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The ErdősTurán 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 for some additional background.
One of his more interesting conjectures is his “conjecture on arithmetic progressions,” sometimes referred to as the “ErdősTurán 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 any length, or, in other words, $A$
Continue reading Two mathematicians’ triple play
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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
The scales of justice (courtesy Wikimedia)
Covid19’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 Covid19 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 Covid19 and the worth of a human life
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
The Covid19 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 Covid19 pandemic. International travel has been greatly curtailed worldwide; many businesses, large and small, have shut their doors; many K12 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
We are happy to announce the publication of “From Analysis to Visualization: A Celebration of the Life and Legacy of Jonathan M. Borwein”, a compilation of research papers devoted to the memory of Jonathan Borwein. The book is the proceedings of a conference held in Borwein’s honor in September 2017 at Newcastle, Australia, near where Prof. Borwein taught for several years before passing away in August 2016.
The volume has been published by Springer, and is available for purchase from the Springer website, or from Amazon.com.
The individual papers are authored by many of Jonathan Borwein’s colleagues and collaborators. Here
Continue reading “From Analysis to Visualization: A Celebration of the Life and Legacy of Jonathan M. Borwein”
An age of unparalleled progress
Though many do not recognize the fact, behind the disturbing headlines that dominate the news today, scientific progress marches forward, unabated and undiminished. Just within the past 100 years, researchers have discovered the theory of relativity, quantum mechanics and the standard model; unraveled the structure of DNA; sequenced the human genome; discovered the accelerating universe; observed extrasolar planets orbiting thousands of distant stars; and detected the collisions of black holes. See this Math Scholar article for additional details.
Spurred by these scientific advances, human technology has advanced at an astonishing pace: advances in medical
Continue reading Why are people embracing astrology in an age of science?
I have prepared a new paper containing a catalogue of 72 summation formulas, integral formulas and iterative algorithms for Pi. The catalogue contains both classical and modern formulas, ranging from Archimedes’ 2200yearold algorithm to intriguing formulas found by Ramanujan and the quadratic, cubic, quartic and nonic algorithms of Jonathan Borwein and Peter Borwein, the latter of which double, triple, quadruple and ninetimes, respectively, the number of correct digits with each iteration.
The catalogue of formulas and iterative algorithm is followed by results of carefully designed computer implementations, which enable one to compare the relative speed of these formulas.
Continue reading PiDay 2020: A catalogue of formulas involving pi, with analysis
Yes, it is that time of year — Pi Day (March 14, or 3/14 in North American month/day date notation) is here.
So in honor of the occasion, I have constructed a new crossword puzzle — see below. This puzzle honors several of the key persons through history who have made significant contributions to the theory and computation of Pi.
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]).
Continue reading Pi Day 2020: A new crossword puzzle
Credit: Quanta Magazine
Introduction
A growing controversy over the multiverse and the anthropic principle has exposed a major fault line in modern physics and cosmology. Some researchers see the multiverse and the anthropic principle as inevitable, others see them as an abdication of empirical science. The controversy spans quantum mechanics, inflationary Big Bang cosmology, string theory, supersymmetry and, more generally, the proper roles of experimentation and mathematical theory in modern science.
The “many worlds interpretation” of quantum mechanics
Since the 1930s, when physicists first developed the mathematics behind quantum mechanics, researchers have found that this theory appears to
Continue reading Universe or multiverse? The war rages on

