Mathematicians Settle Erdős Coloring Conjecture | Combinatorics - Quanta Magazine

Kelsey Houston-Edwards, Ph.D Student observes, Fifty years ago, Paul Erdős and two other mathematicians came up with a graph theory problem that they thought they might solve on the spot. A team of mathematicians has finally settled it.

The authors ​combined​ many techniques​ ​to create an algorithm that covers all types of linear hypergraphs. Above, notes they made during the process.

The authors ​combined​ many techniques​ ​to create an algorithm that covers all types of linear hypergraphs. Above, notes they made during the process.

In the fall of 1972, Vance Faber was a new professor at the University of Colorado. When two influential mathematicians, Paul Erdős and László Lovász, came for a visit, Faber decided to host a tea party. Erdős in particular had an international reputation as an eccentric and energetic researcher, and Faber’s colleagues were eager to meet him.

“While we were there, like at so many of these tea parties, Erdős would sit in a corner, surrounded by his fans,” said Faber. “He’d be carrying on simultaneous discussions, often in several languages about different things.”

Erdős, Faber and Lovász focused their conversation on hypergraphs, a promising new idea in graph theory at the time. After some debate they arrived at a single question, later known as the Erdős-Faber-Lovász conjecture. It concerns the minimum number of colors needed to color the edges of hypergraphs within certain constraints...

The extreme generality of the Erdős-Faber-Lovász conjecture makes it challenging to prove. As you move to hypergraphs with more and more vertices, the ways of arranging their looping edges multiply as well. With all these possibilities, it might seem likely that there is some configuration of edges that requires more colors than it has vertices.

“There are so many different types of hypergraphs that have completely different flavors,” said Abhishek Methuku, one of the authors of the new proof, along with Dong-yeap Kang, Tom Kelly, Daniela Kühn and Deryk Osthus, all of the University of Birmingham. “It is surprising that it is true.”

...Their strategy for coloring the large edges relied on a simplification. They reconfigured these edges as the vertices of an ordinary graph (where each edge only connects two vertices). They colored them using established results from standard graph theory and then transported that coloring back to the original hypergraph.

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Source: Quanta Magazine

Klarman fellow bridges divide between math and philosophy | Arts & Humanities - Cornell Chronicle

James Walsh will spend three years tapping into Cornell’s robust resources in the field of logic, combining the precision and methods of math with the interests of philosophy by Kate Blackwood, writer for the College of Arts and Sciences.

In mathematics, axioms are statements that don’t need to be proved; they are truths one can assume, such as the axioms “for any number x, x + 0 = x” or “Between any two points is a line.”

Working in the field of logic, James Walsh, a Klarman Postdoctoral Fellow in philosophy, studies the axiomatic method, a central methodology in mathematics whereby claims are proven from axioms.

Based in the Sage School of Philosophy in the College of Arts and Sciences, Walsh is tapping into Cornell’s robust logic resources in philosophy, mathematics and linguistics to accomplish three years of study under the Klarman Postdoctoral Fellowship, working closely with Alexander (Arc) Kocurek, assistant professor of philosophy.

Walsh is trying to get to the heart of a discrepancy between the natural theories, which arise in mathematical practice, and unnatural ones that do not...

Walsh stands out for his ability to bridge the divide between mathematics and philosophy, Kocurek said. “Often, researchers in logic are either really just mathematicians or really just philosophers. But James is really both: He publishes serious mathematical work while also being able to isolate the philosophically important aspects of that work.”...

Logic appeals to Walsh because the field combines the precision and methods of math with the interests of philosophy. “You prove things that say something important about the scope of what we can know and what we can prove,” he said. “But you do so with mathematical techniques and a lot of precision.”

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Source: Cornell Chronicle

CodeSpark Academy is a fun app that will get kids coding | Parenting - Reviewed

Janelle Randazza, Staff Writer - Parenting recommends,  Coding never looked so cute. 

Photo: CodeSpark Academy

Want to get your kids coding? It can be harder than it looks. Our family has checked out countless video games and apps that profess to be fun while teaching kids the fundamentals of coding. In the end, most are either underwhelming in their concept and playability, or they feel a little too advanced for the 5- to 7-year-old age set.

When we signed up to test CodeSpark Academy through their seven-day free trial, I was happy to discover that there is a well-designed and well-thought-out early coding app out there. This app is made for kids ages 5 to 9; for the past three weeks our seven-year-old has been giving it rave reviews and asking to play with it constantly. To him, it it feels like an entertaining break, all the while it teaches advanced math concepts (like sequencing and order of operations) in such a way that even if coding is somehow replaced with some other arcade skill by the time he enters the workplace, he’ll still be building valuable math and logic skills to help him in myriad ways... 

Is CodeSpark worth it?

Yes. The most common complaint I read about CodeSpark is about the $7.99 monthly price tag. While that can add up, we think it’s worth it. There is no advertising to pause play and frustrate young users, and there are no micro-transactions that pop up unexpectedly to unlock additional areas. I find with so many “free” apps, my son stops using them the minute we hit a paywall. I don’t mind the $7.99 monthly fee because what you see is what you get. It’s also easy to cancel at any time.  

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Source: Reviewed

New statistical method eases data reproducibility crisis | Statistics - ScienceDaily and Pennsylvania State University

Summary:

A reproducibility crisis is ongoing in scientific research, where many studies may be difficult or impossible to replicate and thereby validate, especially when the study involves a very large sample size. Now researchers have developed a statistical tool that can accurately estimate the replicability of a study, thus eliminating the need to duplicate the work and effectively mitigating the reproducibility crisis. _____________________________________________________

The new tool enhances the replicability of large genomic datasets, as ScienceDaily and Pennsylvania State University reports.

Researchers have developed a new statistical tool that can accurately estimate the replicability of a study, thus eliminating the need to duplicate the work. The method will be particularly useful for large genome-wide association studies.
Photo: National Cancer Institute, Unsplash

A reproducibility crisis is ongoing in scientific research, where many studies may be difficult or impossible to replicate and thereby validate, especially when the study involves a very large sample size. For example, to evaluate the validity of a high-throughput genetic study's findings scientists must be able to replicate the study and achieve the same results. Now researchers at Penn State and the University of Minnesota have developed a statistical tool that can accurately estimate the replicability of a study, thus eliminating the need to duplicate the work and effectively mitigating the reproducibility crisis.

The team used its new method, which they describe in a paper publishing today (March 30) in Nature Communications, to confirm the findings of a 2019 study on the genetic factors that contribute to smoking and drinking addiction but noted that it also can be applied to other genome-wide association studies -- or studies that investigate the genetic underpinnings for diseases...

Liu noted that the method can be applied to genome-wide association studies focused on a wide variety of traits. "I think in the next decade or so, an essential focus of biology will be to interpret and make sense of those genome-wide association study discoveries and whether we can translate some of them into medications to facilitate personalized medicine," he said. "We are excited to be able to offer this statistical approach as a service to the research community."

Other authors on the paper include graduate students Daniel McGuire, Yu Jiang, J. Dylan Weissenkampen, Scott Eckert, and Lina Yang; Postdoctoral Scholar Fang Chen; and Associate Professor of Public Health Sciences and Statistics Arthur Berg, all at Penn State. Mengzhen Liu and Scott Vrieze at the University of Minnesota also are authors on the paper.

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Journal Reference:

Daniel McGuire, Yu Jiang, Mengzhen Liu, J. Dylan Weissenkampen, Scott Eckert, Lina Yang, Fang Chen, Arthur Berg, Scott Vrieze, Bibo Jiang, Qunhua Li, Dajiang J. Liu. Model-based assessment of replicability for genome-wide association meta-analysis. Nature Communications, 2021; 12 (1) 

DOI: 10.1038/s41467-021-21226-z

Source: ScienceDaily and Pennsylvania State University

The Platonic Academy of Athens: The World’s First University | Platonic Academy - Greek Reporter

Τhe Platonic Academy, or simply, ”The Academy”, was a famous school in ancient Athens founded by Plato in 428/427 BC and located a couple of miles outside the ancient city named Akademeia, after the legendary hero, Akademos by Nick Kampouris - GreekReporter.com

“The School of Athens,” by Raphael. Vatican Museums.
Photo: Public domain

Plato is the one figure who must receive the credit for giving birth to this unique institution. He inherited the land on which the Academy was eventually built, and began holding informal gatherings there to discuss philosophical issues with some of his friends.

The gatherings included thinkers such as Theaetetus of Sunium, Archytas of Tarentum, Leodamas of Thasos, and Neoclides...

The Platonic Academy is considered the world’s first university

The Platonic Academy was not an educational institution as we know it in modern times, but because it had the characteristics of a school and covered a wide variety of topics such as philosophy, astronomy, mathematics, politics, physics and more, it is considered to be the first university in the entire world...

One of them, Aristotle, came to be one of the world’s most influential philosophers of all time.

The teaching methods used by Plato, including both lectures and seminars, focused on his instructions, in addition to dialogue between teachers and students.

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Source: Greek Reporter

What Are Prime Numbers, and Why Do They Matter? | Math Concepts - HowStuffWorks

Patrick J. Kiger, HowStuffWorks observes, You may remember from math class that a prime number is a number that can only be divided by 1 and itself. But why are they important anyway?

What do these numbers have in common? They're all prime!
Photo: geralt/Pixabay

If you only vaguely remember your elementary school mathematics class, you may not remember what a prime number is. That's a pity, because if you're trying to keep your emails safe from hackers or surf the web confidentially on a virtual private network (VPN), you're using prime numbers without even realizing it.

That's because prime numbers are a crucial part of RSA encryption, a common tool for protecting information, which uses prime numbers as keys to unlock the messages hidden inside gigantic amounts of what's disguised as digital gibberish. Additionally, prime numbers have other applications in the modern technological world, including an important role in defining the color intensity of the pixels on the computer screen that you're staring at now.

So, what are prime numbers, anyway? And how did they get to be so important in the modern world?...

Mark Zegarelli, author of numerous books on math in the popular "For Dummies" series who also teaches test prep courses, offers an illustration involving coins that he uses with some of his students to explain the difference between primes and composite numbers, which can be divided by other numbers besides one and themselves. (Composite numbers are the opposite of primes.)...

That's why mathematicians have continued to labor to come up with increasingly bigger primes, in an ongoing project called the Great Internet Mersenne Prime Search. In 2018, that project led to the discovery of a prime number that consisted of 23,249,425 digits, enough to fill 9,000 book pages, as University of Portsmouth (England) mathematician Ittay Weiss described it in The Conversation. It took 14 years of computations to come up with the gigantic prime, which is more than 230,000 times bigger than the estimated number of atoms in the observable universe!

You can imagine how impressed Euclid might be by that.

Read more... 

Source: HowStuffWorks

The Beautiful Consistency of Mathematics — Alexander Yessenin-Volpin | Philosophy Of Mathematics - Medium

Mathematics is often believed to bring people to madness. We hear many stories like those about Gödel, Cantor, Nash, and Grothendieck, describing geniuses haunted by insanity that is developing along with their mathematics, explains Jan Gronwald, published in Cantor’s Paradise.

Photo: Medium

And there is something to it. A certain psychologist said that

A paranoid person is irrationally rational. . . . Paranoid thinking is characterized not by illogic, but by a misguided logic, by logic run wild

Mathematics is the paradigm of rationality and maybe if the rationality takes over all of the aspects of life, we can talk of a mental issue. But this time I want to bring to light an opposite example. This time I want to share a story about a mathematician who was the voice of reason and sanity in the world that has run wild. And one whose mathematics was the model of his approach in social life. 

Meet Alexander Yessenin-Volpin (1924–2016)...

His Mathematics

Yessenin-Volpin believed that the traditional style of making mathematics is similarly hypocritical to the style of handling legal issues in the Soviet Union. He claimed that the unreasonable and careless inclusion of the concept of infinity into mathematical discourse is the culprit of depriving it of exactness it was actually to grant.

Therefore he urged for a radical revision of foundations of mathematics, based on the claim that the concept of infinity, both potential and actual, is utterly nonsensical.

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Source: Medium

7 Books By Women In STEM That Will Blow Your Mind | Books - Bustle

Alice Broster, Freelance journalist recommends, For International Women’s Month 2021 Bustle UK is shining a light on the women in STEM  (science, technology, economics, and math.) 

From climate change to AI, women are making serious moves in STEM.
Photo: Getty Images

There are many people who identify as women working in these industries, making groundbreaking discoveries and changing the way we look at the world. One of the best ways to support them is to know their names and buy their work, where possible. So here’s seven of the best books written by women in STEM. Prepare to have your eyes opened.

As Catalyst reports there’s a gender gap in STEM that persists across the world. Studies have highlighted that the women who begin careers in STEM face male-dominated workplaces with high rates of discrimination. However, that isn’t to say that there aren’t amazing mathematicians, scientists, statisticians, engineers, and women working in tech. But that more needs to be done to support them and encourage others to join them.  

...There’s never been a better time to get inspired by the work of women in STEM. And whether you work in the field or just want to get lost in the amazing efforts of others, these books will leave you floored.

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Source: Bustle

Book review: First Person Singular, by Haruki Murakami | Books - The Scotsman

The deceptively simple short stories in this new collection by Haruki Murakami offer many pleasures along with occasional irritations, writes Allan Massie, Scottish journalist.

Haruki Murakami pictured outside Hans Christian Andersen's house in Odense
Photo: Henning Bagger / Scanpix Denmark / AFP via Getty Images

Haruki Murakami is a writer of considerable charm. His books are often like that sort of sweet, even sugary music described as being “easy listening.” This collection of eight short stories, each – as the title suggests – written in the first person, certainly offers easy reading. The voice is the same in each story, so one may assume each has the same narrator, and may suspect that there is some element of autobiography. Certainly it is difficult not to identify the narrator with the author.

Most are trivial. That’s to say, success depends on the manner, not the material. There is very little narrative interest. The stories meander like a river through gentle countryside. Some are whimsical. One has the narrator engaged in conversation with a monkey who tells him that he is attracted to women not to female monkeys, and that being unable to fulfil his desires, he “started stealing the names of women I fell for.” This is an example of what one reviewer has called Murakami’s “beguiling simplicity.”.

Some of the stories work even for the unbeguiled. “Charlie Parker Plays Bossa Nova,” for example. As a student and jazz enthusiast, the narrator writes an article about a record that was never made and never could have been made, the great saxophone player Charlie “Bird” Parker being dead aged only 34, several years before that style of samba music was developed...

This new book will surely please those who already know and delight in his work, and serve as an enjoyable introduction for those unfamiliar with it. Sometimes the faux-naif tone may be tiresome, but mostly he offers agreeable comfort reading. Some will read it as pure fiction, more perhaps as a lightly fictionalised memoir. It doesn’t matter which it is. The pleasures and occasional irritations will be the same.

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Source: The Scotsman

How Alan Turing and a university in Yorkshire are helping inspire the next generation of mathematicians | Education - Yorkshire Post

His genius was to prove vital in bringing victory over Germany during the Second World War after he famously helped crack the Enigma code, Paul Jeeves, Head of News reports.

Academics from the University of Sheffield are now launching a project to allow school pupils across the country the chance to explore how maths can be used to understand the world and nature, centred on Alan Turing’s own theories.

Mathematician Alan Turing is known as the father of modern computing and as the codebreaker responsible for breaking the Germans’ ciphers in the 1940s.

But his talents also extended to a theory of using mathematics to explain biology – showing how animals get their distinctive markings.

Academics from the University of Sheffield are now launching a project to allow school pupils across the country the chance to explore how maths can be used to understand the world and nature, centred on Turing’s own theories...

The high-level maths which Turing used can be broken down into thousands of smaller calculations using the methods of adding, subtracting, multiplying and dividing.

As part of the lessons, children will complete calculations – cracking various codes – before submitting their answers to the next school to perform the next set of calculations.

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Source: Yorkshire Post