PiDay 2021 crossword puzzle

Yes, it is that time of year — Pi Day (March 14, or 3/14 in North American month/day date notation) is approaching. So in honor of the occasion, I have constructed a new crossword puzzle — see below. This puzzle employs a certain pi-related feature that will become evident as you solve it.

This puzzle conforms to the New York Times crossword conventions. As far as difficulty level, it would be comparable to the NYT Tuesday or Wednesday puzzles (the NYT puzzles are graded each week from Monday [easiest] to Saturday [most difficult]).

If you would like a full-page version

Continue reading PiDay 2021 crossword puzzle

Machine-learning breakthrough in protein folding

Structure of the Nsp 15 hexamer, a component of Covid-19 (Courtesy Argonne Natl Lab)

Protein folding

Proteins are the workhorses of biology. A few examples in human biology include actin and myosin, the proteins that enable muscles to work; keratin, which is the basis of skin and hair; hemoglobin, the basis of red blood that carries oxygen to cells throughout the body; pepsin, an enzyme that breaks down food for digestion; and insulin, which controls metabolism. A protein known as “spike” is the key for the coronavirus to invade healthy cells. And for every protein in human biology, there

Continue reading Machine-learning breakthrough in protein folding

Do odd perfect numbers exist? New results on an old problem

Euclid, from Rafael’s “School of Athens”, Vatican Museum, Rome, photo courtesy Clay Mathematics Institute

MathJax.Hub.Config({tex2jax: {inlineMath: [[‘$’,’$’], [‘\\(‘,’\\)’]]}});

Perfect numbers

A perfect number is a positive integer whose divisors (not including itself) add up to the integer. The smallest perfect number is $6$, since $6 = 1 + 2 + 3$. The next is $28 = 1 + 2 + 4 + 7 + 14$, followed by $496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248$ and $8128 = 1 + 2 + 4 + 8 + 16 + 32

Continue reading Do odd perfect numbers exist? New results on an old problem

Can computers do mathematical research?

Courtesy Adarsh Pathak

AI comes of age

The modern field of artificial intelligence (AI) began in 1950 with Alan Turing’s landmark paper Computing machinery and intelligence, which outlined the principles of AI and proposed the Turing test. Although early researchers were confident that AI systems would soon be a reality, inflated promises and expectations led to disappointment in the 1970s and again in the 1980s.

A breakthrough of sorts came in the late 1990s and early 2000s with the emergence of Bayes-theorem-based methods, which quickly displaced the older methods based mostly on formal reasoning. When combined with steadily advancing

Continue reading Can computers do mathematical research?

Peter Borwein dies at 67

Peter Borwein (sitting), with his brother Jonathan Borwein (courtesy Canada Foundation for Innovation)

Peter Borwein, retired Professor of Mathematics at Simon Fraser University (British Columbia, Canada) and former Director of SFU’s Centre for Interdisciplinary Research in the Mathematical and Computational Sciences (IRMACS), died on August 23, 2020, at the age of 67, of pneumonia, after courageously battling multiple sclerosis for over 20 years.

Peter was a prolific mathematician, with over 200 publications, including several books. His research included works in classical analysis, computational number theory, Diophantine number theory and symbolic computing. Many of these papers were co-authored with his brother

Continue reading Peter Borwein dies at 67

Two mathematicians’ triple play

MathJax.Hub.Config({tex2jax: {inlineMath: [[‘$’,’$’], [‘\\(‘,’\\)’]]}});

The Erdős-Turán conjecture

Paul Erdős, one of the twentieth century’s most unique mathematicians, was known to travel from colleague to colleague, often arriving unannounced, and to immediately delve into some particularly intriguing research problem. See this article and this book for some additional background on this influential mathematician.

One of his more interesting conjectures is his “conjecture on arithmetic progressions,” sometimes referred to as the “Erdős-Turán conjecture.” It can be simply stated as follows: If $A$ is a set of positive integers such that $$\sum_{k \in A} \frac{1}{k} = \infty,$$ then $A$ contains arithmetic progressions of

Continue reading Two mathematicians’ triple play

How old is the universe? New results clash

MathJax.Hub.Config({tex2jax: {inlineMath: [[‘$’,’$’], [‘\\(‘,’\\)’]]}});

The standard model

The standard model of physics, namely the framework of laws at the foundation of modern physics, has reigned supreme since the 1970s, confirmed to great precision in a vast array of experimental tests. Among other things, the standard model predicted the existence of the Higgs boson, which was experimentally discovered in 2012, nearly 50 years after it was first predicted.

Yet physicists have recognized for many years that the standard model cannot be the final answer. For example, quantum theory and general relativity are known to be mathematically incompatible. String theory and

Continue reading How old is the universe? New results clash

Covid-19 and the worth of a human life

The scales of justice (courtesy Wikimedia)

Covid-19’s grim toll

The statistics are staggering: As of 1 June 2020, according to the Johns Hopkins University database, the U.S. had logged over 1.811 million confirmed cases of Covid-19 and over 105,000 deaths. The U.K. was next, with over 277,000 confirmed cases and over 38,000 deaths. Worldwide, over 6.3 million cases had been confirmed, with more than 376,000 deaths. If current trends continue, the U.S. death toll alone will soon exceed that of all wars in its history except for the Civil War and World War II.

The economic costs have been

Continue reading Covid-19 and the worth of a human life

The origin of life in an inflationary universe

RNA (on left) compared with DNA (on right); courtesy Wikimedia

The abiogenesis problem

Exactly how life first emerged on Earth (the “abiogenesis” problem) remains a critical unsolved question in biology. Was it inevitable, given a favorable environment, or was it a fantastically improbable event? All we know for sure is that it occurred at least 3.8 billion years ago and possibly more than 4 billion years ago. The fact that life arose relatively soon after the surface of the Earth solidified indicates to some that abiogenesis was inevitable, but there is no way to know for sure. For further

Continue reading The origin of life in an inflationary universe

Pseudoscience in the age of Coronavirus

The Covid-19 virus (courtesy U.S. Center for Disease Control and Prevention)

A pandemic is upon us

As this is being written (April 2020), the entire world is gripped in the throes of the rapidly spreading and deadly Covid-19 pandemic. International travel has been greatly curtailed worldwide; many businesses, large and small, have shut their doors; many K-12 schools and universities have closed; and entire regions and nations, encompassing well over one billion people, have been ordered to remain in their homes.

As of the current date (28 April 2020), the Johns Hopkins University Coronavirus Resource Center has tallied 3,062,000

Continue reading Pseudoscience in the age of Coronavirus