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The twin prime and prime gap conjectures
Paul Erdos, one of the 20th century’s most famed mathematicians (1913-1996) was well-known for offering prizes for the solution of various mathematical problems. Although most of these prizes were a token USD$25 or the like, he did offer a UDS$10,000 prize for the solution to a certain conjecture of his on prime gaps. Until very recently, it had not been solved.
The twin prime conjecture is that the gap between successive primes, namely gn = pn+1 – pn, is two for infinitely many n. This has not been proven, and is one of
Continue reading New results on the prime gap conjecture
Introduction
The world community is truly at a crossroads like never before faced in the history of our civilization. If we continue business-as-usual with the consumption of fossil fuels, then, according to the 2014 edition of the Intergovernmental Panel on Climate Change (IPCC)’s climate change report, grave consequences will almost surely ensue, including rising sea levels, extreme temperatures, flooding, drought, agricultural losses and, quite likely, violent conflicts among human societies. Delays may have already cost the world society USD$8 trillion.
Some who distrust the scientific consensus on climate change have taken heart at an apparent leveling off of world temperatures
Continue reading Low energy nuclear reactions: Papers and patents
On 17 November 2014, research groups at Google and Stanford University jointly announced significant advances in image recognition software.
Image recognition has been pursued for many years. One of the first and still most widely deployed applications is to recognize faces. Indeed, facial recognition systems have been employed in numerous settings:
In numerous U.S. locations, such as airports; in fact, many presume that such cameras are ubiquitous, as in TV police dramas, although this is not the case. In London, as part of their closed-circuit TV camera crime monitoring system. In various casinos, to recognize “card counters” and other unwanted
Continue reading Big brother is watching, with new image-recognition techniques
June 23, 2014 was a very nice day for five mathematicians: Simon Donaldson, Maxim Kontsevich, Jacob Lurie, Terence Tao and Richard Taylor. They were informed that they would be receiving the inaugural Breakthrough Prizes in Mathematics, each with a cash award of USD$3,000,000. They plan to share their good fortune.
The international, if still Euro-centric, nature of mathematics is obvious when we list where the five were born, educated and currently work:
Name Born Educated Current appointment(s) Donaldson Cambridge, UK Cambridge, UK; Oxford, UK Stony Brook, USA; Imperial College, UK Kontsevich Khimki, Russia Moscow State, Russia; Univ. Bonn, Germany IHES,
Continue reading Big Bucks for Big Breakthroughs: Prize recipients give three million dollar maths talks
When numbers of any sort are presented, whether in mathematics, science, business, government or finance, the default assumption is that the data presented are reasonably reliable to the last digit presented. Thus, if a light bulb is listed as using 3.14 watts, then its actual usage is presumably between 3.13 and 3.15 watts, and certainly not 2.8 or 4.2 watts. Or if the average interest rate paid on a set of securities is listed as 2.718 percent, then a reasonable reader presumes that the actual figure is between 2.717 and 2.719 percent.
The total number of significant digits can vary
Continue reading Dubious digits: Is this data really that accurate?
The IPCC report’s warning
The latest draft edition of the UN’s Intergovernmental Panel on Climate Change (IPCC) bluntly warns that business-as-usual increases in greenhouse gas emissions will cause “further warming and long-lasting changes” in the earth’s the climate system, with increasing likelihood of “severe, pervasive and irreversible impacts for people and ecosystems.” Specific dangers include rising sea levels, more frequent extreme temperatures, flooding, drought, harm to marine life and violent conflicts among human societies in the wake of agricultural changes. Delays may have already cost the world society US$8 trillion.
So what can be done? As we argued in a
Continue reading Fusion energy: Hope or hype?
Don Quixote
September 29, 2014 is the 467th birthday of Spanish author Miguel de Cervantes, whose immortal Don Quixote is widely regarded as one of the greatest works of literature in any language. In fact, in 2002 the Norwegian Book Club named Don Quixote as “best literary work ever written” in their listing of the world’s 100 greatest books.
Don Quixote describes the exploits the protagonist Alonso Quixano, a semi-retired Spanish gentleman who got so carried away with his readings of chivalry and knights-errant exploits that he adopted the name Don Quixote, donned some old armor, named a friend as
Continue reading Tilting at windmills
The latest IPCC report
The latest draft edition of the UN’s Intergovernmental Panel on Climate Change (IPCC) report includes some rather stark language:
Continued emission of greenhouse gases will cause further warming and long-lasting changes in all components of the climate system, increasing the likelihood of severe, pervasive and irreversible impacts for people and ecosystems.
Among the potential sources of havoc are rising sea levels, more frequent extreme temperatures, flooding, drought, harm to marine life and potentially violent conflicts arising from the changing agricultural and meteorological environment.
Skeptics of the scientific consensus have made great hay over the fact
Continue reading To frack or not to frack: That’s not the question
“Experimental mathematics” has emerged in the past 25 years or so to become a competing paradigm for research in the mathematical sciences. An exciting workshop entitled Challenges in 21st Century Experimental Mathematical Computation was held at the Institute for Computational and Experimental Research in Mathematics (ICERM), July 21-25, 2014, which explored emerging challenges of experimental mathematics in the rapidly changing era of modern computer technology. This report summarizes the workshop findings (without mentioning any of the research presentations).
While several more precise definitions have been offered for “experimental mathematics,” we used the informal one given in the book The Computer
Continue reading Opportunities and challenges in experimental mathematics
Introduction
On 10 August 2014, a team led by Thomas Hales of the University of Pittsburgh, USA, announced that their decade-long effort to construct a computer-verified formal proof of the Kepler conjecture was now complete. The project was known as Flyspeck, a rough acronym for “Formal Proof of Kepler.”
The Kepler conjecture is the assertion that the simple scheme of stacking oranges typically seen in a supermarket has the highest possible average density, namely pi/(3 sqrt(2)) = 0.740480489…, for any possible arrangement, regular or irregular. It is named after 17th-century astronomer Johannes Kepler, who first proposed that planets orbited in
Continue reading Formal proof completed for Kepler’s conjecture on sphere packing
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