
Block function approximation to normal distribution
Introduction
Today, arguably more than ever before, the world is governed by the science of probability and statistics. “Big data” is now the norm in scientific research, with terabytes of data streaming into research centers from satellites and experimental facilities, analyzed by supercomputers. “Data mining” is now an essential part of mathematical finance and business management. Numerous public opinion polls, expertly analyzed, guide the political arena. Covid19 infection rates, immunization levels and r0 factors are a staple of nightly newscasts.
Yet the public at large remains mostly ignorant of the basic principles
Continue reading The brave new world of probability and statistics
Courtesy Maria Nguyen, Quanta Magazine
Computer discovery of mathematical theorems
In 1983 the present author recalls discussing the future of mathematics with Paul Cohen, who in 1963 proved that the continuum hypothesis is independent from the axioms of ZermeloFraenkel set theory. Cohen was convinced that the future of mathematics, and much more, lies in artificial intelligence. Reuben Hersch recalls Cohen saying specifically that at some point in the future mathematicians would be replaced by computers. So how close are we to Cohen’s vision?
In fact, computer programs that discover new mathematical identities and theorems are already a staple
Continue reading Computer theorem prover verifies sophisticated new result
Johann Sebastian Bach; credit Wikimedia
OK. Johann Sebastian Bach (16851750) was not a mathematician in a strict sense of the word. There is no “Bach convergence theorem” in real analysis, nor is there a “Bach isomorphism theorem” in algebra. Bach had no formal training in mathematics beyond elementary arithmetic.
But, as we will see, Bach was definitely a mathematician in a more general sense, as a composer whose works are replete with patterns, structures, recursions and other precisely crafted features. There are even hints of Fibonacci numbers and the golden ratio in Bach’s music (see below). Indeed, in this larger
Continue reading Bach as mathematician
Courtesy: towardsdatascience.com
AI comes of age
After several decades of disappointment, effective artificial intelligence (AI) systems emerged in the late 1990s and early 2000s, with the emergence of Bayestheorembased methods, combined with steadily advancing computer technology.
One notable milestone came in March 2016, when a computer program named “AlphaGo,” developed by researchers at DeepMind, a subsidiary of Alphabet (Google’s parent company), defeated champion Go master Lee Sedol, an achievement that many observers had not expected to occur for decades. Then in October 2017, Deep Mind researchers announced a new program, AlphaGo Zero, which was programmed only with the rules
Continue reading AI system finds counterexamples to graph theory conjectures
Muon g2 experiment (courtesy Fermilab)
A new measurement of the magnetic moment of the muon may draw into question the standard model of physics, the reigning theoretical construct describing all known fundamental forces and particles of physics. The new result was released in a paper dated today (7 April 2021) with 240 authors, led by researchers at Fermilab in the U.S., but also including researchers from Italy, Germany, United Kingdom, Russia, South Korea, China and Croatia.
The standard model of physics is arguably is the most successful physical theory ever devised, explaining all known fundamental particles and all known forces
Continue reading Muon result may rewrite standard model of physics
Alfred V. Aho (courtesy ACM)
Jeffrey D. Ullman (courtesy ACM)
The 2020 Alan M. Turing Award, bestowed by the Association for Computing Machinery, has been granted to Alfred V. Aho, Professor Emeritus at Columbia University, and Jeffrey D. Ullman, Professor Emeritus at Stanford University. The ACM Turing Award, which is named after computing pioneer Alan Turing, is widely considered to be the most prestigious award in the field of computer science. Past recipients include many of the most accomplished figures in the field, including Richard Hamming, Donald Knuth, William Kahan, Edward Feigenbaum, Jim Gray, Tim BernersLee, John Hennessy and David
Continue reading Aho and Ullman receive the ACM Turing Award
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The Abel Prize
The 2021 Abel Prize, arguably the closest equivalent in mathematics to the Nobel prize, has been awarded jointly to Avi Widgerson of the Institute for Advanced Study in Princeton, and László Lovász of the Eötvös Loránd University in Budapest, for their research linking discrete mathematics and computer science. The recipients will split the award, which is approximately USD$880,000.
According to Hanz MuntheKaas of the University of Bergen in Norway, who chaired the Abel Prize committee, Widgerson and Lovász “really opened up the landscape
Continue reading Two researchers share Abel prize for work in discrete mathematics and computer science
Credit: U.S. Center for Disease Control and Prevention
Covid19 and the misinformation pandemic
The Covid19 pandemic has disrupted human life like no other event of modern history. As of the present date (1 Mar 2021), over 114,000,000 confirmed cases and 2.5 million deaths have been recorded worldwide, according to the Johns Hopkins University database. The U.S. has recorded over 514,000 deaths, and the U.K. has recorded over 123,000. The U.S. death toll, for instance, exceeds the combined combat death toll of all wars fought in its 245year history (save only the civil war). And for every death there are
Continue reading Pandemics, misinformation and pseudoscience
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 pirelated 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 fullpage version
Continue reading PiDay 2021 crossword puzzle
Structure of the Nsp 15 hexamer, a component of Covid19 (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 Machinelearning breakthrough in protein folding

