The discovery of decimal arithmetic in ancient India, together with the well-known schemes for long multiplication and long division, surely must rank as one of the most important discoveries in the history of science. The date of this discovery, by an unknown Indian mathematician or group of mathematicians, was recently pushed back to the third century CE, based on the recent dating of the Bakhshali manuscript, but it probably happened earlier, perhaps around 0 CE.

Arithmetic on modern computersComputers, of course, do not use

Continue reading An n log(n) algorithm for multiplication

]]>The discovery of decimal arithmetic in ancient India, together with the well-known schemes for long multiplication and long division, surely must rank as one of the most important discoveries in the history of science. The date of this discovery, by an unknown Indian mathematician or group of mathematicians, was recently pushed back to the third century CE, based on the recent dating of the Bakhshali manuscript, but it probably happened earlier, perhaps around 0 CE.

Computers, of course, do not use decimal arithmetic in computation (except input and output for human eyes). Instead they use binary arithmetic, either in an integer format, where the bits in a computer word represent a binary integer (with perhaps one bit as a sign), or else in floating-point format, where part of the computer word represents the data and part represents an exponent — the binary equivalent of writing a value in scientific notation, such as $1.2345 \times 10^{67}$. A widely-used 64-bit integer format can represent integers up to roughly $4.5 \times 10^{18}$, whereas a widely-used 64-bit floating-point format can represent numbers from about $10^{-308}$ to $10^{308}$, with nearly 16-significant-digit accuracy. Arithmetic on such numbers is typically done in computer hardware using binary variations of the basic schemes. For additional details on these formats, see this Wikipedia article.

For some scientific and engineering applications, however, even 16-digit accuracy is not sufficient, and so such applications rely on “multiprecision” arithmetic — software extensions to the standard hardware arithmetic operations. Cryptography, which is typically performed numerous times each day in one’s smartphone or laptop, requires computations to be done on integers as large as several thousand digits. Some computations in pure mathematics and mathematical physics research require extremely high-precision floating-point arithmetic — tens of thousands of digits in some cases (see for example this paper). Researchers exploring properties of the binary and decimal expansions of numbers such as $\pi$ have computed with millions, billions or even trillions of digits — the most recent record computation of $\pi$ was to 31 trillion digits (actually 31,415,926,535,897 decimal digits, which as a point of interest, happens to be the first 14 digits in the decimal expansion of $\pi$).

It was widely believed, even as late as the 1950s, that there was no fundamentally faster way of multiplying large numbers than the ancient scheme we learned in school, or a minor variation such as performing the operations in base $2^{32}$ rather than base ten. Then in 1960, the young Russian mathematician Anatoly Karatsuba found a clever technique for performing very high precision multiplication. By dividing each of the inputs into two substrings, he could find the product, as least to the precision often needed, with only three substring multiplications instead of four. By continuing recursively to break down the input numbers in this fashion, he obtained a scheme whose cost increased only by the approximate factor $n^{1.58}$ instead of $n^2$ with the ordinary scheme, where $n$ is the number of computer words. But this was soon overshadowed by a scheme first formalized by Schonhage and Strassen, which we now describe.

A major breakthrough in performing such computations came with the realization that the fast Fourier transform (FFT), which was originally developed to process signal data, could be used to dramatically accelerate high-precision multiplication. The fast Fourier transform is merely a clever computer algorithm to rapidly calculate the frequency spectrum of a string of data interpreted as a signal. In effect, one’s ear performs a Fourier transform (in analog, not in digital) when it distinguishes the pitch contour of a musical note or a person’s voice.

The idea for FFT-based multiplication is, first of all, to represent a very high precision number as a string of computer words, each containing, say, 32 successive bits of its binary expansion (i.e., each entry is an integer between $0$ and $2^{32} – 1$). Then two such multiprecision numbers can be multiplied by observing that their product computed the old-fashioned way (in base $2^{32}$ instead of base ten) is nothing more than the “linear convolution” of the two strings of computer words, which convolution can be performed efficiently using an FFT. In particular, the FFT-based scheme for multiplying two high-precision numbers $A = (a_0, a_1, a_2, \cdots, a_{n-1})$ and $B = (b_0, b_1, b_2, \cdots, b_{n-1})$ is the following:

- Extend each of the $n$-long input strings $A$ and $B$ to length $2n$ by padding with zeroes.
- Perform an FFT on each of the extended $A$ and $B$ strings to produce $2n$-long strings denoted $F(A)$ and $F(B)$.
- Multiply the strings $F(A)$ and $F(B)$ together, term-by-term.
- Perform an inverse FFT on the resulting $2n$-long string to yield a $2n$-long string $C$, i.e., $C = F^{-1}[F(A) \cdot F(B)]$.
- Starting at the end of the resulting $2n$-long string $C$, release carries in each term, i.e., leave only the final 32 bits in each word, with the higher-order bits added to the previous word or words as necessary.
- The result $D$ is the product of $A$ and $B$, represented as a $2n$-long string of 32-bit integers.

Note, however, that several important details were omitted here. For instance, the product of two 32-bit numbers can be computed exactly using a 64-bit hardware multiply operation, but numerous such 64-bit values must be added together when computing an FFT, requiring somewhat more than 64-bit precision. Also, for optimal performance it is important to take advantage of the fact that both input strings $A$ and $B$ are real numbers rather than complex. Finally, the outline above omitted mention on how to manage roundoff error when performing these operations. One remedy is to perform these computations not in the field of complex numbers using floating-point arithmetic, but instead in the field of integers modulo a prime number, so that roundoff error is not a factor. For additional details, see this paper.

With a fast scheme such as FFT-based multiplication in hand, division can be performed by Newton iterations, with a level of precision that approximately doubles with each iteration. This scheme reduces the cost of high-precision division to only somewhat more than twice that of high-precision multiplication. A similar scheme can be used to compute high-precision square roots. See this paper for details.

Schonhage and Strassen, in carefully analyzing a certain variation of the FFT-based multiplication scheme, found that for large $n$ (where $n$ is the number of computer words) the total computational cost, in terms of hardware operations, scales as $n \cdot \log(n) \cdot \log(\log(n))$. In practice, the FFT-based scheme is extremely fast for high-precision multiplication, since one often can utilize highly tuned library routines to perform the FFT operations.

Ever since the Schoenhage-Strassen scheme was discovered and formalized, researchers have wondered if this was truly the end of the road. For example, can we remove the final $\log(\log(n))$ factor and just have $n \cdot \log(n)$ asymptotic complexity?

Well that day has now arrived. In a brilliant new paper by David Harvey of the University of New South Wales, Australia, and Joris van der Hoeven of the French National Centre for Scientific Research, France, the authors have defined a new algorithm with exactly that property.

The Harvey-van der Hoeven paper approaches the problem by splitting the problem into smaller problems, applying the FFT multiple times, and replacing more multiplications with additions and subtractions, all in a very clever scheme that eliminates the nagging $\log(\log(n))$ factor. The authors note that their algorithm does not prove that there is no algorithm even faster than theirs. So maybe an even faster one will be found. Or not.

Either way, don’t put away your multiplication tables quite yet. The Harvey-van der Hoeven scheme is only superior to FFT-based multiplication for exceedingly high precision. But it is an important theoretical result that has wide-ranging implications in the field of computational complexity.

For some additional details, see this well-written Quanta article by Kevin Hartnett.

]]>But

Continue reading LENR: A skeptical perspective

]]>As chronicled in the earlier Math Scholar blog, a community of researchers has been pursuing a new green energy source, known as “low energy nuclear reaction” (LENR) physics, or, variously, “lattice-assisted nuclear reaction” (LANR) physics or “condensed matter nuclear reaction” (CMNR) physics. This work harkens back to 1989, when University of Utah researchers Martin Fleischmann and Stanley Pons that they had achieved desktop “cold fusion,” in a hastily called news conference. After several other laboratories failed to reproduce these findings, the scientific community quickly concluded that the Utah researchers were mistaken, to put it mildly.

But a number of other researchers did find excess heat and other signatures of interesting physics, and these researchers have continued to explore the phenomenon to the present day. If anything, this community, representing numerous nations and research institutions, has significantly grown in the past few years, with researchers claiming better results. The larger scientific community, however, continues to reject this work.

So what is the truth here? Is LENR a real phenomenon, one that could lead to a remarkable new clean, radiation-free form of energy? Or is it mirage?

In an attempt to further explore this issue, the earlier Math Scholar blog presented a bibliography of 39 recent papers published by the LENR community reporting experimental work, and discussed the status of the field. It concluded that we are left with three rather stark choices: (a) well over 100 qualified researchers from around the world, representing universities, government laboratories and private firms, each have made some fundamental errors in their experimental work; (b) at least some of these researchers are colluding to cover less-than-forthright scientific claims; or (c) these researchers are making important discoveries that are not accepted by the larger scientific community.

So which is it? Wishful thinking or reality?

While many in the LENR community remain optimistic that they have identified and real and promising phenomenon, some other researchers who have been following the field remain highly skeptical of these results. These researchers have expressed their disagreements in a number of published papers, and also in a variety of online discussion forums.

Here is a bibliography of papers that present some of these skeptical assessments, and some responses by the LENR community. This listing is in rough chronological order (most recent last) and, for each entry, includes literature citation, PDF link and a brief synopsis.

- Kirk L. Shanahan, “A systematic error in mass flow calorimetry demonstrated,”
*Thermochimica Acta*, vol. 387 (2002), 95-100, PDF. Synopsis: This paper re-analyzed the cold fusion study of E. Storms (presented in 2000 at ICCF8), which suggested a non-nuclear explanation of the signals. - W. Brian Clarke, “Search for He and He in Arata-Style palladium cathodes I: A negative result,”
*Fusion Science and Technology*, vol. 40 (2001), 147-151, PDF. Synopsis: A study of 4 Arata-style Pd cathodes found no evidence of 4He production. - W. Brian Clarke, Brian M. Oliver, Michael C. H. McKubre, Francis L. Tanzella and Paolo Tripodi, “Search for He and He in Arata-Style palladium cathodes II: Evidence for tritium production,”
*Fusion Science and Technology*, vol 40 (2001), 152-167, PDF. Synopsis: Researchers found 3He (a tritium decay product) in Arata-style cathodes (same as in [2]). - W. Brian Clarke, “Production of He in D2-loaded palladium-carbon catalyst I,”
*Fusion Science and Technology*, vol. (2003), 122-127, PDF. Synopsis: A failed replication attempt of Case-method cold fusion. - W. Brian Clarke, Stanley J. Bos and Brian M. Oliver, “Production of He in D2-Loaded palladium-carbon catalyst II,”
*Fusion Science and Technology*, vol. 43 (2003), 250-255, PDF. Synopsis: A study of 4 Case-method samples showing strong evidence of air ingress, leading to artificial 4He signals, and thus drawing into question earlier claims of 4He production. - S. Szpak, P. A. Mosier-Boss, M. H. Miles and M. Fleischmann, “Thermal behavior of polarized Pd/D electrodes prepared by co-deposition,”
*Thermochimica Acta*, vol. 410 (2004), 101-107, PDF. Synopsis: An experimental report of LENR, with negative comments regarding [1]. - Kirk L. Shanahan, “Comments on ‘Thermal behavior of polarized Pd/D electrodes prepared by co-deposition’,”
*Thermochimica Acta*, vol. 428 (2005), 207-212, PDF. Synopsis: Shanahan’s response to [6], showing how data in [6] supports his skeptical conclusion. - Edmund Storms, “Comment on papers by K. Shanahan that propose to explain anomalous heat generated by cold fusion,”
*Thermochimica Acta*, vol. 441 (2006), 207-209, PDF. Synopsis: A response by Storms to the 2002 Shanahan paper that suggested Shanahan erred. - Kirk L. Shanahan, “Reply to ‘Comment on papers by K. Shanahan that propose to explain anomalous heat generated by cold fusion’, E. Storms,
*Thermochimica Acta*, 2006,”*Thermochimica Acta*, vol 441 (2006), 210-214, PDF. A response to [8] delineating errors in [8]. - S. B. Krivit and J. Marwan, “A new look at low-energy nuclear reaction research,”
*Journal of Environmental Monitoring*vol. (2009), 1731, PDF. Synopsis: A positive overview article on the status of LENR (cold fusion). - Kirk L. Shanahan, “Comments on ‘A new look at low-energy nuclear reaction research’,”
*Journal of Environmental Monitoring*vol. 12 (2010), 1756-1764, PDF. Synopsis: A review of some of the problems and difficulties omitted from [10]. - J. Marwan, M. C. H. McKubre, F. L. Tanzella, P. L. Hagelstein, M. H. Miles, M. R. Swartz, Edmund Storms, Y. Iwamura, P. A. Mosier-Boss and L. P. G. Forsley, “A new look at low-energy nuclear reaction (LENR) research: a response to Shanahan,”
*Journal of Environmental Monitoring*, vol. 12 (2010), 1765-1770, PDF. Synopsis: A response to [11]. Note: Shanahan in reply has argued that this response employs a faulty logical device (strawman argument) to conclude Shanahan [11] is wrong. - Kirk L. Shanahan, “A realistic examination of cold fusion claims 24 Years later. A whitepaper on conventional explanations for ‘cold fusion’,”
*SRNL Technical Report SRNL-STI-2012-00678*, 22 Oct 22, 2012, PDF. Synopsis: Unreviewed whitepaper with sections on: Fleischmann and Pons calorimetry errors, response to [12], response to S. Krivit’s response to [11], unpublished manuscript regarding problems in [14]. - Akira Kitamura, Takayoshi Nohmi, Yu Sasaki, Akira Taniike, A. Tahahaski, R. Seto, Yushi Fujita, “Anomalous effects in charging of Pd powders with high density hydrogen isotopes,”
*Physics Letters A*, vol. 373 (2009), 3109-3112, PDF. Synopsis: Claims of LENR in Pd/ZrO2 systems; a replication of a prior claim by Arata.

The award cited her work in geometric analysis, gauge theory and global analysis, which has application across a broad range of modern mathematics and mathematical physics, including models for particle physics, string theory and general relativity.

Her career began in the mid-1960s, under the advisor Richard Palais. Palais had been exploring some connections between analysis (generalizations of calculus) and topology and geometry (the mathematical theory of

Continue reading Karen Uhlenbeck wins the Abel Prize

]]>The award cited her work in geometric analysis, gauge theory and global analysis, which has application across a broad range of modern mathematics and mathematical physics, including models for particle physics, string theory and general relativity.

Her career began in the mid-1960s, under the advisor Richard Palais. Palais had been exploring some connections between analysis (generalizations of calculus) and topology and geometry (the mathematical theory of shapes and continuous deformations). Palais and noted mathematician Stephen Smale had recently found some new results for harmonic maps, which can be thought of as generalizations of the calculus of variations (techniques for finding maxima and minima according to certain criteria). When the geometric/topological shape of a space is complicated, such as in a very high-dimensional object, determining the range of possible harmonic maps is quite difficult. Palais and Smale devised a condition that guarantees that at least some deformations of these spaces will converge.

In the 1970s, while at the University of Illinois, Urbana-Champaign, Uhlenbeck sought to understand more clearly the conditions under which these harmonic deformation processes would fail to converge. Working together with Jonathan Sacks, she found that these processes do converge at almost all points, but at certain points the maps may develop a “bubble singularity.” Their result has now been applied in numerous areas of mathematics.

Among other things, Uhlenbeck applied her approach to the mathematics behind the standard model of physics. Among other things, she discovered a new coordinate system for which the equations behind the standard model could be studied more easily. Then she proved her remarkable “removable singularity” theorem, which demonstrated that for four-dimensional shapes, the bubbling mentioned earlier cannot occur at isolated points. Thus any finite-energy solution to the Yang-Mills equations behind the standard model that is well-defined in a neighborhood of a point will also extend smoothly to the point.

Uhlenbeck’s results “underpin most subsequent work in the area,” according to Simon Donaldson of Imperial College in London.

Uhlenbeck has broken ground in more than one way. First of all, she is the first woman to receive the Abel award in its seventeen-year history. In 2007, she was the first woman to receive the American Mathematical Society’s Steele Prize. In 1990, she was the second woman to present a plenary lecture at the International Congress of Mathematicians (the first was Emmy Noether, in 1932). Interestingly, she also is a counterexample to the stereotype of the child prodigy that many associate with mathematicians — she didn’t really become very interested in mathematics until her freshman year at the University of Michigan.

Uhlenbeck has long found deep satisfaction and fulfillment in mathematical research. As she wrote in accepting the Leroy P. Steele Prize from the American Mathematical society,

Along the way I have made great friends and worked with a number of creative and interesting people. I have been saved from boredom, dourness, and self-absorption. One cannot ask for more.

Additional details can be read in a very nice Quanta Magazine article by Erica Klarreich, from which part of the above note was summarized.

]]>The threat of climate change is emerging as the premier global issue of our time. As a recent report by the Intergovernmental Panel on Climate Change (IPCC) grimly warns, even a 1.5 degree C (2.7 degree F) rise in global temperatures would have “substantial” consequences, in terms of extreme weather, damage to ecosystems and calamitous impact on human communities. But limiting the increase to 1.5 degree C will still require a wrenching change away from fossil fuels and an equally wrenching realignment of global economies, all over the next decade or as

Continue reading LENR energy: Science or pseudoscience?

]]>The threat of climate change is emerging as the premier global issue of our time. As a recent report by the Intergovernmental Panel on Climate Change (IPCC) grimly warns, even a 1.5 degree C (2.7 degree F) rise in global temperatures would have “substantial” consequences, in terms of extreme weather, damage to ecosystems and calamitous impact on human communities. But limiting the increase to 1.5 degree C will still require a wrenching change away from fossil fuels and an equally wrenching realignment of global economies, all over the next decade or as soon as humanly possible. The longer-term economic costs of not making these wrenching changes are almost certain to dwarf the near-term costs of taking decisive action.

It has now been 30 years since University of Utah researchers Martin Fleischmann and Stanley Pons announced that they had achieved desktop “cold fusion,” in a hastily called and clearly premature news conference on March 23, 1989. After several other laboratories failed to reproduce these findings, the scientific community quickly concluded that the Utah researchers were mistaken, to put it mildly. At an American Physical Society meeting a few weeks after the announcement, Steven Koonin of Cal Tech called the Utah claims the result of the incompetence and delusion of Pons and Fleischmann, to which the audience applauded in approval. John Huizenga, who later chronicled the episode in a book, called it the scientific fiasco of the century.

But a funny thing happened on the way to the public dunking of Fleischmann and Pons: A number of other researchers found excess heat and other signatures of interesting physics, and these researchers, tilting against the scientific establishment, have continued to explore the phenomenon to the present day. If anything, this community has significantly grown in the past few years — several hundred researchers worldwide are now involved.

So what in the world is going on here? Is this science or pseudoscience?

One can certainly excuse the physics community for reticence in taking these claims seriously. After all, nuclear fusion, such as occurs in the center of stars and is the source of the light they produce, requires enormous temperatures (millions of degrees) for atoms of hydrogen, say, to acquire sufficient energy to overcome the Coulomb repulsion between protons and undergo fusion.

In response, the low-energy nuclear reaction (LENR) community, as they prefer to be named, propose that the reaction is not “fusion” in the classical sense, but some related nuclear phenomenon that occurs in a condensed matter regime at modest temperatures; hence the name LENR, or, alternatively, “lattice-assisted nuclear reaction” (LANR) or “condensed matter nuclear reaction” (CMNR). The phenomenon they claim to observe produces measurable heat, but does not produce measurable amounts of ionizing radiation of the sort that causes cancer and other health problems in living organisms.

If true, the claimed phenomenon would appear to be an ideal energy source:

- In some experiments, commercially significant heat is output, and there is even hope of direct production of electricity.
- Systems can be scaled up from very small units to very large systems.
- There is no greenhouse gas emission or pollution of any kind.
- The “fuel” for these reactions is cheap and effectively unlimited in supply — a few grams of inexpensive materials such as hydrogen, lithium and nickel should be sufficient to provide steady output energy for a year or more (palladium, an expensive metal, is used in some of these experiments, but only as a catalyst).
- In most experimental setups, there is no measurable ionizing radiation above normal background levels.
- There is no radioactive fuel or radioactive byproducts.
- Unlike solar and wind power, LENR is not dependent on the vagaries of weather and climate.

So who is right: the LENR community, which claims steadily improving results, or the mainstream physics community, most of whom insist that “cold fusion” was debunked long ago?

As someone who has followed these developments for several years, the present author is as perplexed as anyone. By the established standards of scientific consensus, not to mention the lack of a well-understood theoretical framework, the LENR researchers should have surrendered long ago. But how can one explain ongoing results by well-qualified experimenters, and the rapidly growing interest among researchers, investors and entrepreneurs? Is this a classic example of wishful-thinking pseudoscience run amok? Or is this the start of a new era in energy technology?

In an attempt to cast some light on this paradox, the present author has collected a set of 39 recent representative published papers in the field — see the Appendix at the end of this article. These papers: (a) have appeared in some credible, peer-reviewed source within the past five years (except [Levi2014]); (b) present or summarize experimental results, as opposed to purely theoretical studies; and (c) are available online in PDF form. Each entry includes a citation, a PDF link and a brief synopsis. Here are some overall observations:

- Almost every paper mentions “excess heat” — measured output heat energy exceeding the total energy input. In most cases the excess energy is a few watts, but [Mizuno2017], for instance, reports roughly 300 watts, and [Parkhomov2016] reports roughly 600 watts, or a total of roughly 40 kWh over three days.
- Other recorded effects include neutrons, energized particle tracks, ultraviolet emanations and nuclear transmutations, which are characteristics of nuclear rather than chemical processes — see [Mills2018], [Rajeev2017], [Roussetski2017], [Violante2016], [Valay2016], [Mills2015], [Levi2014] and others.
- Some papers mention specific techniques, such as stimulation by lasers of a certain frequency, that enhance the measured effects — see [Letts2015] for instance.
- Most experiments use relatively sophisticated equipment, such as mass-flow calorimetry, CR-39 energetic particle detectors and time-of-flight secondary ion mass spectrometers — see [Mosier-Boss2017], [Rajeev2017], [Roussetski2017], [Kitamura2015], [Swartz2015b], [Aizawa2014] and others.
- Reproducibility appears to be significantly improved compared to earlier years. For example, [Letts2015] reports excess power in 161 of 170 experimental runs.
- Most of these researchers appear to be well-qualified. For instance, the SRI International team, led by Francis Tanzella (see [Tanzella2018], [Mosier-Boss2015], [Godes2014], [McKubre2014]) claims 75 person-years of experience in calorimetry experimentation of the type required for these experiments — they are literally the world’s experts.
- Most papers have multiple authors. [Celani2018], [Kitamura2017b] and [Celani2014] each have 15 authors; [Kitamura2018], [Iwamura2017b] and [Kitamura2017a] each have 17.
- The collection features 119 distinct authors, representing seven different nations (USA, Japan, Italy, Sweden, China, Russia and Ukraine) and 39 different institutions, ranging from universities such as MIT and Kobe University to NASA, the U.S. Navy, China’s Institute of Atomic Energy, Italy’s National Institute for Nuclear Physics and Sweden’s Royal Institute of Technology, as well as several private entities such as SRI International, Nissan Motors, Brillouin Energy and JET Energy.

From a first look at these papers there does not appear to be any easy way to dismiss them. For the most part, these experiments are meticulously documented and performed with up-to-date equipment; results are carefully recorded and analyzed; and proper attention is paid to reproducibility, all as far as the present author can determine, although he does not pretend to be an expert in this particular discipline.

On the downside, whereas nearly all of the publications and conference proceedings listed below are peer-reviewed, top-tier journals (e.g., the Journal of the American Chemical Society and the Physical Review journals) are conspicuously missing. Many of these LENR papers are in the *Journal of Condensed Matter Nuclear Science*, a publication that was formed and is edited by persons in the LENR field. Researchers in the field acknowledge that their work remains stuck in what Cambridge philosopher Huw Price calls a reputation trap, because of its association with the “cold fusion” fiasco. As LENR researcher Michael KcKubre explains [McKubre2016],

This timidity and unwillingness to re-investigate the data of the [Fleishmann-Pons heat effect] and related effects by editors of mainstream journals, effectively ostracizes the [condensed matter nuclear science] community from the main body of science and may be the single most effective barricade to progress.

Along this line, the present author is impressed by several instances (see [Alabin2018] for instance) of authors reporting that they were unable to reproduce a finding of a previous study. From the present author’s experience, such forthright reporting is not the mark of weakness but instead is the mark of professional, seasoned experimental science. The present author, who has openly called out inflated claims and poor reproducibility in his own research field, dearly wishes his colleagues would be more open in reporting failures and disappointments.

It should be noted that the bibliography below is but a subset of the published literature in this area. For example, the paper [Mosier-Boss2019] (the first item listed below) presents a synopsis of 60 published papers in the field that appeared from 1991-2018, not just in the past five years. Many others could be listed.

It should be added that while many in the LENR community remain optimistic that they have identified and real and promising phenomenon, some other researchers who have been following the field remain highly skeptical of these results. These researchers have expressed their disagreements in a number of published papers, and also in a variety of online discussion forums.

A bibliography of papers that present some of these skeptical assessments, together with some responses by the LENR community, is available Here.

Several investors and entrepreneurs are now sufficiently confident in LENR that they are pursuing commercial ventures.

One such venture is Brillouin Energy Corp. of Berkeley, California, which was founded by physicist Robert Godes. Brillouin is developing what they term a controlled electron capture reaction (CECR) process. In their experiments, ordinary hydrogen is loaded into a nickel lattice, and then an electronic pulse is passed through the system, using a proprietary control system. They claim that their device converts H-1 (ordinary hydrogen) to H-2 (deuterium), then to H-3 (tritium) and H-4 (quatrium), which then decays to He-4 and releases energy. Additional technical details are given at the Brillouin Energy website, and in a patent application. Brillouin is preparing an international commercial roll-out.

Their work is backed by significant published experimental work — see, for example, [Roussetski2017], [McKubre2016], [Mosier-Boss2015], [Godes2014] and [McKubre2014] in the bibliography below. [McKubre2016], for instance, cites results for hundreds of experiments on Brillouin-like power cells, all performed independently at SRI International by highly qualified researchers.

Another, more controversial, commercial venture has been founded by Andrea Rossi, a colorful (and controversial) Italian-American engineer-entrepreneur. Rossi claims that he has developed and patented a reactor (named the “E-Cat SK”), roughly the size of a suitcase, that produces up to 22 KWatt output power in the form of heat; thus 50 such units, which would fit very comfortably in a shipping container, could produce more than 1 MWatt heat. His company (Leonardo Corporation) is now inviting orders for a heat service based on the E-Cat SK — a customer pays a discounted rate for the net heat (output heat minus input electricity) produced by a E-Cat SK system sited on the customer’s premises but managed by Leonardo. See Leonardo Corporation in the U.S. or Hydro Fusion in the U.K.

On the downside, Rossi’s work leaves much to be desired from a scientific point view. Rossi remains very tight-lipped, preferring to protect his company’s intellectual property through silence, and thus has not released any details on exactly how his systems are constructed and operated. For the most part, we have only Rossi’s word about his system’s performance and commercial prospects, and thus many remain highly skeptical of his operation.

There is some experimental backing for Rossi’s approach, although not nearly as much as for Brillouin. In October 2014, a team of Italian and Swedish researchers released a paper (see [Levi2014] below) that reported a “coefficient of performance” (ratio of output heat to input power) of up to 3.6, based on experiments with a Rossi-type reactor performed at a laboratory in Lugano, Switzerland. More recently, Russian physicist Alexandre Parkhomov and others have reported success in performing Rossi-like experiments — see the papers [Alabin2018], [Zelensky2017], [Parkhomov2016] and [Valay2016] below for further details.

As the present author and a colleague observed in an earlier blog, the current LENR research work leaves us with three stark choices: (a) well over 100 qualified researchers from around the world, representing universities, government laboratories and private firms, each have made some fundamental errors in their experimental work, perhaps out of wishful thinking; (b) at least some of these researchers are colluding to cover less-than-forthright scientific claims; or (c) these researchers are making important discoveries that are not accepted by the larger scientific community.

With each passing month, and with more researchers publishing similar results, option (a), while still possible, looks less likely. With regards to (b), while a small-scale collusion is definitely possible, a collusion involving well over 100 researchers seems most unlikely. As Benjamin Franklin wrote in his Poor Richard’s Almanack, “Three may keep a secret, if two of them are dead,” or as physicist David Robert Grimes observed in a 2016 study, the chances of maintaining a conspiracy decline sharply with time and exponentially with the number of persons involved. Along this line, since private firms are nearing commercial roll-out, there would seem little point in major fraud, since, as Mats Lewan has observed, an inexpensive power meter (see this item for instance) can be used measure to input power, and a net heat service would lose money.

And yet, option (c), namely that LENR researchers have found a real phenomenon that the broader scientific community rejects, seems rather unlikely also, to say the least. Among other things, what exactly is the theoretical framework behind LENR? Although numerous LENR researchers have published papers exploring the question, as yet there is no clear consensus, and in the absence of a well-grounded physical theory, considerable caution is certainly justified.

In a recent article, Huw Price points out that the more society needs something, the more prudent it is to make sure we don’t miss it. Wrongly dismissing a potential new source of carbon-free energy could be catastrophic, so LENR merits serious attention, even if it seems quite unlikely. The present author agrees that although skepticism is still in order, LENR research deserves more than the contemptuous brush-off it typically receives today, such as in this 2016 New Scientist editorial.

Needless to say, if LENR is upheld the stakes are very high:

*An environmental windfall*: A non-carbon, non-polluting energy source whose “fuel” costs virtually nothing and is unlimited in supply, which can be scaled from very small units to very large systems, which does not produce radioactive byproducts or harmful ionizing radiation, and which, unlike solar or wind power, is not dependent on vagaries of weather or climate.*Financial repercussions*. According to a 2015 Bloomberg report, at least one-half trillion dollars of bonds are at risk if oil prices drop further.*International repercussions*: According to a separate Bloomberg report, Saudia Arabia is having difficulty keeping its economy operating at full steam with the current drop in oil prices and its own longer term goals. Venezuela is already in a humanitarian crisis brought on in part by the recent drop in oil prices (but due also, of course, to corrupt and incompetent government).*Scientific repercussions*: If LENR is confirmed, the broader scientific community would be embarrassed, to put it mildly, since it has overlooked and discouraged research in the area for three decades, very likely delaying the commercial roll-out of such technology.*Political repercussions*: Some right-leaning political groups may oppose LENR, in spite of its potential economic benefit, since it may threaten existing fossil fuel industries. Some left-leaning political groups may oppose LENR, in spite of its potential environmental benefit, because nuclear energy has traditionally been anathema.

However this turns out, the present author only wishes to add the following: If the LENR researchers truly have something, they have a *moral obligation to the world society* to come public with their methods and make this technology available worldwide as soon as possible. The author obviously recognizes the need for reasonable privacy of intellectual property, but as Huw Price has noted, missing a desperately needed new energy source could be catastrophic.

So is LENR a revolutionary new form of energy, or merely a new chapter in the long and sordid annals of pseudoscience? Read some of the papers below and decide for yourself. One way or the other, the next few months will be very interesting. Hold on to your hats!

These are listed by year, most recent (2019) first; within a year, listed alphabetically by first author. Note that in some instances, the PDF file linked to below is the entire volume of the journal in question, so that one must search in the file for the indicated page numbers of the particular article. In this listing, “low-energy nuclear reaction” (LENR), “lattice-assisted nuclear reaction” (LANR) and “condensed matter nuclear reaction” (CMNR) should be considered as largely synonymous, since these effects are most likely connected.

A bibliography of papers that present some more skeptical assessments, together with some responses by the LENR community, is available Here.

- [Mosier-Boss2019] P. A. Mosier-Boss and L. P. Forsley, “A synopsis of nuclear reactions in condensed matter,”
*Global Energy Corp. Technical Report*, 2019, PDF. Synopsis: Presents a synopsis of 60 condensed matter nuclear reaction papers that have appeared from 1991-2018, including tabulations of journals, authors, affiliations and nationalities. - [Alabin2018] K. A. Alabin, S. N. Andreev, A. G. Sobolev, S. N. Zabavin, A. G. Parkhomov and T. R. Timerbulatov, “Isotopic and elemental composition of substance in nickel-hydrogen heat generators,”
*Journal of Condensed Matter Nuclear Science*, vol. 26 (2018), pg. 32-44, PDF. Synopsis: Results are presented for Rossi-type Ni-Li-H experiments, which produced up to 790 MJ excess energy. No change was seen in the isotopic composition of Ni or Li, but a significant increase was seen in several other nuclides. - [Celani2018] F. Celani, B. Ortenzi, S. Pella and A. Spallone, G. Vassallo, E. Purchi, S. Fiorilla, L. Notargiacomo, C. Lorenzetti, A. Calaon, A. Spallone, M. Nakamura, A. Nuvoli, P. Cirilli and P. Boccanera, “Improved stability and performance of surface-modified constantan wires, by chemical additions and unconventional geometrical structures,”
*Journal of Condensed Matter Nuclear Science*, vol. 27 (2018), pg. 9-21, PDF. Synopsis: The anomalous heat effect observed by others in Cu-Ni-Mn (“constantan”) materials with H2-D2 absorption is enhanced by high temperatures and the addition of Fe-K-Mn. - [Kitamura2018] A. Kitamura, A. Takahashi, K. Takahashi, R. Setoa, T. Hatanoa, Y. Iwamura, T. Itoh, J. Kasagi, M. Nakamura, M. Uchimurac, H. Takahashi, S. Sumitomo, T. Hioki, T. Motohiro, Y. Furuyama, M. Kishida and H. Matsune, “Excess heat evolution from nanocomposite samples under exposure to hydrogen isotope gases,”
*International Journal of Hydrogen Energy*, vol. 43, no. 33 (16 Aug 2018), pg. 16187-16200, PDF. Synopsis: At 200-300 C, binary metal nanocomposite samples produced excess power of 3-24 W; the excess power was observed not only in D-Pd-Ni but also in H-Pd-Ni and H-Cu-Ni systems. - [Mills2018] R. Mills, Y. Lu, R. Frazer, “Power determination and hydrino product characterization of ultra-low field ignition of hydrated silver shots,”
*Chinese Journal of Physics*, vol. 56, (2018), pg. 1667-1717, DOI: 10.1016/j.cjph.2018.04.015, preprint at PDF. Synopsis: A hydrino reactor consisting of a water-entrained injected molten silver matrix produced a plasma that delivered 400 KW power in EUV and UV light. - [Tanzella2018] F. Tanzella, “Isoperibolic hydrogen hot tube reactor studies,”
*SRI International Technical Report*, Mar 2018, PDF. Synopsis: Over 100 runs were performed on 34 different Ni-coated cores, with experiments operated in H2 or He gas from 200 C to 600 C. The measured coefficient of performance (ratio of output power to input power) ranged from 1.27 to 1.41. - [Beiting2017] E. J. Beiting, “Investigation of the nickel-hydrogen anomalous heat effect,”
*Aerospace Technical Report*, 15 May 2017, PDF. Synopsis: Experiments with Ni-Pd materials produced 7.5% more power than the input power over a period of 1000 hours (40 days). - [Itoh2017] T. Itoh, Y. Iwamura, J. Kasagi and H. Shishido, “Anomalous excess heat generated by the interaction between nano-structured Pd/Ni surface and D2 gas,”
*Journal of Condensed Matter Nuclear Science*, vol. 24 (2017), pg. 179-190, PDF. Synopsis: In experiments with nano-structured Pd and Ni, a 123 C increase in temperature was observed compared to a control experiment. - [Iwamura2017a] Y. Iwamura, T. Itoh, J. Kasagi, A. Kitamura, A. Takahashi and K. Takahashi, “Replication experiments at Tohoku University on anomalous heat generation using nickel-based binary nanocomposites and hydrogen isotope gas,”
*Journal of Condensed Matter Nuclear Science*, vol. 24 (2017), pg. 191-201, PDF. Synopsis: In one experiment with D2 gas, excess heat up to 10 W was observed. The amount of excess energy reached 2.5 MJ, corresponding to 14.9 eV per absorbed D. - [Iwamura2017b] Y. Iwamura, T. Itoh, J. Kasagi, A. Kitamura, A. Takahashi, K. Takahashi, R. Seto, T. Hatano, T. Hioki, T. Motohiro, M. Nakamura, M. Uchimura, H. Takahashi, S. Sumitomo, Y. Furuyama, M. Kishida, and H. Matsune, “Reproducibility on anomalous heat generation by metal nanocomposites and hydrogen isotope gas,”
*Proceedings of the 18th Meeting of Japan CF Research Society (JCF18)*, Tohoku University, Japan (24-25 Nov 2017), PDF. Synopsis: Researchers obtained significant anomalous heat generation in experiments with Pd-Ni-Zr and Cu-Ni-Zr samples, at elevated temperatures. - [Kitamura2017a] A. Kitamura, A. Takahashi, K. Takahashi, R. Seto, T. Hatano, Y. Iwamura, T. Itoh, J. Kasagi, M. Nakamura, M. Uchimura, H. Takahashi, S. Sumitomo, T. Hioki, T. Motohiro, Y. Furuyama, M. Kishida and H. Matsune, “Comparison of excess heat evolution from zirconia-supported Pd-Ni nanocomposite samples with different Pd/Ni ratio under exposure to hydrogen isotope gases,”
*Proceedings of the 18th Meeting of Japan CF Research Society (JCF18)*, Tohoku University, Japan (24-25 Nov 2017), pg. 1-13, PDF. Synopsis: Anomalous heat effect was seen in the interaction of hydrogen isotope gas and nanoparticles; excess power of 3 to 24 watts continued for several weeks. - [Kitamura2017b] A. Kitamura, A. Takahashi, K. Takahashi, R. Seto, Y. Matsuda, Y. Iwamura, T. Itoh, J. Kasagi, M. Nakamura, M. Uchimura, H. Takahashi, T. Hioki and T. Motohiro, Y. Furuyama and M. Kishida, “Collaborative examination on anomalous heat effect using nickel-based binary nanocomposites supported by zirconia,”
*Journal of Condensed Matter Nuclear Science*, vol. 24 (2017), pg. 202-213, PDF. Synopsis: At temperatures 200-300 C, samples of Pd-Ni-Zr and Cu-Ni-Zr showed excess energy of 5-10 W for periods of several days. - [Mizuno2017] T. Mizuno, “Observation of excess heat by activated metal and deuterium gas,”
*Journal of Condensed Matter Nuclear Science*, vol. 25 (2017), pg. 1-25, PDF. Synopsis: In a series of Ni-D2 experiments, with reactor temperature up to 300 C, running up to 30 days, typically 300 W excess heat was produced. - [Mosier-Boss2017] P. A. Mosier-Boss, F. E. Gordon, L. P. Forsley, D. Zhou, “Detection of high energy particles using CR-39 detectors part 1: Results of microscopic examination, scanning, and LET analysis,”
*International Journal of Hydrogen Energy*, vol 42 (2017), pg. 416-428,

PDF. Synopsis: During Pd/D deposition, high-energy particles, 133 times background, were detected corresponding to tracks created in the Pd deposit. - [Rajeev2017] K. P. Rajeev and D. Gaur, “Evidence for nuclear transmutations in Ni-H electrolysis,”
*Journal of Condensed Matter Nuclear Science*, vol. 24 (2017), pg. 278-283, PDF. Synopsis: Using a time-of-flight secondary ion mass spectrometer (ToF-SIMS), a Ni-H system produced isotopes 58-Ni, 60-Ni and 62-Ni, with ratios differing significantly from natural abundances. - [Roussetski2017] A. S. Roussetski, A. G. Lipson, E. I. Saunin, F. Tanzella and M. McKubre, “Detection of high energy particles using CR-39 detectors Part 2: Results of in-depth destructive etching analysis,”
*International Journal of Hydrogen Energy*, vol. 42 (2017), pg. 429-436,

PDF. Synopsis: Researchers performed Pd electrodeposition in D2O and H2O electrolytes in the presence of a magnetic field, and found a statistically significant increase in the number of tracks measured in the D2O runs. - [Zelensky2017] V. F. Zelensky, V. O. Gamov, A. L. Ulybkin and V. D. Virich, “Experimental device of cold HD-fusion energy development and testing (verification experiment),”
*Journal of Condensed Matter Nuclear Science*, vol. 24 (2017), pg. 168-178, PDF. Synopsis: Results generally confirmed earlier results obtained by an independent group in Lugano, Switzerland, for the HTE-Cat reactor. See [Levi2014] below. - [Kitamura2016] A. Kitamura, E. F. Marano, A. Takahashi, R. Seto, T. Yokose, A. Taniike and Y. Furuyama, “Heat evolution from zirconia-supported Ni-based nano-composite samples under exposure to hydrogen isotope gas,”
*Proceedings of the 16th Meeting of Japan CF Research Society (JCF16)*, (2016) pg. 1-16, PDF. Synopsis: Excess power of 11-12 W was recorded repeatedly in both Pd-Ni-ZrO2 and Cu-Ni-ZrO2 sample runs with both H2 and D2 gas. - [McKubre2016] M. C. H. McKubre, “Cold Fusion-CMNS-LENR: Past, present and projected future status,”
*Journal of Condensed Matter Nuclear Science*, vol. 19 (2016), pg. 183-191, PDF. Synopsis: Provides an overview and status of the field as of early 2016, including a brief summary of hundreds of experiments performed at SRI International. - [Mosier-Boss2016] P. A. Mosier-Boss, L. P. G. Forsley, and P. K. McDaniel, “Investigation of nano-nuclear reactions in condensed matter: Final report,”
*Defense Threat Reduction Agency Report*(2016), PDF. Synopsis: This summarizes over 20 years of experimental work and analysis in LENR at the U.S. Navy’s Space and Naval Warfare Systems Center Pacific and some related centers. - [Parkhomov2016] A. G. Parkhomov and E. O. Belousova, “Research into heat generators similar to high-temperature Rossi reactor,”
*Journal of Condensed Matter Nuclear Science*, vol. 19 (2016), pg. 244-256, PDF. Synopsis: Devices similar to those of Rossi, using a mixture of Ni and Li-AlH4, produced more than 40 KwH or 150 MJ heat in excess energy when the system was heated to 1100 C or higher. - [Violante2016] V. Violante, E. Castagna, S. Lecci, G. Pagano, M. Sansovini and F. Sarto, “Heat production and RF detection during cathodic polarization of palladium in 0.1 M LiOD,”
*Journal of Condensed Matter Nuclear Science*, vol. 19 (2016), pg. 319-324, PDF. Synopsis: Radio-frequency emissions were observed in several experiments with Pd-based cathodes. - [Valay2016] M. Valat, A. Goldwater, R. Greenyer, R. Higgins and R. Hunt, “Investigations of the Lugano HotCat reactor,”
*Journal of Condensed Matter Nuclear Science*, vol. 21 (2016), pg. 81-96, PDF. Synopsis: In an attempt to replicate the Lugano report of Rossi’s HotCat reactor, using a Ni-Li-H compound, a team found that the earlier report had overestimated the results, but still found a significant amount of excess thermal energy and low energy gamma radiation. - [Kitamura2015] A. Kitamura, A. Takahashi, R. Seto, Y. Fujita, A. Taniike and Y. Furuyama, “Brief summary of latest experimental results with a mass-flow calorimetry system for anomalous heat effect of nano-composite metals under D(H)-gas charging,”
*Current Science*, vol. 108, no. 4 (Feb 2015), pg. 589-593, PDF. Synopsis: An experimental run with a Cu-Ni-ZrO2 sample produced 15 watts for three days, and 10 watts for three weeks. - [Letts2015] D. Letts, “Highly reproducible LENR experiments using dual laser stimulation,”
*Current Science*, vol. 108 (2015), pg. 559-561, PDF. Synopsis: Excess power was observed in 161 out of 170 Pd-D tests stimulated by dual lasers at 8, 15 and 20 THz. - [Mills2015] R. Mills and J. Lotoski, “H2O-based solid fuel power source based on the catalysis of H by HOH catalyst,”
*International Journal of Hydrogen Energy*, vol. 40 (2015), pg. 25-37, PDF. Synopsis: When high current was applied, a water-based solid fuel element exploded with a large burst of ultraviolet light. - [Mosier-Boss2015] P. A. Mosier-Boss, L. P. Forsley, F. E. Gordon, D. Letts, D. Cravens, M. H. Miles, M. Swartz, J. Dash, F. Tanzella, P. Hagelstein, M. McKubre and J. Bao, “Condensed matter nuclear reaction products observed in Pd/D co-deposition experiments,”
*Current Science*, vol. 108 (2015), pg. 656-659, PDF. Synopsis: A Pd/D co-deposition experiment produced heat, transmutation, tritium, neutrons and energetic charged particles. - [Swartz2015a] M. Swartz, G. Verner, J. Tolleson, L. Wright, R. Goldbaum, P. Hagelstein, “Amplification and restoration of energy gain using fractionated magnetic fields on ZrO2-PdD nanostructured components,”
*Journal of Condensed Matter Nuclear Science*, vol. 15 (2015), pg. 66-80, PDF. Synopsis: Experiments with nanocomposite ZrO2-PdNiD materials exhibited significant energy gain over long periods of time, with fractionated magnetic fields having a significant amplification effect. - [Swartz2015b] M. R. Swartz, G. Verner, J. Tolleson, L. Wright, R. Goldbaum, P. Mosier-Boss and P. L. Hagelstein, “Imaging of an active NANOR-type LANR component using CR-39,”
*Journal of Condensed Matter Nuclear Science*, vol. 15 (2015), pg. 81-91, PDF. Synopsis: Experiments with CR-39 energized particle detectors found tracks corresponding to components exhibiting energy production, strongly suggesting that the operation is a nuclear process. - [Aizawa2014] H. Aizawa, K. Mita, D. Mizukami, H. Uno and H. Yamada, “Detecting energetic charged particle in D2O and H2O electrolysis using a simple arrangement of cathode and CR-39,”
*Journal of Condensed Matter Nuclear Science*, vol. 13 (2014), pg. 6-12, PDF. Synopsis: A CR-39 energetic particle detector with Ni film found tracks in one out of seven experimental runs with D2O and two out of five runs with H2O. - [Celani2014] F. Celani, E. F. Marano, B. Ortenzi, S. Pella, S. Bartalucci, F. Micciulla, S. Bellucci, A. Spallone, A. Nuvoli, E. Purchi, M. Nakamura, E. Righi, G. Trenta, G. L. Zangari, A. Ovidi, “Cu-Ni-Mn alloy wires, with improved sub-micrometric surfaces, used as LENR device by new transparent, dissipation-type calorimeter,”
*Journal of Condensed Matter Nuclear Science*, vol. 13 (2014), pg. 56-67, PDF. Synopsis: An experimental run with Ni-Cr wire achieved 21 W excess power. - [Godes2014] R. Godes, R. George, F. Tanzella and M. McKubre, “Controlled electron capture and the path toward commercialization,”
*Journal of Condensed Matter Nuclear Science*, vol. 13 (2014), pg. 127-137, PDF. Synopsis: In over 150 experimental runs, using two different cell/calorimeter designs, excess energies of up to 100% (i.e., heat output twice the input energy) have been observed in runs with Q pulses tuned to the resonance of Pd and Ni hydrides in pressurized vessels. - [Jiang2014] S. Jiang, X. Xu, L. Zhu, S. Gu, X. Ruan, M. He and B. Qi, “Neutron burst emissions from uranium deuteride and deuterium-loaded titanium,”
*Journal of Condensed Matter Nuclear Science*, vol. 13 (2014), pg. 253-263, PDF. Synopsis: Bursts of up to 2800 neutrons occurred in less than 30 seconds in an experiment with deuterium-loaded titanium and uranium deuteride samples at room temperatures, suggesting nuclear reactions. - [Levi2014] G. Levi, E. Foschi, B. Hoistad, R. Pettersson, L. Tegner and H. Essen, “Observation of abundant heat production from a reactor device and of isotopic changes in the fuel,”
*University of Bologna*, 2014, PDF. Synopsis: Researchers running a version of Rossi’s E-Cat found a coefficient of performance of 3.2 (for the 1260 C run) and 3.6 (for the 1400 C run). The total net energy output was about 1.5 MWh. Changes were also observed in the isotopic composition of lithium and nickel. Note: Strictly speaking this is not a peer-reviewed publication (and some weaknesses have subsequently been identified in the analysis), but it is included here because several others reference it. - [McKubre2014] M. McKubre, J. Bao and F. Tanzella, “Calorimetric studies of the destructive stimulation of palladium and nickel fine wires,”
*Journal of Condensed Matter Nuclear Science*, vol. 13 (2014), pg. 337-345, PDF. Synopsis: Thirty experiments, some with Pd and others with Ni, demonstrated excess energy, with the active region primarily in the co-deposited material. - [Miles2014] M. H. Miles, “Co-deposition of palladium and other transition metals in H2O and D2O solutions,”
*Journal of Condensed Matter Nuclear Science*, vol. 13 (2014), pg. 401-410, PDF. Synopsis: Experiments studying co-deposition of Pd, Ru, Re, Ni and Ir in H2O and D2O ammonium systems found significant excess in the Pd-D2O system but none with Ru, Re or Ni systems. - [Mosier-Boss2014] P. A. Mosier-Boss, “It is not low energy — but it is nuclear,”
*Journal of Condensed Matter Nuclear Science*, vol. 13 (2014), pg. 432-442, PDF. Synopsis: Pd/D co-deposition experiments produced energetic particles: 2.45 MeV neutrons, 3-10 MeV protons, 2-15 MeV alphas and 14.1 MeV neutrons. - [Sakoh2014] H. Sakoh, Y. Miyoshi, A. Taniike, Y. Furuyama, A. Kitamura, A. Takahashi, R. Seto, Y. Fujita, T. Murota and T. Tahara, “Hydrogen isotope absorption and heat release characteristics of a Ni-based sample,”
*Journal of Condensed Matter Nuclear Science*, vol. 13 (2014), pg. 471-484, PDF. Synopsis: Hydrogen isotope absorption runs using Cu-Ni-ZrO2 and Ni-ZrO2 nano-powders produced long-lasting temperature changes corresponding to high levels of energy output per atom of Ni. - [Swartz2014] M. R. Swartz and P. L. Hagelstein, “Demonstration of energy gain from a preloaded ZrO2-PdD nanostructured CF/LANR quantum electronic Device at MIT,”
*Journal of Condensed Matter Nuclear Science*, vol. 13 (2014), pg. 516-527, PDF. Synopsis: Several experiments with a ZrO2-PdD device found energy gain ranging from 5 to 16.

In 1957, British-American mathematician Louis Mordell asked whether, given some integer $k$, there are integers $x, y, z$ such that $x^3 + y^3 + z^3 = k$. Like Fermat’s last theorem, this problem is very easily stated but very difficult to explore, much less solve definitively.

Some solutions are easy. When $k = 3$, for instance, there are two simple solutions: $1^3 + 1^3 + 1^3 = 3$ and $4^3 + 4^3 + (-5)^3 = 3$. It is also known that there are no solutions in other cases, including

Continue reading New result for Mordell’s cube sum problem

]]>In 1957, British-American mathematician Louis Mordell asked whether, given some integer $k$, there are integers $x, y, z$ such that $x^3 + y^3 + z^3 = k$. Like Fermat’s last theorem, this problem is very easily stated but very difficult to explore, much less solve definitively.

Some solutions are easy. When $k = 3$, for instance, there are two simple solutions: $1^3 + 1^3 + 1^3 = 3$ and $4^3 + 4^3 + (-5)^3 = 3$. It is also known that there are no solutions in other cases, including $4, 5, 13, 14$ and others.

Significant progress was made in 2016 by Sander Huisman. Employing an algorithm due to Noam Elkies, he found a total of 966 new solutions for $k$ in the range $1 \le k \lt 1000$. To begin with, Huisman found the first known solution for $k = 74$, namely $$74 = (−284650292555885)^3 + 66229832190556^3 + 283450105697727^3.$$ He also found a second known solution for each of these three cases: $$606 = (−170404832787569)^3 + (−16010802062873)^3 + 170451934224718^3,$$ $$830 = (−947922123009026)^3 + (−335912682279105)^3 + 961779444965911^3,$$ $$966 = (−1134209166959435)^3 + 291690681248788^3 + 1127741630138089^3.$$ His work left 13 cases less than 1000 for which no solutions had been found: $k = 33, 42, 114, 165, 390, 579, 627, 633, 732, 795, 906, 921, 975$.

The latest news here is that Andrew Booker, who confessed he was inspired by watching the Numberphile video The untracked problem with 33, has found a new solution for $k = 33$, one of the unsolved cases remaining from Huisman’s work. Whereas Huisman employed Elkies’ algorithm, Booker employed a scheme based on the fact that in any solution $k – z^3 = x^3 + y^3$ must have $x + y$ as a factor. Booker found that the running time for this scheme is very nearly linear in the height bound of $x, y, z$.

This is Booker’s result: $$33=8866128975287528^3 +(−8778405442862239)^3 +(−2736111468807040)^3.$$ The run required approximately 15 core-years of computation over three weeks. This leaves $k = 42$ as the only case less than 100 for which no solution is known, and 12 cases less than 1000.

But who knows? Maybe others will soon find solutions for these as well…

For some additional details, see this New Scientist article by Donna Lu.

]]>San Francisco’s Exploratorium is featuring several events, culminating with a “Pi Procession” at 1:59pm Pacific Time (corresponding to 3.14159) and pie served at 2:15pm. The website teachpi.org lists 50 ideas to make Pi Day “entertaining, educational, tasty and fun.”

For this year’s Pi Day festivities, the Math Scholar blog presents a Pi Day crossword puzzle (see below), created by the present author. We will announce the first correct solver! Send

Continue reading A Pi Day crossword puzzle

]]>San Francisco’s Exploratorium is featuring several events, culminating with a “Pi Procession” at 1:59pm Pacific Time (corresponding to 3.14159) and pie served at 2:15pm. The website teachpi.org lists 50 ideas to make Pi Day “entertaining, educational, tasty and fun.”

For this year’s Pi Day festivities, the Math Scholar blog presents a Pi Day crossword puzzle (see below), created by the present author. We will announce the first correct solver! Send a photo of your completed puzzle to the email given at the top right of the author’s home page.

[Added 15 Mar 2019: I am pleased to announce that Ross Blocher, co-host of the popular podcast “Oh no, Ross and Carrie!”, has posted the first correct solution. Congratulations, Ross!]

Here is the solution: PDF.

]]>Introduction

Archimedes is widely regarded as the greatest mathematician of antiquity. He was a pioneer of applied mathematics, for instance with his discovery of the principle of buoyancy, and a master of engineering designs, for instance with his “screw” to raise water from one level to another. But his most far-reaching discovery was the “method of exhaustion,” which he used to deduce the area of a circle, the surface area and volume of a sphere and the area under a parabola. Indeed, with this method Archimedes anticipated, by nearly

Continue reading Simple proofs: Archimedes’ calculation of pi

]]>**Introduction**

Archimedes is widely regarded as the greatest mathematician of antiquity. He was a pioneer of applied mathematics, for instance with his discovery of the principle of buoyancy, and a master of engineering designs, for instance with his “screw” to raise water from one level to another. But his most far-reaching discovery was the “method of exhaustion,” which he used to deduce the area of a circle, the surface area and volume of a sphere and the area under a parabola. Indeed, with this method Archimedes anticipated, by nearly 2000 years, the development of calculus in the 17th century by Leibniz and Newton. For additional details, see the Wikipedia article.

In this article, we present Archimedes’ ingenious method to calculate the perimeter and area of a circle, while taking advantage of a much more facile system of notation (algebra), a much more facile system of calculation (decimal arithmetic and computer technology), and a much better-developed framework for rigorous mathematical proof. For a step-by-step presentation of Archimedes’ actual computation, see this article by Chuck Lindsey.

One motivation for presenting this material is that a surprisingly large fraction of treatments the present author has seen are either incomplete, deficient in rigor or assume some concept or technique (such as radian measure or other facts about trig functions) that presupposes properties of $\pi$. This presentation aims to avoid such missteps.

**Pi denial**

But another motivation is to respond to those writers who reject basic mathematical theory and the accepted value of $\pi$, claiming instead that they have found $\pi$ to be some other value. For example, one author asserts that $\pi = 17 – 8 \sqrt{3} = 3.1435935394\ldots$. Another author asserts that $\pi = (14 – \sqrt{2}) / 4 = 3.1464466094\ldots$. A third author promises to reveal an “exact” value of $\pi$, differing significantly from the accepted value. For other examples, see this Math Scholar blog. Of course, $\pi$ cannot possibly be given by any algebraic expression such as these, since $\pi$ was proven transcendental by Lindemann in 1882, and his proof has been checked carefully by many thousands of mathematicians since then.

Thus the material below will demonstrate, as simply and concisely as possible, why $\pi = 3.14159\ldots$ and certainly not any of these variant values. To that end, this material requires no mathematical background beyond very basic algebra, trigonometry and the Pythagorean theorem, and scrupulously avoids calculus, advanced analysis or any reasoning that depends on prior knowledge about $\pi$. Along this line, traditional degree notation is used for angles instead of radian measure customary in professional research work, both to make the presentation easier follow and also to avoid any concepts or techniques that might be viewed as dependent on $\pi$.

We start by establishing some basic identities. These proofs assume only the definitions of the trigonometric functions, namely $\sin(\alpha)$ (= opposite side / hypotenuse in a right triangle), $\cos(\alpha)$ (= adjacent side / hypotenuse) and $\tan(\alpha)$ (= opposite / adjacent), together with the Pythagorean theorem. Note, by these definitions, that $\tan(\alpha) = \sin(\alpha) / \cos(\alpha)$, and $\sin^2(\alpha) + \cos^2(\alpha) = 1$. Readers who are familiar with the following well-known identities may skip to the next section.

**LEMMA 1 (Double-angle and half-angle formulas)**: The double angle formulas are $\sin(2\alpha) = 2 \cos(\alpha) \sin(\alpha), \; \cos(2\alpha) = 1 – 2 \sin^2(\alpha)$ and $\tan(2\alpha) = 2 \tan(\alpha) / (1 – \tan^2(\alpha))$. The corresponding half-angle formulas are $$\sin(\alpha/2) = \sqrt{(1 – \cos(\alpha))/2}, \;\; \cos(\alpha/2) = \sqrt{(1 + \cos(\alpha))/2}, \;\; \tan(\alpha/2) = \frac{\sin(\alpha)}{1 + \cos(\alpha)} = \frac{\tan(\alpha)\sin(\alpha)}{\tan(\alpha) + \sin(\alpha)},$$ however note that the first two of these are valid only for $0 \le \alpha \leq 180^\circ$, because of the ambiguity of the sign when taking a square root.

**Proof**: We first establish some more general results: $$\sin (\alpha + \beta) = \sin (\alpha) \cos (\beta) + \cos (\alpha) \sin (\beta),$$ $$\cos (\alpha + \beta) = \cos (\alpha) \cos (\beta) – \sin (\alpha) \sin (\beta),$$ $$\tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 – \tan(\alpha)\tan(\beta)}.$$ The formula for $\sin(\alpha + \beta)$ has a simple geometric proof, based only on the Pythagorean formula and simple rules of right triangles, which is illustrated to the right (here $OP = 1$). First note that $RPQ = \alpha, \, PQ = \sin(\beta)$ and $OQ = \cos (\beta)$. Further, $AQ/OQ = \sin(\alpha)$, so $AQ = \sin(\alpha) \cos(\beta)$, and $PR/PQ = \cos(\alpha)$, so $PR = \cos(\alpha) \sin(\beta)$. Combining these results, $$\sin(\alpha + \beta) = PB = RB + PR = AQ + PR = \sin(\alpha) \cos(\beta) + \cos(\alpha) \sin(\beta).$$ The proof of the formula for the cosine of the sum of two angles is entirely similar, and the formula for $\tan(\alpha + \beta)$ is obtained by dividing the formula for $\sin(\alpha + \beta)$ by the formula for $\cos(\alpha + \beta)$, followed by some simple algebra. See this Wikipedia article, from which the above illustration and proof were taken, for additional details.

Setting $\alpha = \beta$ in the above formulas yields $\sin(2\alpha) = 2 \cos(\alpha) \sin(\alpha), \, \cos(2\alpha) = \cos^2(\alpha) – \sin^2(\alpha) = 1 – 2 \sin^2(\alpha)$, and $\tan(2\alpha) = \sin(\alpha)/(1 + \cos(\alpha)) = \tan(\alpha)\sin(\alpha)/(\tan(\alpha) + \sin(\alpha))$. The half-angle formulas can then easily be derived by simple algebra. For example, from $\cos(\alpha) = 1 – 2 \sin^2(\alpha/2)$ we can write $2 \sin^2(\alpha/2) = 1 – \cos(\alpha)$, from which we deduce $\sin(\alpha/2) = \sqrt{(1 – \cos(\alpha))/2}$ (however, as noted before, this formula is only valid for $0 \leq \alpha \leq 180^\circ$, because of the ambiguity in the sign when taking a square root).

**Archimedes’ algorithm for approximating Pi**

With this background, we are now able to present Archimedes’ algorithm for approximating $\pi$. Consider the case of a circle with radius one (see diagram). We see that each side of a regular inscribed hexagon has length one, and thus, of course, each half-side has length one-half. This is reflected in the formula $\sin(30^\circ) = 1/2$, a formula which in effect is proven by this diagram. Note that by applying the identity $\cos^2(\alpha) = 1 – \sin^2(\alpha)$, we obtain $\cos(30^\circ) = \sqrt{3}/2 = 0.866025\ldots$, and also that $\tan(30^\circ) = \sin(30^\circ)/\cos(30^\circ) = \sqrt{3}/3 = 0.577350\ldots$.

Let $a_1$ be the semi-perimeter of the regular circumscribed hexagon of a circle with radius one, and let $b_1$ denote the semi-perimeter of the regular inscribed hexagon. By examining the figure, we see each of the six equilateral triangles in the circumscribed hexagon has base $= 2 \tan{30^\circ} = 2 \sqrt{3}/3$. Thus $a_1 = 6 \tan(30^\circ) = 2\sqrt{3} = 3.464101\ldots$. Each of the six equilateral triangles in the inscribed hexagon has base $= 2 \sin(30^\circ) = 1$, so that $b_1 = 6 \sin(30^\circ) = 3$. In a similar fashion, let $c_1$ be the area of the regular circumscribed hexagon of a circle with radius one, and let $d_1$ denote the area of the regular inscribed hexagon. Since the altitude of each section of the circumscribed hexagon is one, $c_1 = a_1 = 2\sqrt{3} = 3.464101\ldots$. Since the altitude of each section of the inscribed hexagon is $\cos(30^\circ)$, $d_1 = 6 \sin(30^\circ) \cos(30^\circ) = 2.598076\ldots$.

Now consider a $12$-sided regular circumscribed polygon of a circle with radius one, and a $12$-sided regular inscribed polygon. Their semi-perimeters will be denoted $a_2$ and $b_2$, respectively, and their full areas will be denoted $c_2$ and $d_2$, respectively. The angles are halved, but the number of sides is doubled. Thus $a_2 = 12 \tan(15^\circ), \, b_2 = 12 \sin(15^\circ), \, c_2 = a_2 = 12 \tan(15^\circ)$ and $d_2 = 12 \sin(15^\circ) \cos(15^\circ)$, the latter of which, by applying the double angle formula for sine from Lemma 1, can be written as $d_2 = 6 \sin(30^\circ) = b_1$. Applying the half-angle formulas from Lemma 1, we obtain $a_2 = 12 (2 – \sqrt{3}) = 3.215390\ldots, \; b_2 = 3 (\sqrt{6} – \sqrt{2}) = 3.105828\ldots, \; c_2 = a_2 = 3.215390\ldots$ and $d_2 = b_1 = 3$.

In general, after $k$ steps of doubling, denote the semi-perimeters of the regular circumscribed and inscribed polygons for a circle of radius one with $3 \cdot 2^k$ sides as $a_k$ and $b_k$, respectively, and denote the full areas as $c_k$ and $d_k$, respectively. As before, because the altitudes of the triangles in the circumscribed polygons always have length one, $c_k = a_k$ for each $k$. Also, as before, after applying the double-angle identity for sine from Lemma 1, we can write $d_k = 3 \cdot 2^k \sin(60^\circ/2^k) \cos(60^\circ/2^k) = 3 \cdot 2^{k-1} \sin(60^\circ/2^{k-1}) = b_{k-1}$. In summary, let $\theta_k = 60^\circ/2^k$. Then $$a_k = 3 \cdot 2^k \tan(\theta_k), \; b_k = 3 \cdot 2^k \sin(\theta_k), \; c_k = a_k, \; d_k = b_{k-1}.$$

**THEOREM 1 (The Archimedean iteration for Pi)**: Define the sequences of real numbers $A_k, \, B_k$ by the following: $A_1 = 2 \sqrt{3}, \, B_1 = 3$. Then, for $k \ge 1$, set $$A_{k+1} = \frac{2 A_k B_k}{A_k + B_k}, \quad B_{k+1} = \sqrt{A_{k+1} B_k}.$$ Then for all $k \ge 1$, we have $A_k = a_k$ and $B_k = b_k$, as given by the formulas above.

**Proof**: $A_1 = a_1$ and $B_1 = b_1$, so the result is true for $k = 1$. By induction, assume the result is true up to some $k$. Then we can write, recalling the formula $\tan(\alpha/2) = \tan(\alpha)\sin(\alpha)/(\tan(\alpha) + \sin(\alpha))$ from Lemma 1, $$A_{k+1} = \frac{2 A_k B_k}{A_k + B_k} = \frac{2 \cdot 3 \cdot 2^k \tan(\theta_k) \cdot 3 \cdot 2^k \sin(\theta_k)}{3 \cdot 2^k \tan(\theta_k) + 3 \cdot 2^k \sin(\theta_k)} = 3 \cdot 2^{k+1} \tan(\theta_k/2) = 3 \cdot 2^{k+1} \tan(\theta_{k+1}) = a_{k+1}.$$ Similarly, recalling the identity $\sin(2\alpha) = 2 \sin(\alpha) \cos(\alpha)$ from Lemma 1, so that $\sin(\theta_k) = 2 \sin(\theta_{k+1}) \cos(\theta_{k+1})$, we can write $$B_{k+1} = \sqrt{A_{k+1} B_k} = \sqrt{9 \cdot 2^{2k+1} \tan(\theta_{k+1}) \sin(\theta_k)} = \sqrt{9 \cdot 2^{2k+2} \tan(\theta_{k+1}) \sin(\theta_{k+1}) \cos(\theta_{k+1})},$$ $$ = \sqrt{9 \cdot 2^{2k+2} \sin^2(\theta_{k+1})} = 3 \cdot 2^{k+1} \sin(\theta_{k+1}) = b_{k+1}.$$

**Computations using the Archimedean iteration**

We are now able to directly compute some approximations to $\pi$, using only the formulas of Theorem 1. These results are shown in the table to 16 digits after the decimal point, but were performed using 50-digit precision arithmetic to rule out any possibility of numerical round-off error corrupting the table results. Keep in mind that the semiperimeters and areas for circumscribed polygons are over-estimates of $\pi$, and those for the inscribed polygons are under-estimates of $\pi$.

Iteration | Sides | ||||

$k$ | $3 \cdot 2^k$ | ||||

1 | 6 | 3.4641016151377545 | 3.0000000000000000 | 3.4641016151377545 | 2.5980762113533159 |

2 | 12 | 3.2153903091734724 | 3.1058285412302491 | 3.2153903091734724 | 3.0000000000000000 |

3 | 24 | 3.1596599420975004 | 3.1326286132812381 | 3.1596599420975004 | 3.1058285412302491 |

4 | 48 | 3.1460862151314349 | 3.1393502030468672 | 3.1460862151314349 | 3.1326286132812381 |

5 | 96 | 3.1427145996453682 | 3.1410319508905096 | 3.1427145996453682 | 3.1393502030468672 |

6 | 192 | 3.1418730499798238 | 3.1414524722854620 | 3.1418730499798238 | 3.1410319508905096 |

7 | 384 | 3.1416627470568485 | 3.1415576079118576 | 3.1416627470568485 | 3.1414524722854620 |

8 | 768 | 3.1416101766046895 | 3.1415838921483184 | 3.1416101766046895 | 3.1415576079118576 |

9 | 1536 | 3.1415970343215261 | 3.1415904632280500 | 3.1415970343215261 | 3.1415838921483184 |

10 | 3072 | 3.1415937487713520 | 3.1415921059992715 | 3.1415937487713520 | 3.1415904632280500 |

11 | 6144 | 3.1415929273850970 | 3.1415925166921574 | 3.1415929273850970 | 3.1415921059992715 |

12 | 12288 | 3.1415927220386138 | 3.1415926193653839 | 3.1415927220386138 | 3.1415925166921574 |

13 | 24576 | 3.1415926707019980 | 3.1415926450336908 | 3.1415926707019980 | 3.1415926193653839 |

14 | 49152 | 3.1415926578678444 | 3.1415926514507676 | 3.1415926578678444 | 3.1415926450336908 |

15 | 98304 | 3.1415926546593060 | 3.1415926530550368 | 3.1415926546593060 | 3.1415926514507676 |

16 | 196608 | 3.1415926538571714 | 3.1415926534561041 | 3.1415926538571714 | 3.1415926530550368 |

17 | 393216 | 3.1415926536566377 | 3.1415926535563709 | 3.1415926536566377 | 3.1415926534561041 |

18 | 786432 | 3.1415926536065043 | 3.1415926535814376 | 3.1415926536065043 | 3.1415926535563709 |

19 | 1572864 | 3.1415926535939710 | 3.1415926535877043 | 3.1415926535939710 | 3.1415926535814376 |

20 | 3145728 | 3.1415926535908376 | 3.1415926535892710 | 3.1415926535908376 | 3.1415926535877043 |

As can be easily seen, each of these columns converges quickly to the well-known value of $\pi$. In the final row of the table, which presents results for circumscribed and inscribed polygons with 3,145,728 sides, all four entries agree to ten digits after the decimal point: $3.1415926535\ldots$. Note, by the way, that *both of the two variant values of $\pi$ mentioned in the Pi denial section above are excluded by iteration four*. There is no escaping these calculations — these and other variant values for $\pi$ are simply wrong.

We will now rigorously prove that the Archimedean iteration converges to $\pi$ in both the circumference and area senses, again relying only on first-principles reasoning.

**AXIOM 1 (Completeness axiom):** Every set of reals that is bounded above has a least upper bound; every set of reals that is bounded below has a greatest lower bound.

**Comment**: This fundamental axiom of real numbers merely states the property that the set of real numbers, unlike say the set of rational numbers, has no “holes.” See the Wikipedia article Completeness of the real numbers and this Chapter for details.

**THEOREM 2 (Pi as the limit of of circumscribed and inscribed polygons)**:

**Theorem 2a**: As the index $k$ increases, the limit of semi-perimeters of circumscribed and inscribed regular polygons with $3 \cdot 2^k$ sides, for a circle of radius one, is a common value, which we may define as $\pi$.

**Theorem 2b**: As the index $k$ increases, the limit of areas of circumscribed and inscribed regular polygons with $3 \cdot 2^k$ sides, for a circle of radius one, is a common value, which value is exactly equal to $\pi$ as defined in Theorem 2a.

**Proof**: Recall from above that $$a_k = 3 \cdot 2^k \tan(\theta_k), \; b_k = 3 \cdot 2^k \sin(\theta_k), \; c_k = a_k, \; d_k = b_{k-1}.$$ First note that since all $\theta_k \gt 0$, all $\cos(\theta_k) \lt 1$ or, in other words, $1 – \cos(\theta_k) \gt 0$. Then we can write $$a_{k} – a_{k+1} = 3 \cdot 2^k \tan(\theta_k) – 3 \cdot 2^{k+1} \tan(\theta_{k+1}) = 3 \cdot 2^k \left(\tan(\theta_k) – \frac{2 \sin(\theta_k)}{1 + \cos(\theta_k)}\right) = \frac{3 \cdot 2^k \tan(\theta_k) (1 – \cos(\theta_k))}{1 + \cos(\theta_k)} \gt 0, $$ $$b_{k+1} – b_k = 3 \cdot 2^{k+1} \sin(\theta_{k+1}) – 3 \cdot 2^k \sin(\theta_k) = 3 \cdot 2^{k+1} (\sin(\theta_{k+1}) – \sin(\theta_{k+1}) \cos(\theta_{k+1})) = 3 \cdot 2^{k+1} \sin(\theta_{k+1})(1 – \cos(\theta_{k+1})) \gt 0,$$ $$a_k – b_k = 3 \cdot 2^k (\tan(\theta_k) – \sin(\theta_k)) = 3 \cdot 2^k \tan(\theta_k) (1 – \cos(\theta_k)) \gt 0.$$ Thus $a_k$ is a strictly decreasing sequence, $b_k$ is a strictly increasing sequence, and each $a_k \gt b_k$. If $k \le m$, then $a_k \ge a_m \gt b_m$, so $a_k \gt b_m$. Thus all $a_k$ are strictly greater than all $b_k$. In particular, since $a_1 = 2 \sqrt{3} \lt 4$, this means that all $a_k \lt 4$ and thus all $b_k \lt 4$. Similarly, since $b_1 = 3$, all $b_k \ge 3$ and thus all $a_k \gt 3$. Also, since $\theta_1 = 30^\circ$ and all $\theta_k$ for $k \gt 1$ are smaller than $\theta_1$, this means that $\cos(\theta_k) \gt 1/2$ for all $k$. Now we can write, starting from the expression a few lines above for $a_k – b_k$, $$a_k – b_k = 3 \cdot 2^k \tan(\theta_k) (1 – \cos(\theta_k)) = \frac{3 \cdot 2^k \tan(\theta_k) \sin^2(\theta_k)}{1 + \cos(\theta_k)} \le 3 \cdot 2^k \tan(\theta_k) \sin^2(\theta_k)$$ $$= \frac{3 \cdot 2^k \sin^3(\theta_k)}{\cos(\theta_k)} \le 2 \cdot 3 \cdot 2^k \sin^3(\theta_k) = \frac{2 (3 \cdot 2^{k})^3 \sin^3(\theta_k)}{(3 \cdot 2^{k})^2} = \frac{2 b_k^3}{9 \cdot 4^k} \le \frac{128}{9 \cdot 4^k},$$ so that the difference between the circumscribed and inscribed semi-perimeters decreases by roughly a factor of four with each iteration (as is also seen in the table above).

Recall from the above that all $a_k \gt 3$, so that the sequence $(a_k)$ of circumscribed semi-perimeters is bounded below. Thus by Axiom 1 the sequence $(a_k)$ has a greatest lower bound $B_1$. Also, all $b_k \lt 4$, so that the sequence $(b_k)$ of inscribed semi-perimeters is bounded above, and thus has a least upper bound $B_2$. Since for any $\epsilon \gt 0$ and all sufficiently large $k$, $a_k – b_k \lt \epsilon$, it follows that $B_1 = B_2$ and this common limit can be defined as $\pi$.

For Theorem 2b, the difference between the circumscribed and inscribed areas is $$c_k – d_k = 3 \cdot 2^k (\tan(\theta_k) – \sin(\theta_k)\cos(\theta_k)) = 3 \cdot 2^k \left(\frac{\sin(\theta_k)}{\cos(\theta_k)} – \sin(\theta_k) \cos(\theta_k)\right) $$ $$= \frac{3 \cdot 2^k \sin(\theta_k) (1 – \cos^2(\theta_k))}{\cos(\theta_k)} = \frac{3 \cdot 2^k \sin^3(\theta_k)}{\cos(\theta_k)} \le \frac{128}{9 \cdot 4^k},$$ since the final inequality was established a few lines above. As before, it follows that the greatest lower bound of the circumscribed areas $c_k$ is exactly equal to the least upper bound of the inscribed areas $d_k$. Furthermore, since the sequence $a_k$ of semi-perimeters of the circumscribed polygons is *exactly the same* as the sequence $c_k$ of areas of the circumscribed polygons, we conclude that the common limit of the areas is identical to the common limit of the semi-perimeters, namely $\pi$. This completes the proof.

**Other formulas and algorithms for Pi**

We note in conclusion that Archimedes’ scheme is just one of many formulas and algorithms for $\pi$. See for example this collection. One such formula, for instance, is the Borwein quartic algorithm: Set $a_0 = 6 – 4\sqrt{2}$ and $y_0 = \sqrt{2} – 1$. Iterate, for $k \ge 0$, $$y_{k+1} = \frac{1 – (1 – y_k^4)^{1/4}}{1 + (1 – y_k^4)^{1/4}},$$ $$a_{k+1} = a_k (1 + y_{k+1})^4 – 2^{2k+3} (1 + y_{k+1} + y_{k+1}^2).$$ Then $1/a_k$ converges quartically to $\pi$: each iteration approximately *quadruples* the number of correct digits. Just three iterations yield 171 correct digits, which are as follows: $$3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482$$ $$534211706798214808651328230664709384460955058223172535940812848111745028410270193\ldots$$

**Other posts in the “Simple proofs” series**

The other posts in the “Simple proofs of great theorems” series are available Here.

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Introduction: The fundamental theorem of calculus, namely the fact that integration is the inverse of differentiation, is indisputably one of the most important results of all mathematics, with applications across the whole of modern science and engineering. It is not an exaggeration to say that our entire modern world hinges on the fundamental theorem of calculus. It has applications in astronomy, astrophysics, quantum theory, relativity, geology, biology, economics, just to name a few fields of science, as well as countless applications in all types of engineering — civil,

Continue reading Simple proofs: The fundamental theorem of calculus

]]>**Introduction:**

The fundamental theorem of calculus, namely the fact that integration is the inverse of differentiation, is indisputably one of the most important results of all mathematics, with applications across the whole of modern science and engineering. It is not an exaggeration to say that our entire modern world hinges on the fundamental theorem of calculus. It has applications in astronomy, astrophysics, quantum theory, relativity, geology, biology, economics, just to name a few fields of science, as well as countless applications in all types of engineering — civil, mechanical, electrical, electronic, chip manufacture, aerospace, medical and more.

The fundamental theorem of calculus was first stated and proved in rudimentary form in the 1600s by James Gregory, and, in improved form, by Isaac Barrow, while Gottfried Leibniz coined the notation and theoretical framework that we still use today.

But it was Isaac Newton who grasped the full impact of the theorem and applied it to unravel both the cosmos and the everyday world. In particular, Newton’s third law of motion states that force is the product of mass acceleration, where acceleration is the second derivative of distance. The third law can then be solved using the fundamental theorem of calculus to predict motion and much else, once the basic underlying forces are known. Ernst Mach, writing in 1901, graciously acknowledged Newton’s contributions to science in general, and his to calculus in particular, in these terms:

All that has been accomplished in mathematics since his day has been a deductive, formal, and mathematical development of mechanics on the basis of Newton’s laws.

We present here a *rigorous* and *self-contained* proof of the fundamental theorem of calculus (Parts 1 and 2), including proofs of necessary underlying lemmas such as the fact that a continuous function on a closed interval is integrable. These proofs are based only on elementary algebra and some basic completeness axioms of real numbers, and thus are suitable for anyone with a high school background in mathematics, although some familiarity with limits, inequalities, derivatives and integrals is required. We believe this exposition to be significantly more concise than most textbook treatments, and thus easier to grasp.

**Definitions: Continuous functions, derivatives and integrals.**

A function $f(t)$ defined on a closed interval $[a,b]$ is *continuous* at a point $t \in [a,b]$ if, given any $\epsilon \gt 0$, there is a $\delta \gt 0$ such that $|f(s) – f(t)| \lt \epsilon$ for all $s \in [a,b]$ with $|s – t| \lt \delta$. The function $f(t)$ is *uniformly continuous* on the closed interval $[a,b]$ if, given any $\epsilon \gt 0$, there exists a $\delta \gt 0$ (independent of $t$) such that $|f(s) – f(t)| \lt \epsilon$ for all $s, t \in [a,b]$ with $|s – t| \lt \delta$.

Given a continuous function $f(t)$ on $[a,b]$, the *derivative* $f'(t)$ is defined for $t \in (a,b)$ as the limit, if it exists,

$$f'(t) = \lim_{h \to 0} \frac{f(t + h) – f(t)}{h}.$$ Note that it follows immediately from this definition that if $f(t) = g(t) + h(t)$ for all $t \in [a,b]$, then $f'(t) = g'(t) + h'(t)$ for all $t \in (a,b)$, and if $f(t) = C$ for all $t \in [a,b]$, then $f'(t) = 0$ for all $t \in (a,b)$. These facts will be used in Theorems 1 and 2 below.

The *Riemann integral* $\int_{a}^{b}f(t)\,{\rm d}t$ is defined informally as the signed area of the region in the $xy$-plane that is bounded by the graph of $f(t)$ and the $x$-axis between $x = a$ and $x = b$. Note that the area above the $x$-axis is positive and adds to the total area, while the area below the $x$-axis is negative and subtracts from the total area.

More formally, given the closed interval $[a, b]$, define a “tagged partition” of $[a,b]$ as a pair of sequences $(x_i, 0 \leq i \leq n)$ and $(t_i, 1 \leq i \le n)$, such that

$$a = x_0 \leq t_1 \leq x_1 \leq t_2 \leq x_2 \leq \cdots \leq x_{n-1} \leq t_n \leq x_n = b.$$ Note that the tagged partition $P = \{x_i, t_i\}$ divides the interval $[a,b]$ into $n$ sub-intervals $[x_{i−1}, x_i]$, each of which includes a point $t_i$. Let $d_i = x_i – x_{i-1}$, and let $D(P) = \max_i d_i$. The *Riemann sum* of the tagged partition $P$ is defined as $R(P) = \sum _{i=1}^{n}f(t_{i}) \, d_i$. The *Riemann integral* of $f(t)$ on $[a,b]$ is then defined as $I = \int_a^b f(t) \, {\rm d}t$, provided that, given any $\epsilon \gt 0$, there is a $\delta \gt 0$ such that for any tagged partition $P$ of $[a,b]$ with $D(P) \lt \delta$, the condition $|I – R(P)| \lt \epsilon$ is satisfied. Note that it follows immediately from this definition that for any $c \in (a,b)$, we have $\int_a^b f(t) \, {\rm d}t = \int_a^c f(t) \, {\rm d}t + \int_c^b f(t) \, {\rm d}t$. This fact will be used in Theorems 1 and 2 below.

**AXIOM 1 (Completeness axioms)**:

Axiom 1a (The Heine-Borel theorem): Any collection of open intervals covering a closed set of real numbers has a finite subcover.

Axiom 1b (Intermediate value theorem): Every continuous function on a closed real interval attains each value between and including its minimum and maximum.

Axiom 1c (Least upper bound / greatest lower bound theorem): Every set of reals that is bounded above has a least upper bound; every set of reals that is bounded below has a greatest lower bound.

**Comment**: These are not really “theorems” but instead are merely equivalent axioms of the property of *completeness* for real numbers — i.e., the property that the set of real numbers, unlike say the set of rational numbers, has no “holes.” Each of these axioms can be proven from the others. See the Wikipedia article Completeness of the real numbers and this Chapter for details.

**LEMMA 1 (Continuity on a closed interval implies uniformly continuity)**: Let $f(t)$ be a continuous function on the closed interval $[a,b]$. Then $f(t)$ is uniformly continuous on $[a,b]$.

**Proof**: By the definition given above for a continuous function, for every point $t \in [a,b]$, given $\epsilon \gt 0$, there is some $\delta_t \gt 0$ such that for all $s$ in the region $|s – t| \lt \delta_t$, we have $|f(s) – f(t)| \lt \epsilon$. Then consider the collection of intervals $C(\epsilon) = \{(t – \delta_t, t + \delta_t), t \in [a,b]\}$. Clearly $C(\epsilon)$ is an open cover of $[a,b]$. Now by Axiom 1a, this collection has a finite subcover $C_n(\epsilon) = \{(t_1 – \delta_1, t_1 + \delta_2), (t_2 – \delta_2, t_2 + \delta_2), \cdots, (t_n – \delta_n, t_n + \delta_n)\}$. Let $\delta = \min_i \delta_i$. Now consider any $x, y \in [a,b]$ with $|x – y| \lt \delta$. If both $x$ and $y$ are in the same interval of the collection, we have $|f(x) – f(y)| \lt 2 \epsilon$. If they lie in adjacent intervals, then $|f(x) – f(y)| \lt 4 \epsilon$. Either way, the condition for uniform continuity is satisfied.

**LEMMA 2 (A continuous function on a closed interval is integrable)**: If the function $f(t)$ is continuous on $[a,b]$, then the integral $\int_a^b f(t) \, {\rm d}t$ exists.

**Proof**: For any tagged partition $P = \{x_i, t_i\}$ of $[a,b]$, define $u_i, v_i \in [x_{i-1}, x_i]$ as the values, guaranteed to exist by Axiom 1b above, such that $f(u_i) = \min_{x \in [x_{i-1},x_i]} f(x)$ and $f(v_i) = \max_{x \in [x_{i-1},x_i]} f(x)$. By Lemma 1, given any $\epsilon \gt 0$, there is some $\delta \gt 0$ such that whenever $D(P) \lt \delta$, (so that $|u_i – v_i| \lt \delta$), it follows that $|f(u_i) – f(v_i)| \lt \epsilon / (b – a)$. Define $U(P) = \sum_{i=1}^n f(u_i) d_i$ and $V(P) = \sum_{i=1}^n f(v_i) d_i$, and let $R(P) = \sum_{i=1}^n f(t_i) d_i$ be the Riemann sum as above. Clearly $U(P) \leq R(P) \leq V(P)$. However, $$V(P) – U(P) = \sum_{i=1}^n (f(v_i) – f(u_i)) d_i \lt \frac{\epsilon}{b-a} \sum_{i=1}^n d_i = \epsilon.$$ Since the set of $U(P)$ over all tagged partitions $P$ is bounded above by $(b-a) \max_{t \in [a,b]} f(t)$ or by any $V(P)$, by Axiom 1c there is a least upper bound $U$. Similarly, since the set of $V(P)$ over all tagged partitions $P$ is bounded below by $(b-a) \min_{t \in [a,b]} f(t)$ or by any $U(P)$, there is a greatest lower bound $V$. In summary, we have shown that given any $\epsilon \gt 0$, there is a $\delta \gt 0$ such that for any tagged partition $P$ with $D(P) \lt \delta$, we have $U(P) \leq R(P) \leq V(P)$; $U(P) \leq U \leq V(P)$; $U(P) \leq V \leq V(P)$; and $V(P) \lt U(P) + \epsilon$. It follows that $U = V = \int_a^b f(t) \, {\rm d}t$.

**LEMMA 3 (Extreme value theorem)**: If $f(t)$ is continuous on $[a,b]$ and differentiable on $(a,b)$, and if $f(a) = f(b)$, then there is some $c \in (a,b)$ such that $f'(c) = 0$.

**Proof**: By Axiom 1b, $f(t)$ has a maximum and a minimum in $[a,b]$. Since $f(a) = f(b)$, then if $f(t)$ is not constant for all $t \in [a,b]$ it must have either a minimum or a maximum at some point $c$ in the *interior* of $[a,b]$. Say $c$ is an interior maximum, but that $g'(c) \gt 0$. Recall that $$g'(c) = \lim_{h \to 0} \frac{g(c+h) – g(c)}{h}.$$ Thus given $\epsilon$ with $0 \lt \epsilon \lt g'(c)$, there is a $\delta \gt 0$ such that for all $h$ with $0 \lt h \lt \delta$ we have that $g(c+h) \gt g(c) + h g'(c) – h \epsilon \gt g(c)$, contradicting the assumption that $c$ is an interior maximum. A similar contradiction results if one assumes that $g'(c) \lt 0$, or assumes that $c$ is an interior minimum and $g'(c) \ne 0$.

**LEMMA 4 (Zero derivative implies constant)**: If $f(t)$ is continuous on $[a,b]$ and $f'(t) = 0$ for all $t \in (a,b)$, then $f(t) = C$ for some constant $C$.

**Proof**: If $f(t)$ is not constant, then by Axiom 1b it has a minimum at some point $c_1$ and a maximum at some point $c_2$. Let $d_1 = f(c_1)$ and $d_2 = f(c_2)$, and assume, for convenience, that $c_1 \lt c_2$. Now define, on the closed interval $[c_1, c_2]$, the function $g(t) = f(t) – (d_2 – d_1) (t – c_1) / (c_2 – c_1) – d_1$. Its derivative is $g'(t) = f'(t) – (d_2 – d_1) / (c_2 – c_1) = – (d_2 – d_1) / (c_2 – c_1)$, since by hypothesis $f'(t) = 0$ for all $t \in (a,b)$. Now note that $g(c_1) = g(c_2) = 0$. Thus by Lemma 3 there is some $c \in (c_1, c_2)$ such that $g'(c) = 0$. But since $g'(c) = – (d_2 – d_1) / (c_2 – c_1)$, this implies that $d_1 = d_2$ and in fact that $f(t)$ is constant on $[a,b]$.

**THEOREM 1 (Fundamental theorem of calculus, Part 1)**: Let $f(t)$ be continuous on $[a,b]$, and define the function $g(x) = \int_a^x f(t) \, {\rm d}t$. Then $g(x)$ is differentiable on $(a,b)$, and for every $x \in (a,b), \, g'(x) = f(x)$.

**Proof**: By Lemma 2, for any $x \in (a,b)$ and any $h \gt 0$ with $x + h \lt b$, we have have that the integrals $\int_a^x f(t) \, {\rm d}t$, $\int_a^{x+h} f(t) \, {\rm d}t$ and their difference, namely $\int_x^{x+h} f(t) \, {\rm d}t$, each exist. By Lemma 1, given any $\epsilon \gt 0$, there is some $\delta \gt 0$ such that for any $x \in (a,b)$, any $h > 0$ such that $(x-h, x+h) \subset (a,b)$, and any $t \in (x-h, x+h)$, the inequality $f(x) – \epsilon \le f(t) \le f(x) + \epsilon$ holds. Thus for any tagged partition $P$ of $[x,x+h]$ with $D(P) \lt \delta$, we can write $$h (f(x) – \epsilon) \le \sum_{i=1}^n f(t_i) \, d_i \le h (f(x) + \epsilon),$$ which then means that $$h (f(x) – \epsilon) \le \int_x^{x+h} f(t) \, {\rm d}t \le h (f(x) + \epsilon).$$ Now note that $\int_x^{x+h} f(t) \, {\rm d}t = g(x+h) – g(x)$. Thus $$f(x) – \epsilon \le \frac{g(x+h) – g(x)}{h} \le f(x) + \epsilon.$$ A similar argument for $\int_{x-h}^x f(t) \, {\rm d}t$ gives the desired result $g'(x) = f(x)$. We note in passing that the condition for continuity of $g(x)$ on $[a,b]$ is easily met: Given $\epsilon \gt 0$, define $\delta \lt \epsilon / (\max_{t \in [a,b]} |f(t)|)$, and then $|g(x) – g(y)| \lt \epsilon$ whenever $|x – y| \lt \delta$.

**THEOREM 2 (Fundamental theorem of calculus, Part 2)**: Let $f(t)$ be continuous on $[a,b]$, and let $G(x)$ be any continuous function on $[a,b]$ that satisfies $G'(x) = f(x)$ on $(a,b)$. Then $\int_a^b f(t) \, {\rm d}t = G(b) – G(a)$.

**Proof**: The function $g(x)$ of Part 1 is continuous on $[a,b]$ and also satisfies $g'(t) = f(x)$ for $x \in (a,b)$. Then $h(x) = G(x) – g(x)$ is continuous and has zero derivative on $(a,b)$. Thus by Lemma 4, $h(x) = C$ for some constant $C$, so that $G(x) = g(x) + C$. Now we can write $$G(b) – G(a) = (g(b) + C) – (g(a) + C) = g(b) – g(a) = \int_a^b f(t) \, {\rm d}t.$$

**Other posts in the “Simple proofs” series**

The other posts in the “Simple proofs of great theorems” series are available Here.

]]>The fact that scientific research has made immense progress over the past years, decades and centuries is taken for granted among professional scientists and most of the lay public as well. But there are others, from both the left wing and the right wing of society, who question, dismiss or even reject the notion that science progresses. One group, which is mostly rooted in the right wing of society, rejects the scientific consensus on evolution, as with the creationism and intelligent

Continue reading Is scientific progress real?

]]>The fact that scientific research has made immense progress over the past years, decades and centuries is taken for granted among professional scientists and most of the lay public as well. But there are others, from both the left wing and the right wing of society, who question, dismiss or even reject the notion that science progresses. One group, which is mostly rooted in the right wing of society, rejects the scientific consensus on evolution, as with the creationism and intelligent design movements, or the scientific consensus on global warming. The other group, namely the “postmodern science studies” movement, which is mostly rooted in the left wing of society, questions, in a very fundamental sense, whether scientific research even uncovers truth at all, much less progresses to an ever sharper view of nature.

Much of today’s postmodern science studies literature is rooted in the writings of Karl Popper and Thomas Kuhn, although the postmodern writers go much further than either Popper or Kuhn. British economist and philosopher Karl Popper was struck by the differences in approach that he perceived at the time between the writings of some popular Freudians and Marxists, who saw “verifications” of their theories in every news report and clinical visit, and the writings of Albert Einstein, who for instance acknowledged that if the predicted red shift of spectral lines due to gravitation were not observed, then his general theory of relativity would be untenable. Popper was convinced that *falsifiability* was the key distinguishing factor, a view he presented in his oft-cited book *The Logic of Scientific Discovery* [Popper1959, pg. 40-41]. His basic view is widely accepted in modern scientific thinking.

In the 1970s, Thomas Kuhn’s work *The Structure of Scientific Revolutions* analyzed numerous historical cases of scientific advancements, and then argued key paradigm shifts did not come easily [Kuhn1970]. As a trained scientist, Kuhn was able to bring significant scientific insight into his analyses of historical scientific revolutions. Unfortunately, his work includes some very dubious and immoderate analysis, such as at one point where he denies that paradigm shifts carry scientists closer to fundamental truth [Kuhn1970, pg. 170], or when he argues that paradigm shifts often occur due to non-experimental factors [Kuhn1970, pg. 135]. For one thing, Kuhn’s “paradigm shift” model has not worked as well in recent years. As a single example, the “standard model” of physics within just a few years completely displaced previous theories of particle physics, after a very orderly transition [Tipler1994, pg. 88-89].

More recent writings in the postmodern science studies field have greatly extended the scope and sharpness of these critiques, declaring that much of modern science, like literary and historical analysis, is “socially constructed,” dependent on the social environment and privileged power structures of the researchers, with no claim whatsoever to fundamental truth or progress [Koertge1998, pg. 258; Madsen1990, pg. 471; Sokal1998, pg. 5-91, 229-258]. Here are just a few of the many examples that could be cited:

- “The validity of theoretical propositions in the sciences is in no way affected by the factual evidence.” [Gergen1988, pg. 258; Sokal2008, pg. 230].
- “The natural world has a small or non-existent role in the construction of scientific knowledge.” [Collins1981; Sokal2008, pg. 230].
- “Since the settlement of a controversy is the
*cause*of Nature’s representation, not the consequence, we can never use the outcome — Nature — to explain how and why a controversy has been settled.” [Latour1987, pg. 99; Sokal2008, pg. 230]. - “For the relativist [such as ourselves] there is no sense attached to the idea that some standards or beliefs are really rational as distinct from merely locally accepted as such.” [Barnes1981, pg. 27; Sokal2008, pg. 230].
- “Science legitimates itself by linking its discoveries with power, a connection which
*determines*(not merely influences) what counts as reliable knowledge.” [Aronowitz1988, pg. 204; Sokal2008, pg. 230].

In a curious turn of events, these postmodern science writers, by undermining scientists’ claim to objective truth, have unwittingly provided arguments and talking points for the creationism, intelligent design and climate change denial movements [Otto2016a; Otto2016b].

In a recently published interview of Kuhn by *Scientific American* writer John Horgan, Kuhn was deeply upset that he has become a patron saint to this type of would-be scientific revolutionary: “I get a lot of letters saying, ‘I’ve just read your book, and it’s transformed my life. I’m trying to start a revolution. Please help me,’ and accompanied by a book-length manuscript.” Kuhn emphasized that in spite of the often iconoclastic way his writings have been interpreted, he remained “pro-science,” noting that science has produced “the greatest and most original bursts of creativity” of any human enterprise [Horgan2012].

While philosophers and postmodern writers may debate whether science fundamentally progresses, what are the facts? Even after properly acknowledging the tentative, falsifiable nature of science as taught by writers such as Popper and Kuhn, it is clear that modern science has produced a sizable body of broad-reaching theoretical structures that describe the universe and life on Earth ever more accurately with each passing year. Keep in mind that each year approximately two million new peer-reviewed scientific research papers are published worldwide [Ware2012]. This rapidly expanding corpus of scientific work is an undeniable testament to the progress of modern science.

It is easy to be blase and dismissive of this progress, but consider for a moment a few of the remarkable developments of the past 120 years:

**Relativity**. In 1905, Albert Einstein published what is now known as the special theory of relativity, which extended the classical Newtonian physics, a theory that had reigned supreme for over 250 years, to the realms of very fast moving objects and systems. Then in 1917, Einstein’s theory of general relativity further extended the theory to accelerating systems, and in the process explained gravity and a host of other phenomena in a very mathematically elegant framework. Relativity has now passed a full century of the most exacting tests, including measurements of the change over time in Mercury’s perihelion motion, measurements of the periodic changes in frequency of binary stars (which agree with theory to astonishing precision), predictions of exotica such as gravitational lenses and black holes, and, in the process, has pushed aside several other competing theories [General2019; Leach2018]. Today’s GPS technology crucially relies on both Einstein’s general relativity and special relativity, and if either of these theories were in error to any significant degree, the entire GPS system and applications that rely on it would quickly fail [Global2019].**Quantum physics**. Another 1905 paper by Einstein proposed that light shine in discrete packets, now called “quanta,” rather in continuous beams. Subsequently physicists such figures as Max Planck, Niels Bohr, Paul Dirac, Werner Heisenberg and Erwin Schrodinger developed what is now known as quantum physics, which governs phenomena at the submicroscopic realm. Quantum physics has also passed a full century of the most exacting tests imaginable, from confirmations of the perplexing predicted behavior of electrons traveling through slits to numerous measurements of fundamental chemical and nuclear properties. As a single example of thousands that could be mentioned here, the numerical value of the magnetic dipole moment of the electron (in certain units), calculated from the present-day theory of quantum electrodynamics on one hand, and calculated from best available experimental measurements on the other hand, are [Sokal1998, pg. 57]: Theoretical: 1.001159652201 (plus or minus 30 in the last two digits); Experimental: 1.001159652188 (plus or minus 4 in the last two digits). Is this agreement, to within one part in 70 billion, just a coincidence? Quantum physics is the basis for chemistry, semiconductor technology and materials science, and thus has far-reaching and absolutely indispensable applications in today’s world. It is not an exaggeration to say that quantum physics underlies virtually every electronic circuit, device and system in use today, and they would quickly fail to work if these theories were even slightly in error.**Standard model**. In the 1970s, quantum electrodynamics (QED) was extended to what is now known as quantum chromodynamics (QCD), and, together with relativity, constitute what is known as the “standard model” of modern physics. Perhaps the most dramatic confirmation of the standard model was the 2012 discovery of the Higgs boson [Overbye2012a]. In the past 30 years other attempts have been made to extend the frontiers of physics, including supersymmetry and string theory, but so far the standard model continues to reign supreme, even though researchers recognize that ultimately either relativity or quantum physics or both must give way to a more fundamental theory.**Structure of DNA**. Surely the discovery of the structure and function of DNA by Francis Crick and James Watson (with assistance from several others, notably Rosalind Franklin) must rank as one of the most significant discoveries of the 20th century, and arguably the single most significant discovery in molecular biology of all time. As the two researchers modestly observed at the conclusion of their original paper, “It has not escaped our notice that the specific pairing we have postulated immediately suggests a possible copying mechanism for the genetic material.” [Watson1953; Pray2008]. The full impact of this discovery in medicine and biology is only now being realized, with the advent of inexpensive full-genome sequencing and gene editing (see below).**Accelerating universe**. Astronomers and physicists were startled when in 1998, two different observational teams found that the expansion of the universe, long assumed to be gradually slowing due to gravitation, was actually accelerating [Wilford1998]. This finding has enormous impact on cosmological models of the universe, and has caused considerable consternation in the field, yet latest studies continue to confirm this finding [Susskind2005, pg. 22, 80-82, 154; Amit2017].**Extrasolar planets**. Following some initial discoveries by astronomers in the 1990s, astronomers have discovered thousands of planets orbiting other stars, in a development that has significant implications for the existence of life outside the Earth. As a single example of these discoveries, in 2017 astronomers found seven roughly Earth-size planets orbiting a star named Trappist-1, about 40 light-years away. At least one or two of these planets appear to have a temperature regime that would support life, although it is still much too soon to say whether or not any life actually exists there [Chang2017].**Gravitational wave astronomy**. In 2016, a team of researchers operating the new Long Interferometer Gravitational Observatory (LIGO) system announced that they had detected a brief chirp, the “sound” in the fabric of the universe of two black holes colliding, as predicted by Einstein’s general relativity. This discovery heralds the start of a new era of astronomy, one where optical and radio telescopes are combined with gravitational wave detections to explore the universe [Overbye2016].

Of course, this is just the briefest summary of highlights. For every item listed above, 1000 very significant other items could have been included. Scientific progress is very real.

Similarly, it is very easy to take for granted our current technology, which is merely the endpoint of a tidal wave of scientific and technological advances in our modern era. But consider just for a moment what has been accomplished:

**Medical technology**. Eyeglasses, from their widespread adoption in the 19th century, have restored clear vision to billions of otherwise blind or nearly blind persons. A recent analysis listed eyeglasses as the fifth most significant invention of all time, one that has “dramatically raised the collective human IQ” [Fallows2013]. Yet vision is just one detail in a huge body of medical technology, mostly developed in the 20th and 21st century, including: (a) vaccination and antibiotics, which have saved billions of persons from otherwise debilitating and deadly disease; (b) x-rays and magnetic resonance imaging; (c) surgical procedures; (c) effective painkillers and pharmaceuticals; and (d) dental procedures that have saved the teeth of billions of persons worldwide. As a result of this technology, worldwide life expectancy has soared from 29 in 1880 to 71 today [Pinker2018, Chap. 5].**Transportation**. Today’s worldwide rail network, which serves a large fraction of humanity, has grown from a few miles in England, Europe and the U.S. in 1830 to millions of miles today. Even more amazing is the growth, beginning in the late 1800s and early 1900s, in highways and automobiles, with over one billion vehicles in use today [Motor2019]. Additionally, a whopping 3.5 billion airplane trips are taken each year, and although 80% of the world’s population has never flown, each year 100 million fly for the first time [Gurdus2017]. More remarkably, very likely within 25 years or so, human passengers (not just a handful of astronauts) will travel to the Moon and Mars, an achievement that only a few years ago was the realm of fantasy [Drake2017]. Firms pursuing commercial space transport include Blue Origin (founded by Jeff Bezos), Boeing, Orbital Sciences, SpaceX (founded by Elon Musk) and Virgin Galactic (founded by Richard Branson).**Moore’s Law and computer technology**. No other single statistic is as compelling a demonstration of technological progress as Moore’s Law, namely the observation that beginning in 1965, when Intel pioneer Gordon Moore first noted it [Moore1965], the number of transistors that can be crammed onto a single integrated circuit roughly doubles every 18-24 months. Moore’s Law has now continued unabated for over 50 years, and the end is not yet in sight. As of 2019, state-of-the-art devices typically have more than 20 billion transistors, an increase by a factor of 80 million over the best 1965-era devices [Transistor2019]. This staggering number of on-chip transistors translates directly into memory capacity and processing speed, endowing a broad range of high-tech devices with capabilities unthinkable even a few years ago. For example, 2019-era supercomputers compute one million times faster, and include one million times as much memory, compared to supercomputers just 25 years ago [Top500]. With technologies such as nanotechnology and quantum computing in development, even more futuristic applications are in the works.**Communication**. Human society leaped forward in the 15th century with the printing press, which provided public access to many literary and scientific works and directly contributed to the birth of modern science. Similarly, the development of the telegraph and telephone facilitated the huge technological boom of the 19th and 20th century. Now we are seeing the effects of an even more far-reaching communication revolution, namely the Internet, which quite literally brings the entire world’s cumulative knowledge to one’s computer or smartphone. Some of the older generation may recall when telephone service was first provided to individual homes, via “party lines,” back in the 1930s, 1940s and 1950s. Long-distance calls were possible, but only at very high rates — typically 50 cents to $1.00 per minute within first-world countries, and $3.00 to $5.00 per minute to foreign countries. Nowadays, via the Internet and services such as Apple’s FaceTime and Microsoft’s Skype, one can communicate by high-resolution color video, for free, to virtually anywhere worldwide.**Smartphones**. No first-world teenager needs to be lectured about the miracle of a smartphone, which quite literally connects nearly the entire world’s population in a communications network, provides full access to Internet resources and includes a GPS mapping facility that by itself would astound anyone of an earlier era. As of 2019, over five billion persons, or roughly 70% of the entire world population, own at least a cell phone, and 2.71 billion, or nearly half the world’s population, own a smartphone [Smartphone2019]. As of 2019, state-of-the-art smartphones typically include touchscreens with over three million pixel resolution, at least two cameras with over ten million pixel resolution, and up to 512 Gbyte storage. Beginning with the 2018-2019 models, Apple smartphones also include special hardware and software that enable 3-D facial recognition. Today’s leading-edge smartphones can perform more than*five trillion*operations per second, a speed faster than that of the most powerful supercomputers of twenty years earlier. Now many of these same smartphone capabilities are being delivered in smartwatches, with built-in wireless facilities, GPS mapping and features for monitoring health and fitness. For example, beginning with the 2018-2019 models, Apple watches can produce a clinical-quality electrocardiogram and can automatically call emergency services if it detects that the wearer has fallen and was not able to get up.**Genome sequencing**. The first complete human genome sequence was completed in 2000, after a ten-year effort that cost approximately $2.7 billion [Genome2010]. But in the wake of several waves of new technology since then, genomes can now be sequenced for less than $1,000, and the cost is expected to drop to only $100 by 2020 or so [Vance2014; Fikes2017]. If the $100 price is achieved by 2020, as expected, the price will have dropped by a stunning factor of 27 million in only twenty years. This is sustained progress by a factor of 2.35 compounded per year, which is significantly faster than the Moore’s Law rate of 1.59 compounded per year. Partial genome sequencing is already wildly popular as a means to identify the national origin of one’s ancestors, and full genome sequencing is being used to identify possible genetic defects. It is inevitable that full genome sequencing will become a standard part of modern medicine in ways that we can only dimly foresee at the present time. What’s more, this same sequencing technology has enabled biologists to study the genomes of thousands of other biological species, producing indisputable evidence of common ancestry between species. In the latest application of DNA technology, researchers have discovered a technique for gene editing (“CRISPR”), a development that is certain to have far-reaching applications in medicine and almost certainly will merit a Nobel Prize [Zimmer2015c].**Artificial intelligence**. Although the notion of artificial intelligence (AI) was first articulated by 1950, early optimism soon faded as researchers realized that AI was very much more difficult than first envisioned. In 1997, in a seminal event for the field, an IBM computer system defeated the world’s champion chess player. In 2011, in a much more impressive AI achievement, an IBM computer named “Watson” defeated two champion contestants on the American quiz game Jeopardy. In 2017, a computer program developed by Google’s DeepMind defeated the world champion Go champion, an achievement that many had thought would not come for decades, if ever. Then later that year, in an even more startling development, DeepMind researchers developed a new program that was merely taught the rules of Go, and then played against itself. Within just three days it had exceeded the skill of the previous program; similar programs then quickly conquered chess and the Japanese game shogi as well. This same AI-machine learning approach is now being applied in numerous commercial developments; indeed, 2018 appears to be the year that AI truly came of age. Among the current participants are Amazon, Apple, Facebook, Google, IBM, Microsoft, Salesforce, and numerous financial firms [Bailey2018]. Closely related is the development of self-driving automobiles and trucks. Waymo has already launched a robot-ride taxi service in Arizona [Davies2018], and self-driving trucks will likely be fielded by 2022 [Freedman2017].

It is a sad commentary on our current society that a large fraction of the populace are so absorbed by day-to-day bad news that they do not recognize unmistakable evidence of longer-term progress, across a broad range of social and economic indicators. Crime is down; life expectancy is up; numerous diseases have been conquered; hundreds of millions fewer worldwide live in poverty; many fewer are dying in military conflicts or in accidents; many more worldwide live in democratic societies where basic human rights are defended, and literacy is on the rise in every nation for which reliable data are available [Pinker2011b; Pinker2018].

At the same time, and in spite of continuing naysaying from both the left wing (in particular, the postmodern science studies movement) and the right wing (in particular, the creationism, intelligent design and climate change denial movements), the engine of scientific and technological progress continues unabated. Just in the past 20 years, scientists have discovered that the universe’s expansion is accelerating, have discovered thousands of planets orbiting other stars, and have catalogued the entire human genome. The latter task cost roughly $2.7 billion when it was completed in 2000, yet dramatic improvements since then have reduced the cost to roughly $1,000, and it will soon fall to $100. Computer and information technologies continue their relentless advance with Moore’s Law. This is perhaps most evident when we see the vast numbers of cell phones and smartphones now in use — more than 5 billion, or roughly 70% of the entire world human population, now own at least a cell phone, and 2.7 billion own a smartphone. The latest smartphones pack worldwide communication, full-fledged Internet facilities, voice recognition, facial recognition and GPS mapping, and feature computing power exceeding the world’s most powerful supercomputers of just 20 years ago. And technologies such as genome sequencing, artificial intelligence, self-driving vehicles and commercial space travel are just getting started.

There is no sign that this torrid rate of progress is slowing down — in 20 years hence we will look back to our own time with just as much disdain as we do today when we recall the world of 20 years ago. So we have much to look forward to. The future is destined to be as exciting as any time in the past. It’s a great time to be alive.

]]>In 1950 Alan Turing’s landmark paper Computing machinery and intelligence outlined the principles of AI

Continue reading 2018: The year that artificial intelligence came of age

]]>The field of artificial intelligence (AI) is actually rather old. Ancient Greek, Chinese and Indian philosophers developed principles of formal reasoning several centuries before Christ. In 1651, British philosopher Thomas Hobbes wrote in *Leviathan* that “reason … is nothing but reckoning (that is, adding and subtracting).” In 1843 century Ada Lovelace, widely considered to be the first computer programmer, ventured that machines such as Babbage’s analytical engine “might compose elaborate and scientific pieces of music of any degree of complexity or extent.”

In 1950 Alan Turing’s landmark paper Computing machinery and intelligence outlined the principles of AI and proposed a test, now known as the Turing test, for establishing whether true AI had been achieved. Early computer scientists were confident that true AI system would soon be a reality. In 1965 Herbert Simon predicted that “machines will be capable, within twenty years, of doing any work a man can do.” In 1970 Marvin Minsky declared “In from three to eight years we will have a machine with the general intelligence of an average human being.”

But this early optimism collided with hard reality. For example, early attempts at producing practical machine translation systems, which were presumed to be imminent, were slammed in a 1966 report. Weizenbaum’s ELIZA program attempted to emulate a psychotherapist, but the resulting dialogue was little more than a reassembly of the user’s input, and there was little indication of how to extend this to true AI. The inevitable backlash against inflated promises and expectations during the 1970s was dubbed the “AI Winter,” a phenomenon that sadly was repeated again, in the late 1980s and early 1990s, when a second wave of AI systems also resulted in disappointment.

In retrospect, these pioneers failed to appreciate the true difficulties of constructing true AI. These include limited computer power, the combinatorial explosion of logical branches, the requirement for commonsense knowledge, and the lack of appreciation for seemingly trivial human capabilities such as visual pattern recognition and physical motion. For additional details, see the Wikipedia article on the history of AI.

A breakthrough of sorts came in the late 1990s and early 2000s with the development of machine learning, Bayes-theorem-based methods, which quickly displaced the older methods based mostly on formal reasoning. In other words, rather than trying to program an AI system as a large web of discrete logical reasoning operations, these researchers were content to use statistical machine learning schemes to automatically produce the reasoning tree. These new methods proved to be superior both in efficiency and in effectiveness.

The other major development was the inexorable rise of computing power and memory, gifts of Moore’s Law that have continued unabated for over 50 years. A typical 2018-2019-era smartphone is based on 8 nm technology, features up to 512 Gbyte flash memory and can perform trillions of operations per second. With such huge computing power and memory, greater in many respects than that of the world’s most powerful 2000-era supercomputers, previously unthinkable AI capabilities, such as 3-D facial recognition, can be provided directly to the consumer.

One highly publicized advance came in 1997, when Garry Kasparov, the reigning world chess champion, was defeated by an IBM-developed computer system named “Deep Blue.” Deep Blue employed some new techniques, but for the most part it simply applied enormous computing power to store openings, look ahead many moves, apply alpha-beta tree pruning and never make mistakes.

This was followed in 2011 with the defeat of two champion contestants on the American quiz show “Jeopardy!” by an IBM-developed computer system named “Watson.” The Watson achievement was significantly more impressive as an AI demonstration, because it involved natural language understanding, i.e. the understanding of ordinary (and often tricky) English text. For example, the “Final Jeopardy” clue at the culmination of the contest, in the category “19th century novelists,” was the following: “William Wilkinson’s ‘An Account of the Principalities of Wallachia and Moldavia’ inspired this author’s most famous novel.” Watson correctly responded “Who is Bram Stoker?” [the author of Dracula], thus sealing the victory.

Legendary Jeopardy champ Ken Jennings conceded by writing on his tablet, “I for one welcome our new computer overlords.”

The ancient Chinese game of Go involves placing black and white beads on a 19×19 grid. The game is notoriously complicated, with strategies that can only be described in vague, subjective terms. For these reasons, many observers did not expect Go-playing computer programs to beat the best human players for many years, if ever. See the earlier MathScholar blog for more details.

This pessimistic outlook changed abruptly in March 2016, when a computer program named “AlphaGo,” developed by researchers at DeepMind, a subsidiary of Alphabet (Google’s parent company), defeated Lee Se-dol, a South Korean Go master, 4-1 in a 5-game tournament. The DeepMind researchers further enhanced their program, which then in May 2017 defeated Ke Jie, a 19-year-old Chinese Go master thought to be the world’s best human Go player.

In developing the program that defeated Lee and Ke, DeepMind researchers fed their program 100,000 top amateur games and “taught” it to imitate what it observed. Then they had the program play itself and learn from the results, slowly increasing its skill.

In an even more startling development, in October 2017, Deep Mind researchers developed from scratch a new program, called AlphaGo Zero. For this program, the DeepMind researchers merely programmed the rules of Go, with a simple reward function that rewarded games won, and then instructed the program to play games against itself. This program was *not* given any records of human games, nor was it programmed with any strategies, general or specific.

Initially, the program merely scattered pieces seemingly at random across the board. But it quickly became more adept at evaluating board positions, and gradually increased in skill. Interestingly, along the way the program rediscovered many well-known elements of Go strategies used by human players, including anticipating its opponent’s probable next moves. But unshackled from the experience of humans, it developed new complex strategies never before seen in human Go games. After just three days of training and 4.9 million training games (with the program playing against itself), the AlphaGo Zero program had advanced to the point that it defeated the earlier Alpha Go program 100 games to zero.

Skill at Go (and several other games) is quantified by the Elo rating, which is based on the record of their past games. Lee’s rating is 3526, while Ke’s rating is 3661. After 40 days of training, AlphaGo Zero’s Elo rating was over 5000. Thus AlphaGo Zero was as far ahead of Ke as Ke is ahead of a good amateur player. Additional details are available in an Economist article, a Scientific American article and a Nature article.

In December 2017, DeepMind announced that they had reconfigured the AlphaGo Zero program, dubbed AlphaZero for short, to play other games, including chess and shogi, a Japanese version of chess that is significantly more complicated and challenging. Recently (December 2018) the DeepMind researchers documented their groundbreaking work in a technical paper published in *Science* (see also this excellent New York Times analysis by mathematician Steven Strogatz).

In the paper, the researchers described various experiments they have run comparing their AlphaZero program to championship-grade software programs, including Stockfish, the 2016 Top Chess Engine Championship champion (significantly more powerful than the 1997 IBM Deep Blue system), and Elmo, the 2017 Computer Shogi Association champion. In a matches against Stockfish in chess, with AlphaZero playing white, AlphaZero won 29%, drew 70.6% and lost 0.4%. In a similar match against Elmo in shogi, with AlphaZero playing white, AlphaZero won 84.2%, drew 2.2% and lost 13.6%. Other comparison results are presented in the technical paper.

Just as impressive as these statistics is the fact that AlphaZero seemed to play with a human-like style. As Strogatz explains, describing the chess program,

[AlphaZero] played like no computer ever has, intuitively and beautifully, with a romantic, attacking style. It played gambits and took risks. … Grandmasters had never seen anything like it. AlphaZero had the finesse of a virtuoso and the power of a machine. It was humankind’s first glimpse of an awesome new kind of intelligence..

AI systems are doing much more than defeating human opponents in games. Here are just a few of the current commercial developments:

- Apple’s Siri and Amazon’s Alexa smartphone-based voice recognition systems are now significantly improved over the earlier versions, and speaker systems incorporating them are rapidly becoming a household staple.
- Facial recognition has also come of age, with Facebook’s facial recognition API and, even more impressively, with Apple’s 3-D facial recognition hardware and software, which is built into the latest iPhones and iPads.
- Self-driving cars are already on the road, and 3.5 million truck driving jobs, just in the U.S., are at risk within the next ten years.
- AI-powered surgical robots may soon perform some procedures faster and more accurately than humans.
- The financial industry already relies heavily on financial machine learning methods, and a huge expansion of these technologies is coming, possibly displacing thousands of highly paid workers.
- Other occupations likely to be impacted include package delivery drivers, construction workers, legal workers, accountants, report writers and salespeople.

So where is all this heading? A recent Time article features an interview with futurist Ray Kurzweil, who predicts an era, roughly in 2045, when machine intelligence will meet, then transcend human intelligence. Such future intelligent systems will then design even more powerful technology, resulting in a dizzying advance that we can only dimly foresee at the present time. Kurzweil outlines this vision in his book The Singularity Is Near.

Futurists such as Kurzweil certainly have their skeptics and detractors. Sun Microsystem founder Bill Joy is concerned that humans could be relegated to minor players in the future, if not extinguished. Indeed, in many cases AI systems already make decisions that humans cannot readily understand or gain insight into. But even setting aside such concerns, there is considerable concern about the societal, legal, financial and ethical challenges of such technologies, as exhibited by the current backlash against technology, science and “elites” today.

One implication of all this is that education programs in engineering, finance, medicine, law and other fields will need to change dramatically to train students in the usage of emerging AI technology. And even the educational system itself will need to change, perhaps along the lines of massive open online courses (MOOC). It should also be noted that large technology firms such as Amazon, Apple, Facebook, Google and Microsoft are aggressively luring top AI talent, including university faculty, with huge salaries. But clearly the field cannot eat its seed corn in this way; some solution is needed to permit faculty to continue teaching while still participating in commercial R&D work.

But one way or the other, intelligent computers are coming. Society must find a way to accommodate this technology, and to deal respectfully with the many people whose lives will be affected. But not all is gloom and doom. Steven Strogatz envisions a mixed future:

]]>Maybe eventually our lack of insight would no longer bother us. After all, AlphaInfinity could cure all our diseases, solve all our scientific problems and make all our other intellectual trains run on time. We did pretty well without much insight for the first 300,000 years or so of our existence as Homo sapiens. And we’ll have no shortage of memory: we will recall with pride the golden era of human insight, this glorious interlude, a few thousand years long, between our uncomprehending past and our incomprehensible future.