🔖 Equilibrium points in n-person games by John Nash | PNAS

Bookmarked Equilibrium points in n-person games by John F. Nash Jr. (PNAS 36 (1) 48-49; https://doi.org/10.1073/pnas.36.1.48)

One may define a concept of an n-person game in which each player has a finite set of pure strategies and in which a definite set of payments to the n players corresponds to each n-tuple of pure strategies, one strategy being taken for each player. For mixed strategies, which are probability distributions over the pure strategies, the pay-off functions are the expectations of the players, thus becoming polylinear forms in the probabilities with which the various players play their various pure strategies.

Any n-tuple of strategies, one for each player, may be regarded as a point in the product space obtained by multiplying the n strategy spaces of the players. One such n-tuple counters another if the strategy of each player in the countering n-tuple yields the highest obtainable expectation for its player against the n − 1 strategies of the other players in the countered n-tuple. A self-countering n-tuple is called an equilibrium point.

The correspondence of each n-tuple with its set of countering n-tuples gives a one-to-many mapping of the product space into itself. From the definition of countering we see that the set of countering points of a point is convex. By using the continuity of the pay-off functions we see that the graph of the mapping is closed. The closedness is equivalent to saying: if P1, P2, … and Q1, Q2, …, Qn, … are sequences of points in the product space where Qn → Q, Pn → P and Qn counters Pn then Q counters P.

Since the graph is closed and since the image of each point under the mapping is convex, we infer from Kakutani’s theorem1 that the mapping has a fixed point (i.e., point contained in its image). Hence there is an equilibrium point.

In the two-person zero-sum case the “main theorem”2 and the existence of an equilibrium point are equivalent. In this case any two equilibrium points lead to the same expectations for the players, but this need not occur in general.

Communicated by S. Lefschetz, November 16, 1949

👓 What Did Ada Lovelace’s Program Actually Do? | Two Bit History

Read What Did Ada Lovelace's Program Actually Do? (twobithistory.org)
In 1843, Ada Lovelace published the first nontrivial program. How did it work?
Interesting that he indicates what may have been one of the first published computer code bugs.

👓 Maryland’s Goucher College eliminating several majors, including math | Baltimore Sun

Read Maryland's Goucher College eliminating several majors, including math (Baltimore Sun)
Math majors at Goucher College will soon be a thing of the past.

👓 ‘Hard Day’s Night’: A Mathematical Mystery Tour | NPR

Read 'Hard Day's Night': A Mathematical Mystery Tour (NPR | Weekend Edition Saturday)
The jangly opening chord of The Beatles' hit "A Hard Day's Night" is one of the most recognizable in pop music. Maybe it sounds like nothing more than a guitarist telling his bandmates, "Hey, we're doing a song here, so listen up." But for decades, guitarists have puzzled over exactly how that chord was played.

👓 A Songwriting Mystery Solved: Math Proves John Lennon Wrote ‘In My Life’ | NPR

Read A Songwriting Mystery Solved: Math Proves John Lennon Wrote 'In My Life' (NPR | Weekend Edition Saturday)

Over the years, Lennon and McCartney have revealed who really wrote what, but some songs are still up for debate. The two even debate between themselves — their memories seem to differ when it comes to who wrote the music for 1965's "In My Life."

Mathematics professor Jason Brown spent 10 years working with statistics to solve the magical mystery. Brown's the findings were presented on Aug. 1 at the Joint Statistical Meeting in a presentation called "Assessing Authorship of Beatles Songs from Musical Content: Bayesian Classification Modeling from Bags-Of-Words Representations."

👓 Make Your Daughter Practice Math. She’ll Thank You Later. | New York Times

Read Opinion | Make Your Daughter Practice Math. She’ll Thank You Later. (nytimes.com)
The way we teach math in America hurts all students, but it may be hurting girls the most.

👓 abc News | Peter Woit

Read abc News by Peter WoitPeter Woit (math.columbia.edu)
The last couple months I’ve heard reports from several people claiming that arithmetic geometers Peter Scholze and Jakob Stix had identified a serious problem with Mochizuki’s claimed proof of the abc conjecture. These reports indicated that Scholze and Stix had traveled to Kyoto to discuss this with Mochizuki, and that they were writing a manuscript, to appear sometime this summer. It seemed best then to not publicize this here, better to give Mochizuki, Scholze and Stix the time to sort out the mathematics and wait for them to have something to say publicly. Today though I saw that Ivan Fesenko has put out a document entitled Remarks on Aspects of Modern Pioneering Mathematical Research.
Peter definitely predicted the Fields medal for Peter Scholze here.

The intrigue of this case is quite interesting. Take a look at some of the comments on these posts. Some border on religious zealotry, and even this when I know Peter heavily curates his comments section to make them useful.

🔖 Gems And Astonishments of Mathematics: Past and Present | Dr. Mike Miller at UCLA Extension

Bookmarked Gems And Astonishments of Mathematics: Past and Present (UCLA Continuing Education)

Mathematics has evolved over the centuries not only by building on the work of past generations, but also through unforeseen discoveries or conjectures that continue to tantalize, bewilder, and engage academics and the public alike.  This course, the first in a two-quarter sequence, is a survey of about two dozen problems—some dating back 400 years, but all readily stated and understood—that either remain unsolved or have been settled in fairly recent times.  Each of them, aside from presenting its own intrigue, has led to the development of novel mathematical approaches to problem solving.  Topics to be discussed include (Google away!): Conway’s Look and Say Sequences, Kepler’s Conjecture, Szilassi’s Polyhedron, the ABC Conjecture, Benford’s Law, Hadamard’s Conjecture, Parrondo’s Paradox, and the Collatz Conjecture.  The course should appeal to devotees of mathematical reasoning and those wishing to keep abreast of recent and continuing mathematical developments.

Suggested prerequisites: Some exposure to advanced mathematical methods, particularly those pertaining to number theory and matrix theory.

 - 
Tuesday 7:00PM - 10:00PM
Location: UCLA
Instructor: Michael Miller
MATH X 451.44 | 362773
Fee: $453.00
I’ve been waiting with bated breath to see what Dr. Miller would be offering in the evenings at UCLA Extension this Fall and Winter quarters. The wait is over, though it’ll be a few days before we can register.

If you’re interested in math at all, I hope you’ll come join the 20+ other students who follow everything that Mike teaches. Once you’ve taken one course from him, you’ll be addicted.

👓 Beauty is truth, truth is beauty, and other lies of physics | Aeon

Read Beauty is truth, truth is beauty, and other lies of physics by Sabine Hossenfelder (Aeon)
After spending billions trying (and failing) to support beautiful ideas in physics, is it time to let evidence lead the way?

👓 Mathematician-M.D. introduces a new methodology suggesting a solution to one of the greatest open problems in the history of mathematics | USC

Read Mathematician-M.D. introduces a new methodology suggesting a solution to one of the greatest open problems in the history of mathematics by Daniel Druhora (USC Viterbi School of Engineering)

A completely new approach suggests the validity of the 110-year-old Lindelöf hypothesis, opening up the possibilities of new discoveries in quantum computing, number theory and cybersecurity

Athanassios Fokas, a mathematician from the Department of Applied Mathematics and Theoretical Physics of the University of Cambridge and visiting professor in the Ming Hsieh Department of Electrical Engineering at the USC Viterbi School of Engineering has announced a novel method suggesting a solution to one of the long-standing problems in the history of mathematics, the Lindelöf Hypothesis.

👓 Stonehenge builders used Pythagoras' theorem 2,000 years before Greek philosopher was born, say experts | The Telegraph

Read Stonehenge builders used Pythagoras' theorem 2,000 years before Greek philosopher was born, say experts  by Sarah Knapton (The Telegraph)
The builders of Britain’s ancient stone circles like Stonehenge were using Pythagoras' theorem 2,000 years before the Greek philosopher was born, experts have claimed.
I’ll be bookmarking the book described in this piece for later. The author doesn’t get into the specifics of the claim in the title enough for my taste. What is the actual evidence? Is there some other geometrical construct they’re using to come up with these figures that doesn’t involve Pythagoras?

Following My Favorite Theorem by Kevin Knudson and Evelyn Lamb

Followed My Favorite Theorem by Kevin Knudson and Evelyn Lamb (kpknudson.com)
University of Florida mathematician Kevin Knudson and I are excited to announce our new math podcast: My Favorite Theorem. In each episode, logically enough, we invite a mathematician on to tell us about their favorite theorem. Because the best things in life are better together, we also ask our guests to pair their theorem with, well, anything: wine, beer, coffee, tea, ice cream flavors, cheese, favorite pieces of music, you name it. We hope you’ll enjoy learning the perfect pairings for some beautiful pieces of math. We’re very excited about the podcast and hope you will listen here, on the site’s page, or wherever you get your podcasts. New episodes will be published approximately every three weeks. We have a great lineup of guests so far and think you’ll enjoy hearing from mathematicians from different mathematical areas, geographic locations, and mathematical careers.

👓 LaTeXiT | chachatelier.fr

Bookmarked LaTeXiT (chachatelier.fr)
Should LaTeXiT be categorized, it would be an equation editor. This is not the plain truth, since LaTeXiT is "simply" a graphical interface above a LaTeX engine. However, its large set of features is a reason to see it as an editor; this is the goal in fact.

👓 Andrew Jordan reviews Peter Woit’s Quantum Theory, Groups and Representations and finds much to admire. | Inference

Read Woit’s Way by Andrew Jordan (Inference: International Review of Science)
Andrew Jordan reviews Peter Woit's Quantum Theory, Groups and Representations and finds much to admire.
For the tourists, I’ve noted before that Peter maintains a free copy of his new textbook on his website.

I also don’t think I’ve ever come across the journal Inference before, but it looks quite nice in terms of content and editorial.