Within arithmetic geometry, the “Cox-Zucker machine” is an algorithm created by David A. Cox and Steven Zucker to determine the intersection numbers of elliptical surface sections.
When Cox and Zucker met as first-year graduate students at Princeton University, “we realized that we had to write a joint paper because the combination of our last names, in the usual alphabetical order, is remarkably obscene.” Five years later, as members of the faculty at Rutgers, the State University of New Jersey, they followed through with their plan.
‘Intersection numbers of sections of elliptic surfaces’, Cox, D.A. & Zucker, S., 1979, Invent Math 53, 1–44. https://doi.org/10.1007/BF01403189
Author bio:
petro c. k. laughs at life’s absurdities. And occasionally writes about them. His work appears in numerous haiku journals and experimental literary/poetry magazines, and he will have debut experimental haiku and dada poetry collections out in 2024. He encourages nanononsense at dadakuku.com You can catch up with petro on Twitter here: @petro_ck
John Hawkhead (@HawkheadJohn) has been writing haiku and illustrating for over 25 years. His work has been published all over the world and he has won a number of haiku competitions. John’s books of haiku and senryu, ‘Small Shadows’ and ‘Bone Moon’, are now available from Alba Publishing (http://www.albapublishing.com/).
Optimal stopping for sexist scenarios – an odd solution.
The Secretary Problem is the ‘fear of missing out’ in mathematical form. It encompasses the idea of deciding when to stop looking through a list of options when you can’t go back to a previously dismissed option and don’t know what future options might hold. It presents the scenario that you’re hiring a secretary and want to hire the very best. You consider each candidate in turn and must decide to hire or reject them immediately after their interview. You know how good they are and how good all those applicants before them were but you’ve no way of knowing how good the remaining applicants are – the best may be yet to come. When do you hire someone, when do you decide which is best?
It’s a problem involving optimal stopping theory – choosing when to take an action to maximise the expected reward and/or minimise the expected cost. When do you stop rejecting applicants to get the best secretary?
The shortest proof to the problem is the odds algorithm devised by F. Thomas Bruss in 2000. The proof equates to the idea that when hiring a secretary under the strict conditions of the problem, you should view a certain number of applicants and then hire the next applicant that scores higher than any of those. The number of applicants to reject first is defined by ~ n/e where n is the total number of applicants and e is the base of the natural logarithm. This proof of the problem is likely to select the single best applicant from the total pool of applicants around 37% of the time, regardless of how many applicants there are.
Curiously The Secretary Problem has been known by a number of different names, most of which involve men picking between women in some way: the marriage problem, the sultan’s dowry problem, the fussy suitor problem, the googol game, and the best choice problem.
I recently started using the Logic2010 software for school and fell, madly, in love with solving simple derivations. These two poems derive from that love:
I Love Derivations, and They Love Me
If I love you well Then you’ll be well loved, my love Love, derivations
This poem is in reference to the basic derivation rule “Modus Ponens” which dictates that a given variable, if equal to an antecedent present in a conditional statement, can be used to derive a consequent. In other symbols: If I have the premises of “X” and “X->Y” I can use Modus Ponens to derive “Y.”
Derivations Hate Me, So I Hate Them
If you are unloved Then I have no choice. I do Not love you either
This poem refers to the derivation inference rule “Modus Tollens.” Logicians can use Modus Tollens to derive a negative antecedent.
An acting inverse of Modus Ponens, Tollens applies to a scenario in which a variable is the negation of a conditional consequent. In other symbols: If I have the premises “~X” and “Y->X” I can use Modus Tollens to derive “~Y.”
Radar obscura Misrepresenting data yet such lovely shapes
Recently I’ve been looking at how to visualise some data in a way that’s engaging for the reader. Radar charts (also known as spider charts, web charts and star plots) seemed to fit the bill. My data with its 6 variables per subject can create a variety of irregular hexagonal shapes that are interesting and informative. Curiosity even had me wondering whether I could look at the surface area of the shapes to compare between different subjects.
Problem solved?
Perhaps not.
In reading up on radar charts I found articles by Chandoo (2008), Odds (2011) and Morin-Chassé (2020) outlining why my plans might not be ideal, and why despite their good looks, comparing between subjects can be less intuitive than simple bar charts. The issue lies with what is added to the data when visualising it. Let’s take my radar chart with 6 variables as an example.
The radar chart consists of a centre point with the 6 variables coming out from it like the spokes of a wheel, with the length of each spoke being the value for that subject in that category. The end of each spoke is then connected to its immediate neighbours with straight lines (although some radar charts are circular and the connecting lines are curved).
Two example radar charts.
The main thing the reader sees and focuses on is those connecting lines and the hexagonal shape they create. But that shape actually means very little – it’s not the actual data, instead it’s a circular sequence of the relationships between pairs of neighbouring variables. Ultimately the reader can easily be distracted from the data by how its visualised.
It gets worse – the shape created depends on a couple of factors:
1. The scale of each spoke.
2. The order of the variables arranged around the graph.
A different scale for one or more of the variables or a different order of the variables around the graph and the resulting shape can look very different. These decisions can also make it harder for readers to interpret the graph – imagine trying to read 6 different axes with their own units and trying to understand what they mean.
What’s more, the areas of the resulting shapes change as the shapes themselves change – simply swapping the location of two variables can result in a different shape and so a different area. And even if the shapes were always regular hexagons, the area doesn’t increase proportionally to the spoke length – a radar graph with longer spokes would have a disproportionally large area compared to one of the same shape with shorter spokes.
These two radar charts are the same subject and underlying data but the positions of two of the variables has been swapped, creating very different shapes with surface areas that differ by about 2%.
All this means that in many situations radar charts can actually cloud interpretation of the data rather than make it clearer. Doesn’t stop them being a good looking graph though!
Curious about what to use instead of radar charts? Check out the articles below for alternatives (including stellar charts and petal charts) and to get a more detailed (and far better written and explained!) understanding of some of the flaws of radar charts.
A note about the sciku: I’ve used the word obscura in the first line. In this case I mean to suggest how the radar chart obscures or obfuscates the data. I could have written obscurer but I wanted to reference the camera obscura that were used from the second half of the 16h century onwards as drawing aids to produce highly accurate representations and were later integral to the development of the camera. I liked the comparison between something that made things clearer and something that purported to make things clearer but often doesn’t.
To avoid striking whales, great creatures of the sea, use the app. Impact!
Blue whales can be injured or killed in collisions with ships, particularly in regions where migration routes cross shipping lanes. Yet because they travel huge distances, predicting where whales will be at any given time is difficult. However, now research by Abrahms et al (2019) suggests that statistical modelling techniques may be able to help.
The researchers used satellite
tracking data from 104 blue whales across 14 years along with daily information
on three-dimensional oceanic habitats to model the whales’ daily distribution.
By using an ensemble modelling approach they were able to produce daily,
year-round predictions of blue whale habitat suitability in the Californian
Current Ecosystem.
The statistical approach allows
the researchers to quantify the spatial and temporal distribution of exposure
to ship strike risk within shipping lanes in the Southern California Bight. The
researchers plan on converting this approach into a downloadable app which
would alert ships to the risks of whale collision and could recommend alternative
shipping lanes or vessel slow-downs.
It’s a truly fascinating piece of research that seems likely to have a huge impact upon a real-world problem – research at its best.
The sciku also includes a line from Mr Scruff’s truly excellent track ‘Shanty Town’ from his ‘Keep It Unreal’ album released in 1999. The full line is ‘Whales! Great creatures of the sea! Please listen to me!’ It’s well worth checking out!
Most of the three line poems I write are zen kōans in the syllabic form of haiku. This one was inspired by the disappearance of some corporate swag, an empty knapsack, taken from my cubicle while I was out of the office. It manages to include a couple of pop culture references, a nod to Zen Buddhism, and a reference to the knapsack problem, a famous optimization problem in computer science that is NP-hard, that is, it cannot be solved in polynomial time.
John Norwood is a Mechanical Engineer working with Carbon, Inc. to revolutionize how things are made. His interests include old houses, yoga, baking, cryptography, and bluegrass music. You can follow him on Twitter under the handle @pryoga
The debate over how knuckles cause a popping sound when cracked has lasted for decades. Now, Chandran Suja and Barakat (2018) have created three equations to mathematically model how the sound is produced. The first equation describes variations in pressure inside the joint, the second describes how pressure variations results in bubble size variations, whilst the third equation links the size variation of bubbles with the production of acoustic pressure waves.
When cracking your fingers the joints are pulled apart, the pressure goes down and bubbles appear in the synovial fluid which lubricates the joint. During knuckle cracking the pressure changes within the joint causing the size of the bubbles to fluctuate quickly resulting in the popping sound. The new model reveals that the bubbles don’t need to completely collapse in order to produce the sound, explaining why bubbles are observed following knuckle cracking.
This poem is about cryptography, which makes reference to the famous unsolved P versus NP problem of computer science. Modern cryptographic techniques rely on problems that are hard to compute (NP, or non-polynomial time) yet easy to verify (Polynomial time). If it were provable that such problems don’t exist, then any cryptography could be easily cracked. Our security is reliant upon an un-provable state and the very nature of its un-provabiity is what makes it secure. This fits my personal definition of God as the unknowable and believe the power of faith is rooted in a healthy relationship with that which cannot be known.
The P versus NP problem was first independently formulated by Stephen Cook and Leonid Levin in 1971, although the underlying ideas were considered earlier by John Nash, Kurt Godel and John von Neumann. It is one of the 7 Millennium Problems identified by the Clay Mathematics Institute with a reward of $1 million for the first to propose a solution.
John Norwood is a Mechanical Engineer working with Carbon, Inc. to revolutionize how things are made. His interests include old houses, yoga, baking, cryptography, and bluegrass music. You can follow him on Twitter under the handle @pryoga
This sciku has to do with probability theory used in data science and machine learning. In a nutshell, the more data you have, the lower the uncertainty of your model and the smaller the bias needed to reliably predict an eventual outcome. This paper by Dolev et al (2010) geeks out on some of the technical details of purifying data and machine learning.
John Norwood is a Mechanical Engineer working with Carbon, Inc. to revolutionize how things are made. His interests include old houses, yoga, baking, cryptography, and bluegrass music. You can follow him on Twitter under the handle @pryoga
Many data objects, such as matrices, which are traditionally described by providing a high precision number for every (row,column) entry, can be represented to usefully high precision by many fewer numbers. This so-called “data sparsity” holds for the mathematical descriptions of many physical and statistical phenomena whose effects or correlations decay smoothly with distance.
An apparently complex interaction or relationship is, with a special perspective, much simpler. In an extreme limit, we can lump the moon’s gravitational effect on the Earth by assuming that all of its distributed mass is concentrated at a single point. The potential to represent dense matrices by products of a small number of vectors (the number needed is called the “rank”) is analogous to this and leads to huge savings in memory and operations when manipulating such objects. The effect of the whole can be represented by a carefully defined abstraction. One version of this technique is called “skeletonization,” which suggests the Sciku above. For an example of this philosophy, see Yokota et al, 2014.
David Keyes directs the Extreme Computing Research Center at KAUST, where he was a founding dean in 2009. He inhabits the intersection of Mathematics, Computer Science, and applications, with a focus on colonizing emerging energy-efficient architectures for scientific computations. He is a Fellow of SIAM and AMS and has received the ACM Gordon Bell Prize and the IEEE Sidney Fernbach Award. As a lover of poetry, he is delighted to discover the Sciku community.
Until recently, the analysis of algorithms emphasized finding the minimum number of operations required to complete a task to a given precision – the algorithmic complexity. This is natural when the operations are both the most time-consuming steps of the computation and a reasonable proxy for all other costs, including data motion.
Today, floating point operations (“flops”) are much cheaper in time and in energy than moving the data into and out of the processing unit while memory hierarchies based on multilevel caches deliver operands at latencies and energy costs that vary by several orders of magnitude, depending upon where the freshest copies of the data are found. This situation results in the resurrection of old algorithms and the design of new ones that may do many more flops than previously “optimal” methods, but move less data back and forth from remote locations, and thus finish in less time, with smaller energy expenditure.
The author’s group has created several pieces of software that have displaced previous choices by optimizing memory transfers rather than flops. An example of a singular value decomposition that overcomes a flops handicap of as much as an order of magnitude is given in Sukkari et al (2016). For a community discussion of new paradigms on the path to exascale, see Dongarra et al (2011).
David Keyes directs the Extreme Computing Research Center at KAUST, where he was a founding dean in 2009. He inhabits the intersection of Mathematics, Computer Science, and applications, with a focus on colonizing emerging energy-efficient architectures for scientific computations. He is a Fellow of SIAM and AMS and has received the ACM Gordon Bell Prize and the IEEE Sidney Fernbach Award. As a lover of poetry, he is delighted to discover the Sciku community.
Academia prides itself on being fair, rational-minded and logical. Yet the practice behind these noble aims is sometimes far from that. A study by Fong & Wilhite (2017) reveals the various manipulations that can take place: from scholars gaining guest authorships on research papers despite contributing nothing to unnecessary reference list padding in an effort to boost citation rate. These instances of misconduct are likely a response to the pressures of an academic career – the demand for high numbers of publications and citation rates.
The survey of approximately 12,000 scholars across 18 disciplines revealed that over 35% of scholars have added an author to a manuscript despite little contribution (with female researchers more likely to add honorary authors than male researchers). 20% of scholars felt someone had been added to one of their grant proposal for no reason. 14% of academics reported being coerced into adding citations to their papers by journals, whilst 40% said they’d padded their reference list to pre-empt any coercion. Whilst changes to aspects of the academic system might help alleviate these issues, it’s likely to be a slow process.
The ossifrage, more commonly known as the bearded vulture, prefers to feed on dead animals, feeding predominantly on the bone marrow as opposed to the meat. It will on occasion kill living animals, with its main prey being tortoises which it drops onto rocks to break them open.
This haiku celebrates a defiant mouse but was inspired by a secondary meaning. In my study of cryptography, I recently de-crypted a challenge with the solution: ‘the magic words are squeamish ossifrage’ which, as it turns out, is a phrase frequently incorporated into the solution of cryptographic puzzles since 1977.
John Norwood is a Mechanical Engineer working with Carbon, Inc. to revolutionize how things are made. His interests include old houses, yoga, baking, cryptography, and bluegrass music. You can follow him on Twitter under the handle @pryoga
Modern science is not perfect, like any area it is subject to human errors, biases and instances of misconduct, accidental or otherwise. The underlying principles of science aim to avoid these problems, trying to achieve the golden ideal of accurate, impartial and trustworthy hypothesis testing. It is crucial then that these weak spots are recognised and addressed in order to avoid these potential pitfalls.
Jelte M. Wicherts (2017) has written a fascinating review of contemporary science, its weak spots and potential solutions. Problems discussed include p-hacking, post-hoc hypothesizing, outcome switching, selective reading and reporting, human error and various biases. Solutions such as increased transparency, data sharing and improved training are suggested. Whilst examples are taken from animal welfare research, the article is well worth a read for all scientists and anyone else interested in the scientific method.
Statistics and animal welfare might seem like unlikely bedfellows but a greater understanding of statistics may actually improve animal welfare. The 3Rs – Replacement, Reduction and Refinement – are critical for the ethical use of animals in experiments, but sometimes the animal species concerned cannot be replaced with a more ethical substitute. Refining procedures and reducing the numbers of animals tested should therefore be a fundamental consideration of any animal experiment.
Determining an appropriate sample size is often done using power analyses based around the P-value, but increasingly there is concern about the validity of this statistical term as a means of accepting or rejecting the experimental hypothesis. Instead, effect sizes and confidence intervals could be used to determine an experiment’s outcome and, in turn, minimum suitable sample size could be calculated using effect size precision. In this way statistics can be used to improve animal welfare by reducing the numbers of animals used. Sneddon et al, 2017.
