To see past entries in the Carnival Of Mathematics and future scheduled hosts, please visit The Aperiodical.

I am honored to again host the Carnival of Mathematics! I learn so much from hosting; things I usually wouldn’t be exposed to are jam packed into every Carnival Of Mathematics post. Be sure to dig into the archive!

Here are the entries. Enjoy!

From Alice To The Mad Hatter: The Road To A PhD

By Storm Bear Williams

Everyone’s path to a PhD is different. Mine is more madcap than others.

LINK to the mayhem.

Editor’s Note: Please upload sanity. Thank you.

Double Maths First Thing

By Colin Beveridge.

Double Maths First Thing is Colin’s weekly news bulletin. Although it’s more like a nerfpelletin, honestly.

Hello! My name is Colin and I am a mathematician on a mission to help everyone find the joy and delight in figuring things out.

LINK

Editor’s Note: Colin is already published four entries!

XKCD Game Theory Question

By Mark Dominus

This post is about the bottom center panel, “Game Theory final exam”.

I don’t know much about game theory and I haven’t seen any other discussion of this question. But I have a strategy I think is plausible and I’m somewhat pleased with.

(I assume that answers to the exam question must be real numbers — not ∞∞ — and that “average” here is short for ‘arithmetic mean’.)

First, I believe the other players and I must find a way to agree on what the average will be, or else we are all doomed. We can’t communicate, so we should choose a Schelling point and hope that everyone else chooses the same one. Fortunately, there is only one distinguished choice: zero. So I will try to make the average zero and I will hope that others are trying to do the same.

LINK

Editor’s Note: Inspired from the XKCD web cartoon.

Planar point sets with forbidden four-point patterns and few distinct distances

by Terence Tao

I’ve just uploaded to the arXiv my paper  “Planar point sets with forbidden 4- point patterns and few distinct distance“. This (very) short paper was a byproduct of my recent explorations of the Erdös problem website in recent months, with a vague emerging plan to locate a suitable problem that might be suitable for some combination of a crowdsourced “Polymath” style project and/or a test case for emerging AI tools. The question below was one potential candidate; however, upon reviewing the literature on the problem, I noticed that the existing techniques only needed one additional tweak to fully resolve the problem. So I ended up writing this note instead to close off the problem.

LINK

Editor’s Note: Terance’s website if loaded with his work in mathematics.

Mathematics With Friends

by Ioanna Georgiou and illustrated by Asuka Young

“Mathematics with Friends” is a blog that takes a
maths-related dialogue from the Friends TV series every month and explores the
maths mentioned further. It comes with its own characters and amazing
illustrations.

LINK to the latest episode

Books That Are About Maths, But Not About Teaching It

By Tom Briggs

In the spirit of getting straight to the point without providing my entire life history, please click below for a list of books that are about maths, but not about teaching it. Read beyond the link if you’d like a bit of context.

The list has been put together at The StoryGraph^[2]: it can be browsed without creating an account and it can be searched, sorted and filtered, which I thought would be helpful as there are over 160 books.

Link to the article.

LINK to the exhaustive list.

The end (or not?)

by Peter Cameron

This is an update from Peter on his retirement from St. Andrews and where he will go from there.

Yes, there is life after retirement. Don’t fret, it will all turn out ok.

LINK

OTHER LINK: Peter also has a nice article on the possibilities of a new mathematics open access journal. Could there be hope ahead?

Saying Goodbye to Tom Blyth

by Peter Cameron

A respectful story about the passing of a colleague. These things are never easy. Also an interesting correlation with Paul Halmos, the Hungarian-American mathematician. The full story is in the link below.

LINK

What’s your mathematical style?

By Writing and Illustrations by Ben Orlin

Math teacher Ben Orlin writes and draws the (aptly named) blog Math With Drawings and is the author of a new book, Math for English Majors: A Human Take on the Universal Language. To mark its publication, he devised this entertaining accompanying quiz. You can read the Ars interview with Orlin here.

Math, we like to say, is math. It often seems that mathematics is something beyond our control—something unyielding, universal, inhuman.

But is it really?

Yes and no. Math may be the language of the cosmos, but it’s also a human language, with its own quirky etymologies, classic literature, and hilarious regional accents. In fact, you could say that math has as many styles—as many idiolects—as it has speakers.

LINK

Editor’s Note: I loved this article!

Viterbo’s conjecture was refuted by Pazit Haim-Kislev and Yaron Ostrover

by Gil Kalai

Claude Viterbo’s 2000 volume-capacity conjecture asserts that the Euclidean (even dimensional) ball maximizes  (every) symplectic capacity  among convex bodies of the same volume. In the recent paper A Counterexample to Viterbo’s Conjecture, Pazit Haim-Kislev and Yaron Ostrover disproved the conjecture. 

There is some background on symplectic geometry in this post.  

LINK

Posted from Gil Kalai’s blog which is very bookmark-worthy

Editor’s Note: This deserves applause I think!

Philip Henslowe’s Card Trick

by Colin Beveridge

In a dimly-lit tavern on the South Bank of the Thames, Philip Henslowe — builder and owner of the Rose Theatre — is celebrating the success of Shakespeare’s latest blockbuster, Henry VI Part I, among the cutthroats, actors and other lowlife of London. He spreads thirteen playing cards on a table in a circle. “Pick a card,” he grins. “Any card.”

And this article is complete with cryptic, vintage explanations!

LINK

Wiki Entry for Philip Henslowe

Editor’s Note: Now I want to visit the Dulwich College Archive!

Book Review: The Best Writing on Mathematics 2021

by Ioanna Georgiou

Mircea Pitici (Editor)
PRINCETON UNIVERSITY PRESS 2022, 320 PAGES
PRICE (PAPERBACK) £20.00 ISBN 978-0-691-22570-8

This book contains articles written in 2020. The editor speaks of the difficulties accessing libraries and resources, something we are all a bit too familiar with. Here I will comment on each of the twenty-six articles selected as the ‘best’ and draw an overall conclusion at the end.

LINK

Editor’s Note: We have to start doing a better job in archiving and making available the collective of human knowledge to everyone. The defunding of libraries has to stop. The abusive research publication industry has to stop.

1. They don’t always do all of their homework.

In college, homework assignments generally make up 5-20% of your grade, but can be the biggest time-suck for most students. Yes, working problems is one of the best ways to turn new concepts into working knowledge, but a large majority of those problems that take you hours and hours to work through, you’ll never see on an exam.

2. They never “read through” the textbook.

Per time spent, reading the textbook is one of the least effective methods for learning new material. Top students use the examples and practice problems, but otherwise use Google, lecture notes, and old exams for study materials.

3. They Google EVERYTHING.

It’s like an automatic reaction. New concept = go to Google for a quick explanation. Don’t think just because your professor gives you a textbook and some examples on the blackboard that you’re limited to that information. You have a massive free search engine at your fingertips, so make use of it.

4. They test themselves frequently.

Testing yourself strengthens your brain’s connections to new material, and gives you immediate and clear feedback on whether you know something or not. Bottom line, repeated self-testing significantly improves long-term retention of new material. 

5. They study in short bursts, not long marathons.

Studying in short bursts tends to help you focus intensely because you know there is at least a short break coming.

This also fits in nicely with our Ultradian Rhythm, the natural activity/rest cycle of our bodies, which makes studying continuously for multiple hours on end counterproductive.

6. They reverse-engineer solved problems.

It’s one thing to follow and memorize a set of steps to solve a calculus problem. It’s an entirely different thing to understand what a derivative is, be able to take derivates of complex functions, know when to use the chain rule vs. the product rule, etc. The problem with simply following the steps the professor provided, or the textbook outlines, is that you’re only achieving a surface-level knowledge of the problem. Top students, instead,take solved problems and work backwards, from solution to question, asking “why.”

Why did this get this value? Why did they simplify this expression? Why did they use that type of derivative rule?

By following this process, you begin to understand the interconnections of the concept, and how to directly apply that to a problem. This “working knowledge” of a concept is key to performing well on exams, especially on problems that you haven’t seen before.

7. They don’t own a highlighter.

Highlighting anything = unengaged reading. If you want to note something that stands out, underline and write a corresponding note to go along with it. Or better yet, write yourself a note summarizing the item in your own words.

8. They sleep–a lot.

The daily routines of top performers, in any field, are characterized by periods of intense work (4-6 hours per day) followed by significant quantities of high-quality sleep (9 hours per night). You see this trend in top violin prodigies and chess champions, as well as elite athletes. The idea is to alternate periods of intense work with rest, so that you create tons of new connections in your nervous system, and then allow adequate time to assimilate those gains.

9. They engage themselves by asking questions.

What happens if I tell you, “Thomas Jefferson almost single-handedly drafted the Delcaration of Independence in 1776.”?

You might say “Hmm.. that’s interesting”, try to remember it for later, maybe even write down a note or two.

But what if I ask you, “Who was Thomas Jefferson?” What changes?

You start searching your memory, sifting through images of old guys, founding fathers, thinking about the Declaration of Independence. You come up with your own narrative, and then realize that you have gaps.

When was he around again? And why was he so important?

You’ll probably find yourself going to Google to fill in the gaps. Through that process your learning will be much more deeply seated in your brain than anything your history teacher ever told you about him. That’s the power of asking questions.

10. They make the best out of lecture.

Yes, your professor sucks. Yes, lectures are boring. Yes, it’s either too fast so you can’t keep up and miss all the important stuff, or it’s way too slow and you start zoning out because you already understand everything.

The best students look at this this way: I’m going to be there no matter what, so what’s the best use of my time while I’m in the classroom? Ask questions, bring the textbook and look stuff up, focus on the important practice problems to copy down in your notes, try to anticipate what the professor is going to say, make note of anything they put emphasis on as a potential exam topic. All of these things make the time you have to spend in lecture more productive and engaging. And that’s less time you have to spend studying later on.

11. They over-learn.

School is hard enough, with the amount of studying and homework you have to do. And on top of all of that Facebooking you have to get done? It might seem ridiculous to suggest learning more than you have to.

What!? Are you insane!?

But this is precisely what top students do. And paradoxically, they end up spending less time trying to understand how to do homework problems, andless time studying for exams because of it. Because when you “over-learn” past what’s presented in class, you build a better framework for the subject.

Think of trying to remember some details about Abraham Lincoln’s life. You try to remember the dates of the Civil War, or what he said in the Emancipation Proclamation. You study the same facts over and over and over again… but it’s just boring, and you quickly forget. But what if you knew his whole life’s story? About how Lincoln suffered from bouts of depression, and his relationship with his wife suffered? You start to learn that the dude was human, and you start to relate to the things he did and the struggles he went through. Now you’ve constructed a story in your head. And studies show that humans learn best through stories. So yes, it’s more information, but your brain knows what to do with it now that all those random facts are linked together. More learning, but less rote memorization and struggling to remember random facts.

12. They immediately study their exam mistakes.

Most students get their exam grade back, flip through to see if the professor made any mistakes they can argue about, and then promptly shove it into their notebook, never to be seen again until the mad scramble at the end of the semester to study for the final.

Instead, top students ignore what they got right, and use their mistakes as an indicator of what to improve on.

13. They’re busy with work and side projects.

Yes, to do well in a course, you need to focus and put in the hours. But like many geniuses throughout history have shown, involvement in a diverse set of subjects, activities, and skill sets keeps you active, and provides you with a rich and diverse set of mental models to pull from.

Also, as they say, “If you need to get something done, give it to the busy person.” If you stay active in multiple areas, you don’t have time to procrastinate, and are forced to be efficient with your study time. This generally translates into quicker learning and better performance throughout the semester.

14. They use lecture as a detective mission.

Though completely unaware of this fact, your professor has tells. Yes, like in poker. Tells during lecture will hint at particular types of concepts and problems that will be emphasized on the midterm or final exam. The best students pay attention to topics professors spend a seemingly inordinate amount of time on and make note. Chances are you’ll see something related on the final.

15. They don’t wait for motivation to strike.

Motivation comes and goes, but studying for a degree requires persistence and consistency. Just like Olympic athletes train even on their worst days, the best students figure out how to get their coursework done when it’s the last thing they want to do.

16. They practice under test conditions. The old adage “practice makes perfect” isn’t totally true. Deliberate practice under the right conditions, with the correct mindset, is more like it. Instead of reading through all of the lecture notes and redoing old homework problems, top students make themselves practice exams, and rehearse their exam performance, under time pressure and in similar conditions (no notes, uncomfortable chair, quiet room, etc.) to what they’ll see on test day.

17. They use old exams.

Professors aren’t the most inventive folk. Along with coming up with lecture material and departmental responsibilities, they’re also primarily concerned with research. So typically midterms and final exams more or less look alike for similar courses year-to-year and even across universities. Because of this, old exams are a gold mine of opportunity for figuring out what problems you should be able to solve and study from.

18. They make their own study guides.

The best students don’t simply use the study guide the teacher provides, they create their own.

Creating the study guide is half the battle, requiring you to go through your notes, consolidate them, and organize them in a way that you understand–all valuable study activities. You’ll also be able to use your equations sheet much more effectively on the exam itself (if allowed) because you know exactly where everything is.

19. They actually write on paper.

Writing out notes on a laptop is efficient. Too efficient. Because it’s so easy to quickly type out exactly what the professor is saying, you don’t have to do the work of trying to figure out how to consolidate the information into your own shorthand. Some also believe that the act of writing helps retain more information.

20. They use the 80/20 rule.

Yes, some students who get good grades do every reading assignment, finish every practice problem, and attend every study session they can get their hands on. But these students are missing the point. There will always be an endless amount of information you could learn given the time and effort, but having the ability to discern what is worth learning will truly set you apart.

Top students identify the 20% of concepts they need to learn deeply, in order to determine 80% of their final grade. They focus intently on those few things, and simply ignore the rest. This is a formula for high performance, without hours and hours of busywork. And it translates seamlessly into the real world too.

21. They don’t complain.

Complaining simply has no place in the smart student’s repertoire. If something sucks, change it or ignore it, but don’t waste your time, energy, and mental state talking about it. Got a crappy professor? Either switch class sections or focus on teaching yourself. Horrible textbook? Find alternate resources (Google is free in case you hadn’t heard).

22. They learn by doing.

Any technical subject can only truly be internalized through use. Just like learning a new language, learning to be fluent in algebra or calculus requires active application of rules and formulas. Top students know there is a big difference between knowledge, and applied knowledge.

23. They take personal responsibility for learning the material.

The best students understand that they, and only they are truly responsible for their own education. So waiting to be spoon-fed by your professor and doing the homework assignments will never be enough. Despite your school’s best intentions, they’ll never be as committed to your academic success as you can be.

24. Following what they love

Those students you admire are passionate about what they are learning. They have the drive to develop their learning further based on their love of what they are discovering. This may not always be the case and is often unavoidable but if you follow what interests you and cultivate a curiosity of this area, your motivation to learn will thrive.

Not every student is the same and many top students don’t follow the status quo. The best way to create good habits for students is to try a variety of techniques and figure out what works for you.

25. Question your teachers Thinking outside the box is a cliche but certainly a reality for students.  They question everything–especially test questions they get wrong.  This attitude is important because it shows a general inquisitiveness that is essential in learning.  As any parent of small children knows, questions are a way to gain knowledge.  Teachers can’t be offended when a student asks a difficult question and parents should encourage this behavior.  

26. They know the best way to study.

It is important to know the best way to study for yourself. Do you need pictures? Sounds? Study better in quiet or noisy environments? Figure out what works best for you so that you can make the most out of your study time.

27. They play hard.

We all know that a balanced lifestyle is the best way to stay mentally and physically healthy! Top students don’t spend all day in the library grafting (contrary to what you might think!) Rather, they take the time to put their studies aside and do something which is fun and exciting!

 Source : 1 2 3 4

I am honored to again host the Carnival of Mathematics! I learn so much from hosting; things I usually wouldn’t be exposed to are jam packed into every Carnival Of Mathematics post. Be sure to dig into the archive!.

See you in October!!!!

July 5th, 2022 was a damn good day. It was the day I moved to Mojave, California for my new job at Virgin Galactic.

It is a dream job! Where else will they allow me to touch spaceships inappropriately? Like anytime I want?

I started out in Quality Assurance finishing up the mods on Eve before she moved to New Mexico. I followed the mothership to the New Mexico Desert and fell in love with Las Cruces.

So yeah, be jealous!

To see past entries in the Carnival Of Mathematics and future scheduled hosts, please visit The Aperiodical.

I am honored to again host the Carnival of Mathematics! I learn so much from hosting; things I usually wouldn’t be exposed to are jam packed into every Carnival Of Mathematics post. Be sure to dig into the archive!.

Here are the entries. Enjoy!

Math Vs Culture

by Storm Bear Williams

A short little rant about all of those forgotten mathematicians from antiquity that hardly ever get the credit they deserve.

Mom Does Math

by Dr. Alyssa J Foss

I’m on the life long journey to recover from math-phobia. Which is why I have thrown myself into the math world head first! Next to momming and mathing, writing is my favorite thing. So, I’m trying to do the blog thing. This is my first post from September.

LINK

Editor’s Note: Her blog, Math Rehab, is really well written. Very worth the read!

What Is the Pascal Matrix?

By Nick Highham

In mathematics, particularly matrix theory and combinatorics, a Pascal matrix is a (possibly infinite) matrix containing the binomial coefficients as its elements. It is thus an encoding of Pascal’s triangle in matrix form. There are three natural ways to achieve this: as a lower-triangular matrix, an upper-triangular matrix, or a symmetric matrix.

Here Professor Higham steps his way through an example.

This article is part of the “What Is” series, available from https://nhigham.com/index-of-what-is-articles/ and in PDF form from the GitHub repository https://github.com/higham/what-is.

LINK

Editor’s Note: For more on Professor Highham; Twitter, Wiki, Google Scholar, Blog.

Ivan Guo: Financial models of the future

by Dr Ivan Guo

How can a 240-year-old logistics problem be used in quantitative finance? Dr Ivan Guo’s research lies predominantly in the areas of stochastic control and financial mathematics. In this interview with the Sydney Mathematical Research Institute, Ivan describes how stochastic transport theory applies in financial maths and how financial models are applied. He also debunks some misunderstandings about his field.

What does craiyon/DALL·E mini ‘think’ mathematics and mathematicians look like?

By Peter Rowlett

“You may have seen DALL·E mini posts appearing on social media for a little while now – it’s been viral for a couple of weeks, according to Know Your Meme. It’s an artificial intelligence model for producing images, operating as an open-source project mimicking the DALL·E system from company OpenAI but trained on a smaller dataset.”

Peter Rowlett presents a good introduction to DALL-E and offers several awesome examples.

LINK

Two Saturday Morning Breakfast Cereal Cartoons

by Zach Weinersmith

The first fun cartoon is titled “Incomplete”. LINK

The second is titled “Mathematics” and we can all feel this one down in our bones! LINK

Editor’s Note: You can find out more about the author here: Twitter, Facebook, Tumblr, Patreon

G0lomb

by Sneak Thief

One of my (many) interest areas is in algorithmic music composition. But it was not until @CarnivalOfMath mentioned that content didn’t need to be blogs, that I thought to submit this.

This piece is one where I use the Golomb ruler to determine the bar position, and length, of notes in a composition. I wrote a short piece of JavaScript which creates the data, incorporate it with a MIDI library which exports a MIDI file, that in turn can be loaded into a sequencer. I then assign each note to specific sound, based on their duration and what my ear tells me is good. (Being a synth-based composer, some sounds change a lot over time and are therefore better for long notes.)

The notes are always determined by me, a human, to match a particular key signature (to they sound in tune) and varied according to previous trial runs of the algorithm. So, for example, if notes of length 10 and 11 do not appear simultaneously I can assign them the notes E and F which (normally) do not sound good together. Similarly, I try to ensure that the start and end of the composition include notes which give an element of “resolution” which is prevalent in most western music. (Just because it’s based on maths, doesn’t mean it has to _sound_ that way!)

Finally, I sprinkle additional sounds generated by a different algorithm.

Click the image within the link to see a representation of the music, and 
you’ll clearly see the ruler being used.

LINK

If you spend just a few minutes on Twitter’s #PhD you will see a range of emotions displayed by those on the path to earning a PhD.

These brave individuals all seem to start out as shining little newbies, fresh out of the newbie bag – with wonder and star-struck innocence in their eyes, eager to start on their journey to increase human knowledge and be known as Doctor.

As they chase the white rabbit of their academic dreams, they soon realize they are spiraling down – not down a rabbit hole, but a deep, festering abyssal where dreams, happiness and wonder are exchanged for stress, tears, fear and abject frustration.

They battle multiple complications and obstacles, all coming at them at once, from all directions. Supervisors seem to produce the most anxiety and dread. Errant supervisors, abusive supervisors, absent supervisors, the handsy supervisor with a furry fetish – they all are problematic and add to the toil of getting a PhD.

Once you FINALLY make your saving throw against the bad supervisor, you get to read. Well, kinda read. Maybe read. You get to read all you want if you have access to Elsevier, if not you either have to get real lucky and find all your relevant research on open servers, find them available on the piratey Sci-Hub or sell a kidney. How science became so closed it has become a real disservice to humanity and modern civilization.

Then the clock gets involved. Days turn to weeks, months to years – and years turn into chapters. You see the end of the tunnel. Light harkens through the darkness, crawling its way to you – showing you a new future… no, fuck that, “major revisions required.”

And just as the funding begins to run out and your Xanax prescription has seen its last refill, you submit FinalFinalFinalThesis-Revision9-Version4-WithChangeNotesSierra.pdf to your supervisor – the second one. The first one took a job at Google. Well, they tell you he went to Google, you can’t exactly recall. You spent a few months in a black out drunk when you heard the news that you would need to reroll the saving throw.

Then…

Finally…

Submitted.

You hear nothing back for a few days.

Panic attacks.

You are now in full Mad Hatter mode. At this point, you are sitting naked, in a room only illuminated by your laptop, tapping violently on the keyboard as you try to explain the scientific method to the QAnon trolls on Facebook.

You wake up on a random Wednesday, fending off a Redbull-induced stroke, to hear that your dissertation defense has been scheduled.

Can this be true? A smell of disbelief fills the room, reminding you you haven’t showered since you submitted.

The door to a conference room opens and you step through to see your second supervisor and other members of the dissertation committee. As the door closes shut you try to recall how you got there.

Time passes.

As the door reopens, the pressure seems to leave the room and the others congratulate you on a job well down. You are now a Doctor Of Philosophy.

This story may not be reality but it seems to be when I read the horror stories of others who have successfully, and sometimes unsuccessfully, traveled this path.

So now, it is my turn. PhD here I come! I have always wanted to earn a PhD ever since I met Carl Sagan. I love science, all aspects of it, all subjects – I cannot get enough. Even as I approach that time where cashiers will give me the “Senior Discount” I continue to be in awe at the things we have discovered. But I could never narrow down what topic to pursue. You can’t just get a PhD in “General Studies,” you have to be startlingly specific. Decades passed and I could never pick something; biology to chemistry, then on to astrophysics, then back to biology… maybe archaeology. I would read a new article or get triggered by advances in understanding of mycelium networks. Who wouldn’t want to do a deep dive on that? It wasn’t until I read the XKCD cartoon titled Purity that I got it.

With all of these subjects that I have an intense interest in, looking at this cartoon made the secluded seem obvious. Mathematics is the base code of the universe and a PhD in math would allow me to peek into the underpinnings of the classic sciences; biology, chemistry and physics. I should have known, but I didn’t. I was too busy in my autism loop process to take the accretion disk point of view.

I have decided that PhD by Previous Publication (PPP) is the best for me. The idea of it doesn’t trigger my autistic predispositions to testing or intellectual judgement. I kinda go at my own pace. I think about my project, write the papers that will take the reader through the story. Rinse and repeat until the body of work is sufficient for defense.

So here I go, one step in front of another, into the future – I mean, Costco to buy a pallet of Red Bull.