 13.

Rules

  • Each clue in this crossnumber contains two statements joined by a logical connective. If the connective is AND, then both the statements are true. If the connective is NAND, then at most one of the statements is true. If the connective is OR, then at least one of the statements is true. If the connective is NOR, then neither of the statements is true. If the connective is XOR, then exactly one of the statements is true. If the connective is XNOR, then either the statements are both true or they are both false.
  • Although many of the clues have multiple answers, there is only one solution to the completed crossnumber. As usual, no numbers begin with 0. Use of Python, OEIS, Wikipedia, etc. is advised for some of the clues.
  • One randomly selected correct answer will win a £100 Maths Gear goody bag, including non-transitive dice, a Festival of the Spoken Nerd DVD, and much, much more. Three randomly selected runners up will win a Chalkdust T-shirt. Maths Gear is a website that sells nerdy things worldwide, with free UK shipping.
  • To enter, enter the sum of the across clues below by 18 September 2021. Only one entry per person will be accepted. Winners will be notified by email and announced on our blog by 1 November 2021.

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Page 3 model: Cows

Have you ‘herd’? The world’s largest cow is over six feet tall and weighs more than 1.3 tonnes. Is a bigger cow possi-bull? Will the future contain infinitely large cows? The steaks have never been higher!

To answer this question, let’s take a look at the cow’s legs. If the main (meaty) bit of the cow has a volume $V$ and density $\rho$ then its weight is $\rho V g$. So each leg supports a load of about
\[N = \frac{\rho V g}{4}.\]
In pursuit of glory, let’s now make the length, height and width of the cow bigger by a factor $a$. The cow’s new volume is $a^3 V$ and so the load on each leg is $a^3N$: it grows cubically as $a$ increases.

Can the legs cope? If we model the legs as cylinders (since they already ‘lactose’…), we can use a 1757 result from the famous cow enthusiast Euler: if a cylinder has height $L$ and radius $r$, the maximum load it can support standing upright is
\[N_\text{max} = \frac{E \pi^3 r^4}{4 L^2}.\]
$E$ here is just a property of the material: its stiffness, or Young’s moo-dulus.

Cow with cylinders for legs

With our scaling, $L$ and $r$ are now $a$ times bigger. Our new maximum load is
\[\frac{E\pi^3a^4r^4}{4a^2 L^2} = a^2 N_\text{max}.\]

Uh oh… this only scales as $a^2$: quadratically.

So even though $N_\text{max}$ starts above $N$ (it has to, given that these cows exist!), there will come a maximum possible $a$, after which there will beef-ar too much cow and its legs will give way… an udder disaster.

This analysis tells us something really important about biology—that there is a natural maximum size for land mammals. But have we reached it for cows? Brody & Lardy’s 1000-page tome Bioenergetics and Growth from 1946 has all the de-tail you need. We’ll leave you to ruminate on the cow-culations.

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Top Ten: Calculator buttons

This issue, Top Ten features the top ten calculator buttons! Then vote here on the waves for issue 14!

At 10, it’s Mambo No. 5 (A Little Bit Of…) by Lou Bega.
At 9, it’s All Apologies by Nirvana.
At 8, it’s Mambo No.5 (A Little Bit Of…) by Lou Bega.
At 7, it’s Up Allnight by Beck.
At 6, it’s Mambo No.5 (A Little Bit Of...) by Lou Bega.
At 5, it’s M+ambo No.5 (A Little Bit Of…) by Lou Bega.
At 4, it’s Mambo No.5 (A Little Bit 0f…) by Lou Bega.
At 3, it’s My Name = by Eminem.
At 2 this issue, it’s Thunderstruck by AC/DC.
At 1, it’s Mambo No.5 (A Little Bit Off…) by Lou Bega.
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Cryptic crossnumber (including hints)

Cryptic clues can sometimes be a little intimidating, especially if you are not familiar with how they work. However, they all share the same basic structure, and have to follow certain rules. Most clues come in two parts: the definition and the word-play. The definition is a word or phrase which simply means the same thing as the answer to the clue, and it can be found almost invariably at the start or end of the clue. The rest of the clue forms the word-play which constructs the answer in some non-literal way. This could be as an anagram, say, or built out of other unrelated words and initialisms.

Some key things to bear in mind: the clue has to be fair, the setter cannot add in irrelevant words to trick you, and once you know the answer it should make sense how the whole clue points to that answer. What the setter is allowed to do is to write the clue so that it’s meaning at face value distracts from it’s cryptic meaning, and this includes the punctuation, so it is usually best to try to ignore these as much as possible. That said, occasionally punctuation can have a cryptic meaning too, for example “?” can indicate that a clue requires lateral thinking, or that the definition does not literally mean the same thing as the answer. Continue reading