At this point in time, the basic facts of climate change are not disputable in the least. Careful planet-wide observations by NASA and others have confirmed that 2018 was the fourth-warmest year in recorded history. The only warmer years were 2016, 2017 and 2015, respectively, and 18 of the 19 warmest years in history have occurred since 2001. Countless observational studies and supercomputer simulations have confirmed both the fact of warming and the conclusion that this warming is principally due to human activity. These studies and computations have been scrutinized in great

Continue reading The scientific debate is over: it is time to act on climate change

]]>At this point in time, the basic facts of climate change are not disputable in the least. Careful planet-wide observations by NASA and others have confirmed that 2018 was the fourth-warmest year in recorded history. The only warmer years were 2016, 2017 and 2015, respectively, and 18 of the 19 warmest years in history have occurred since 2001. Countless observational studies and supercomputer simulations have confirmed both the fact of warming and the conclusion that this warming is principally due to human activity. These studies and computations have been scrutinized in great detail by a climate science community numbering in the thousands, representing all major nations, as summarized in the latest report by the Intergovernmental Panel on Climate Change (IPCC).

Climate change skeptics continue to raise objections, claiming that there is “scientific evidence” that this mainstream consensus on climate change is wrong. But these objections have been debunked many times. Here are some examples:

*Climate change is just part of the natural cycle.***Rejoinder**: Yes, the geologic and paleontological record confirms numerous climate changes in the past. But the changes of the past 150 years, coinciding with the industrial revolution and huge increases in carbon emissions, exceeds that of the past five million years. See also this Scientific American article.*Changes are due to sunspots or galactic cosmic rays.***Rejoinder**: Scientists who have carefully monitored the sun for at least the past 20 years have seen no significant upward trend. Besides, such effects would be seen in the high atmosphere, whereas observed climate change effects are seen in the lower atmosphere. See also this Carbon Brief article.*CO2 is a small part of the atmosphere, so it can’t have a large heating effect.***Rejoinder**: The fact that CO2 results in a greenhouse warming effect has been known since 1856, and countless experimental analyses since then have confirmed heating even with only very small concentrations.*Scientists manipulate datasets to show a warming trend.***Rejoinder**: Yes, scientists adjust data, as in all experimental studies, but these adjustments are performed based on very well-understood effects, and are open to the scrutiny of peers.*Climate models are unreliable and too sensitive to carbon dioxide.***Rejoinder**Climate models are extremely sophisticated computer programs, running on some of the world’s most powerful supercomputers, and based on massive experimental datasets. These models have been produced by numerous independent international teams, which rigorously critique each other’s work, in an ongoing process of peer review. All of the latest models consistently now predict significant long-term global warming. See for example the Community Earth System Model website.

Along this line, it is worth pointing out that some previous climate skeptics have changed their mind. For example, in a remarkable New York Times Op-Ed, former climate change skeptic Richard Muller of the University of California, Berkeley, declared not only that global warming is real, but also that “humans are almost entirely the cause.”

Muller’s Berkeley Earth group approached the problem by rigorously analyzing historic temperature reports. As he described their efforts,

We carefully studied issues raised by skeptics: biases from urban heating (we duplicated our results using rural data alone), from data selection (prior groups selected fewer than 20 percent of the available temperature stations; we used virtually 100 percent), from poor station quality (we separately analyzed good stations and poor ones) and from human intervention and data adjustment (our work is completely automated and hands-off). In our papers we demonstrate that none of these potentially troublesome effects unduly biased our conclusions.

Muller noted that their record of temperatures is long enough that they could search for the fingerprint of variability in the sun’s output reaching the earth. But Muller found no such fingerprint. Global warming is real.

So how much, if any, of this warming can truly be ascribed to human activity? Muller’s Berkeley Earth group found that the record of temperatures over the past 250 years fits the increasing emissions of CO2 better than any other statistic they tried, and the magnitude of the change is entirely consistent with the known greenhouse effect of CO2.

At this point in time, at least 97% of climate science researchers agree with the central conclusion that the Earth is warming and that human activity is the primary cause. This statistic is based on multiple in-depth surveys of thousands of recently published papers in the climate science field. Further, this consensus is supported by official statements from the American Association for the Advancement of Science, the American Chemical Society, the American Geophysical Union, the American Meteorological Society, the American Physical Society, the Geological Society of America, the U.S. National Academy of Sciences and numerous other scientific societies worldwide.

In contrast, large numbers of Americans in particular continue to deny even the most basic facts. In a 2017 Pew Research Center survey, 23% denied that there is any solid evidence that the Earth has been warming, and of those who acknowledge warming, nearly half doubted that it is due to human activities.

So why are so many skeptical of the scientific consensus? According to a separate 2017 Pew Research Center survey, only 27% agreed that “almost all” scientists are in agreement; 35% said only “More than half,” and 35% said half or fewer. But even more disturbingly, only 32% agreed that the “best available scientific evidence” influences the climate scientists’ conclusions; 48% said only “some of the time”, and 18% said “not too often or never.” These results underscore a severe level of distrust of scientists in general and climate scientists in particular by the public.

According to the latest IPCC report, impacts on natural and human systems are already occurring, and even a warming of 1.5 C, which at this point can hardly be averted, will have very serious consequences, including more extreme temperature events, more instances of heavy precipitation, more severe droughts, rising sea levels damaging cities and agricultural lands, as well as enormous stress on ecosystems worldwide.

An October 2019 Scientific American article listed some of the frightening developments, just in the previous 12 months:

- In December 2018, the World Health Organization said that fossil fuel emissions are “a major contributor to health-damaging air pollution, which every year kills over seven million people.” It added that extreme weather events, which have been linked to human-caused climate change, are “a clear and present danger to health security.”
- Also in December 2018, the Global Carbon Project reported that global CO2 emissions reached an all-time high in 2018, up more than two percent after three level years. What’s more, additional increases are likely in 2019.
- In April 2019, a NASA-funded study found the mass loss of ice discharged into the ocean from Greenland glaciers had increased by a factor of six since the 1980s. Partly as a result, mean sea level has risen nearly 14 millimeters since 1972, with 7 millimeters in the past eight years. Subsequently, in July 2019, a severe Arctic heat wave resulted in 12.5 billion tons of ice melting into the ocean on a single day, the largest single-day loss on record.
- In May 2019, a United Nations biodiversity panel reported that over one million animal and plant species are threatened with extinction in the next few decades, and, further, that rates of extinction are “accelerating.”
- In September 2019, an IPCC report concluded that warming oceans, melting ice, and rising sea levels are already affecting 10 percent of the world’s population that lives in low-lying coastal areas, and that negative impacts will greatly worsen in the coming years and decades.
- In October 2019, California staggered through its third consecutive catastrophic wildfire season, with thousands of fires incinerating tens of thousands of acres, and, once again, causing billions of dollars in damages. This is in spite of the unprecedented step of pre-emptively shutting off power to large portions of the state in an effort to prevent downed power lines from generating more fires, a step that in effect reduced the world’s premier high technology leader and fifth largest economy to third-world status. The consensus of scientists is that the dry autumn weather and winds that precipitated these fires are exacerbated by climate change, with the dismal prospect for even more wildfires and power blackouts in the future.

So where do we go from here? Sixteen-year-old Swedish activist Greta Thunberg said it best in her eloquent remarks to the 2019 U.N. Climate Summit (excerpt):

]]>You have stolen my dreams and my childhood with your empty words. And yet I’m one of the lucky ones. People are suffering. People are dying. Entire ecosystems are collapsing. We are in the beginning of a mass extinction, and all you can talk about is money and fairy tales of eternal economic growth. How dare you!

For more than 30 years, the science has been crystal clear. How dare you continue to look away and come here saying that you’re doing enough, when the politics and solutions needed are still nowhere in sight.

You say you hear us and that you understand the urgency. But no matter how sad and angry I am, I do not want to believe that. Because if you really understood the situation and still kept on failing to act, then you would be evil. And that I refuse to believe.

The popular idea of cutting our emissions in half in 10 years only gives us a 50% chance of staying below 1.5 degrees [Celsius], and the risk of setting off irreversible chain reactions beyond human control.

Fifty percent may be acceptable to you. But those numbers do not include tipping points, most feedback loops, additional warming hidden by toxic air pollution or the aspects of equity and climate justice. They also rely on my generation sucking hundreds of billions of tons of your CO2 out of the air with technologies that barely exist.

So a 50% risk is simply not acceptable to us — we who have to live with the consequences.

To have a 67% chance of staying below a 1.5 degrees global temperature rise – the best odds given by the [Intergovernmental Panel on Climate Change] – the world had 420 gigatons of CO2 left to emit back on Jan. 1st, 2018. Today that figure is already down to less than 350 gigatons.

How dare you pretend that this can be solved with just ‘business as usual’ and some technical solutions? With today’s emissions levels, that remaining CO2 budget will be entirely gone within less than 8 1/2 years.

There will not be any solutions or plans presented in line with these figures here today, because these numbers are too uncomfortable. And you are still not mature enough to tell it like it is.

You are failing us. But the young people are starting to understand your betrayal. The eyes of all future generations are upon you. And if you choose to fail us, I say: We will never forgive you.

We will not let you get away with this. Right here, right now is where we draw the line. The world is waking up. And change is coming, whether you like it or not.

For at least three decades, teams of researchers have been exploring quantum computers for real-world applications in scientific research, engineering and finance. Researchers have dreamed of the day when quantum computers would first achieve “supremacy” over classical computers, in the sense that a quantum computer solving a particular problem faster than any present-day or soon-to-be-produced classical computer system.

In a Nature article dated 23 October 2019, researchers at Google announced that they have achieved exactly this.

Google researchers employed a custom-designed quantum processor, named “Sycamore,” consisting of programmable quantum

Continue reading Quantum supremacy has been achieved; or has it?

]]>For at least three decades, teams of researchers have been exploring quantum computers for real-world applications in scientific research, engineering and finance. Researchers have dreamed of the day when quantum computers would first achieve “supremacy” over classical computers, in the sense that a quantum computer solving a particular problem faster than any present-day or soon-to-be-produced classical computer system.

In a Nature article dated 23 October 2019, researchers at Google announced that they have achieved exactly this.

Google researchers employed a custom-designed quantum processor, named “Sycamore,” consisting of programmable quantum computing units, to create quantum states on 53 “qubits,” which, as they point out, corresponds to exploring a space of dimension 2^{53}, or approximately 10^{16}. The task that the researchers chose was one that admittedly was specifically chosen to be well-suited for quantum computers: the team programmed the quantum computer to run a circuit that passes the 53 qubits through a series of random operations, in effect generating a string of pseudorandom zeroes and ones. The computer then calculated the probability distribution generated by these pseudorandom trials, by sampling the circuit task one million times and recording the results.

The Google researchers’ Sycamore quantum computer system completed the task in only 3 minutes 20 seconds. They claimed that the task would require 10,000 years, even on a present-day supercomputer with 1,000,000 processors.

For additional details, see the Google researchers’ Nature paper, this Nature summary, and this New Scientist report.

The ink was barely dry on Google’s announcement when IBM released a technical paper arguing that Google’s comparison to a classical computer is flawed. The IBM researchers argue that the classical computation the Google researchers outlined (but did not actually perform) did not take full advantage of a typical present-day supercomputer’s storage system. When a storage system is strategically employed in the computation, IBM argues that the hypothetical run would complete in only 2.55 days on the “Summit” supercomputer at Oak Ridge National Laboratory, running at 87.4 Pflop/s (i.e., 87.4 quadrillion floating-point operations per second, or 8.74 x 10^{16} floating-point operations per second). IBM further argues that their computation could produce solutions with higher fidelity than the Sycamore quantum computer.

As Ciaran Gilligan-Lee of University College London observes,

Classical computers have such a large suite of things built into them that if you don’t utilise every single thing you leave yourself open for a tweaked classical algorithm to beat your quantum one.

At the least, Gilligan-Lee argues that claims of quantum supremacy “should be taken with a grain of salt.”

Another criticism of the Google team’s claim is that the particular benchmark problem they selected, namely to produce statistics from generating large numbers of pseudorandom numbers, is a highly artificial task, one that in no way reflects present-day real-world high-performance scientific computing. The pseudorandom number task is completely different from problems such as computationally simulating the earth’s climate for several decades into the future, to better understand different strategies for combatting global warming, computationally exploring the space of inorganic materials for improved solar cells or superconductors, or dynamically optimizing an investment portfolio.

One final criticism is that Google’s quantum processor is only one of many similar projects, some of which have already achieved significant success. The Canadian company D-Wave, for instance, has been producing quantum computers for several years, focusing on systems designed specifically for “quantum annealing,” namely a particular type of minimization or maximization calculation that nonetheless has broad applicability to real-world problems.

In the 1970s and 1980s, most high-end supercomputers were based on a pipelined vector architecture, where a processor performed, say, 64 identical arithmetic or logical operations on 64 sets of arguments, in a zipper-like pipelined stream. But beginning in the early 1990s, systems with a highly parallel architecture, characterized by numerous independent processor-memory nodes connected in a network, became commercially available. The high-performance computing world raced to explore these new highly parallel systems for large-scale scientific and engineering computations.

Unfortunately, however, many researchers using these systems, in their zeal to be part of a new computing movement, began to publish performance levels and comparisons with prevailing vector computers that were not carefully done. Questionable performance reports were seen in papers from universities and government laboratories as well as private industry. The present author and some others in the high-performance scientific computing field grew concerned that, in many cases, exuberance was being transformed into exaggeration and even borderline fraud, hardly becoming of rigorous, sober scientific research.

Some of the questionable practices included (see also this paper):

- Performing calculations on a small parallel systems, but reporting results scaled to a full-sized parallel system merely by multiplication (i.e., claiming runs on a system with 4096 processors, but only performing runs on a system with 1024 processors, with performance results multiplied by four).
- Reporting performance in units of operations per second, but employing inefficient algorithms that result in artificially inflated performance levels.
- Comparing highly tuned parallel computer runs with untuned programs on a vector computer system.
- Not actually performing a claimed computation on the system being compared to, but instead employing a highly questionable “rule of thumb” conversion factor.
- Employing questionable performance plots, often with extrapolated data points not clearly acknowledged in the text of the paper.
- Omitting key details in the technical paper, such as whether the computation was performed using 64-bit floating-point arithmetic (approximately 16 decimal digit accuracy) or 32-bit floating-point arithmetic (approximately 7 decimal digit accuracy, which usually runs faster but produces much less accurate final results).

At first, the present author warned the community by means of a widely circulated humorous essay; when this stratagem failed, he published a more serious technical paper that directly quoted from a number of the questionable papers (and he endured considerable sniping from some in the community as a result).

Partly as a consequence of these exaggerated performance claims, many universities and laboratories became disappointed and disillusioned with the parallel computers that they had purchased, and the parallel computing field languished. Only about ten years later, after substantially improved highly parallel computers became available, and after significantly more realistic performance benchmarks and practices were adopted in the high-performance computing community, did highly parallel computing finally gain widespread acceptance.

For additional details, see this talk by the present author.

Spanish-American philosopher George Santayana once quipped, “Those who cannot remember the past are condemned to repeat it.”

With the rise of quantum computers, as they challenge the long-running reign of highly parallel computers, we are at a juncture quite similar to the rise of parallel computer systems in the early 1990s, as they challenged the long-running reign of vector computers. In this era, as in the previous era, it is crucial that researchers, in their enthusiasm to explore the new architectures, not set aside basic principles of rigorous scientific methodology, objectivity and reproducibility.

To that end, here are some suggestions to avoid learning lessons of the past the “hard” way:

- The quantum computing field must establish clear standards for performance analyses.
- Editors of journals and referees of papers must be vigilant in enforcing these standards.
- Government funding agencies must support these standards efforts.
- Community-developed and community-endorsed benchmarks are a must.
- Benchmarks should include problems to measure specific features of a system, as well to assess performance on full-scale real-world problems typical of those that the community agrees are truly representative of computations that will actually be performed on such systems.
- Both leaders and bench-level researchers need to be frank about shortcomings of these new systems — such hard-hitting frankness is essential for the field to move forward.

The present author is as hopeful as anyone that these quantum computing systems will be successful. But let’s explore these systems with our eyes open!

]]>Introduction In a previous Math Scholar blog, we presented Archimedes’ ingenious scheme for approximating $\pi$, based on an analysis of regular circumscribed and inscribed polygons with $3 \cdot 2^k$ sides, using modern mathematical notation and techniques.

One motivation for both the previous blog and this blog is to respond to some recent writers who reject basic mathematical theory and the accepted value of $\pi$, claiming instead that they have found $\pi$ to be a different value. For example, one author asserts that $\pi = 17 – 8 \sqrt{3} =

Continue reading Pi as the limit of n-sided circumscribed and inscribed polygons

]]>**Introduction**

In a previous Math Scholar blog, we presented Archimedes’ ingenious scheme for approximating $\pi$, based on an analysis of regular circumscribed and inscribed polygons with $3 \cdot 2^k$ sides, using modern mathematical notation and techniques.

One motivation for both the previous blog and this blog is to respond to some recent writers who reject basic mathematical theory and the accepted value of $\pi$, claiming instead that they have found $\pi$ to be a different 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.

To that end, the previous blog presented a rigorous proof of Archimedes’ method, demonstrating that $\pi = 3.1415926535\ldots$ and certainly not any variant values. It employed only basic algebra, trigonometry and the Pythagorean theorem, and avoided calculus, analysis or any reasoning that depends on properties of $\pi$.

**N-sided polygons**

As mentioned above, Archimedes’ construction was based on polygons with $3 \cdot 2^k$ sides, and the presentation of this scheme in the previous blog followed this convention.

However, strictly speaking from a modern perspective, $\pi$ is most commonly defined as the circumference of a circle divided by its diameter (or as the semi-circumference of a circle of radius one), where the circumference of a circle is defined as the limit of the perimeters of circumscribed or inscribed regular polygons with $n$ sides, as $n$ increases without bound. Note that this is a somewhat stricter definition than Archimedean definition, which only deals with the special case $n = 3 \cdot 2^k$, leaving open the question of other $n$.

In the spirit of adhering to the modern convention, we present here a complete proof that $\pi$ as defined by Archimedes is the same as $\pi$ based on general $n$-sided regular polygons. As a bonus, we also present a proof that the limits of the areas of these polygons is also equal to $\pi$. These proofs are a bit more difficult than with the Archimedean case $n = 3 \cdot 2^k$ (see the previous blog), but still can be presented entirely using only basic algebra and trigonometry, with no need to resort to calculus, complex numbers or any other topic not familiar to a standard high-school mathematics curriculum. As with the previous blog, traditional degree notation is used here for angles instead of the 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 well-known trigonometric identities. Readers who are familiar with these identities may skip to the next section.

**LEMMA 1 (Sine, cosine and tangent of a sum)**:

$$\sin(\alpha + \beta) = \sin(\alpha) \cos(\beta) + \cos(\alpha) \sin(\beta),$$ $$\cos(\alpha + \beta) = \cos(\alpha) \cos(\beta) – \sin(\alpha) \sin(\beta),$$ $$\tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 – \tan(\alpha)\tan(\beta)}.$$ **Proof**: These identities were proved in the previous article, relying only on simple geometry and the Pythagorean theorem. See also this Wikipedia article.

**LEMMA 2 (Multi-angle inequalities)**:

Let $\alpha$ be any angle and $n \ge 2$ be any positive integer such that $0 \lt n \alpha \lt 45^\circ$. Then $\sin(n \alpha) \lt n \sin(\alpha)$ and $\tan(n \alpha) \gt n \tan(\alpha).$

**Proof**: These identities can be proved most easily using an induction argument: Clearly they hold for $n = 2$, since $\sin(2 \alpha) = 2 \sin(\alpha) \cos(\alpha) \lt 2 \sin\alpha$, and $\tan (2 \alpha) = 2 \tan (\alpha) / (1 – \tan^2(\alpha)) > 2 \tan(\alpha)$. Then assuming the inequalities hold for $n$, note that $$\sin((n+1) \alpha) = \sin(n \alpha) \cos(\alpha) + \cos(n \alpha) \sin(\alpha) \lt n \sin(\alpha) + \sin(\alpha) = (n+1) \sin(\alpha),$$ $$\tan((n+1) \alpha) = (\tan(n \alpha) + \tan(\alpha)) / (1 – \tan(n \alpha) \tan(\alpha)) \gt n \tan(\alpha) + \tan(\alpha) = (n + 1) \tan(\alpha),$$ so that the inequalities hold for $n+1$ also.

** LEMMA 3 (Ratios of multi-angle sines and tangents)**.

Let $\alpha$ be any angle and $n \ge 2$ be any positive integer such that $0 \lt n \alpha \lt 45^\circ$. Then $$\frac{\sin((n+1) \alpha)}{\sin(n \alpha)} \lt \frac{n+1}{n,}$$ $$\frac{\tan((n+1) \alpha)}{\tan(n \alpha)} \gt \frac{n+1}{n}.$$ **Proof**: First note that for $0 \lt n \alpha \lt 45^\circ$, $\sqrt{3}/2 \lt \cos(\alpha) \lt 1$ and $0 \lt \tan(\alpha) \lt 1$ (and also for $n \alpha$ in place of $\alpha$). Then we can write $$\frac{\sin((n+1) \alpha)}{\sin(n \alpha)} = \frac{\sin(n \alpha) \cos (\alpha) + \cos(n \alpha) \sin(\alpha)}{\sin(n \alpha)} = \cos (\alpha)\left(1 + \frac{\tan(\alpha)}{\tan(n \alpha)}\right) \lt 1 + \frac{\tan(\alpha)}{\tan(n \alpha)} \lt 1 + \frac{1}{n},$$ where we applied Lemma 2 at the final step. The second requires a bit more work: $$\frac{\tan((n+1)\alpha)}{\tan(n\alpha)} = \frac{\tan(n \alpha) + \tan(\alpha)}{\tan(n \alpha) (1 – \tan (n \alpha) \tan(\alpha))} = \frac{1 + \tan{\alpha}/\tan(n \alpha)}{1 – \tan(n \alpha) \tan(\alpha)} = \left(1 + \frac{\tan(\alpha)}{\tan(n \alpha)}\right) \cdot \left(1 + \frac{\tan(n \alpha) \tan(\alpha)}{1 – \tan(n \alpha) \tan(\alpha)}\right) $$ $$\gt \left(1 + \frac{\tan(\alpha)}{\tan(n \alpha)}\right) \cdot \left(1 + \tan(n \alpha) \tan(\alpha)\right) = 1 + \frac{\tan(\alpha)}{\tan(n \alpha)} + \tan(n \alpha) \tan(\alpha) + \tan^2(\alpha)$$ $$\gt 1 + \frac{\tan(\alpha)}{\tan(n \alpha)} \left(1 + \tan^2(n \alpha)\right) = 1 + \frac{\tan(\alpha)}{\tan(n \alpha) \cos^2(n \alpha)} = 1 + \frac{\sin(\alpha)}{\sin(n \alpha) \cos(n \alpha) \cos(\alpha)} > 1 + \frac{\sin(\alpha)}{\sin(n \alpha)} \gt 1 + \frac{1}{n},$$ where again we employed Lemma 2 at the final step.

**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.” An equivalent statement of the completeness axiom is “Every Cauchy sequence of real numbers has a limit in the real numbers.” See the Wikipedia article Completeness of the real numbers and this Chapter for details.

**THEOREM 1 (Pi as the limit of n-sided circumscribed and inscribed regular polygons)**:

**Theorem 1a**: For a circle of radius one, as the index $n \ge 6$ increases, the greatest lower bound of the semi-perimeters of circumscribed regular polygons with $n$ sides is exactly equal to the least upper bound of the semi-perimeters of inscribed regular polygons with $n$ sides, which value is exactly the same as $\pi$ as defined by Archimedes as in the previous blog.

**Theorem 1b**: For a circle of radius one, as the index $n \ge 6$ increases, the greatest lower bound of the areas of circumscribed regular polygons with $n$ sides is exactly equal to the least upper bound of the areas of inscribed regular polygons with $n$ sides, which value is exactly equal to $\pi$ as defined in Theorem 1a and in the previous blog.

**Proof strategy**: We will show that (a) the sequence of circumscribed semi-perimeters $(p_n)$ is strictly decreasing; (b) the sequence of inscribed semi-perimeters $(q_n)$ is strictly increasing; (c) all $(p_n)$ are strictly greater than all $(q_n)$; and (d) the distance between $p_n$ and $q_n$ becomes arbitrarily small for large $n$. Thus the greatest lower bound of the circumscribed semi-perimeters is equal to the least upper bound of inscribed semi-perimeters, and this common limit coincides with $\pi$ as defined by Archimedes in terms of polygons with $3 \cdot 2^k$ sides. A similar argument reaches the same conclusion for the sequence of circumscribed and inscribed areas.

**Proof**: For the entirety of this proof, we will assume $n \ge 6$. Let $p_n$ be the semi-perimeter of an $n$-sided regular circumscribed polygon for a circle with radius one, and let $q_n$ be the semi-perimeter for an $n$-sided regular inscribed polygon for a circle with radius one. By examining the figure (see also the previous blog), it is clear that $p_n = n \tan (180^\circ/n)$ and $q_n = n \sin (180^\circ/n)$. Note, for instance, that $p_6 = 6 \tan (30^\circ) = 2 \sqrt{3} = 3.46419\ldots$ and $q_6 = 6 \sin(60^\circ) = 3$. Now let $\alpha = 180^\circ/ (n(n+1))$. By applying Lemma 3, we can then write $$\frac{p_{n+1}}{p_n} = \frac{(n+1) \tan(180^\circ/(n+1))}{n \tan (180^\circ/n)} = \frac{(n+1) \tan (n \alpha)}{n \tan((n+1) \alpha)} \lt 1,$$ $$\frac{q_{n+1}}{q_n} = \frac{(n+1) \sin(180^\circ/(n+1))}{n \sin (180^\circ/n)} = \frac{(n+1) \sin(n \alpha)}{n \sin ((n+1) \alpha)} \gt 1,$$ so that $(p_n)$ is a strictly decreasing sequence and $(q_n)$ is a strictly increasing sequence. Let $\beta = 180^\circ / n$. For $n \ge 6$ note that $1/2 \lt \sqrt{3}/3 \lt \cos(\beta) \lt 1$. Then we can write $$p_n – q_n = n \tan(\beta) – n \sin(\beta) = \frac{n \sin(\beta)}{\cos(\beta)} \left(1 – \cos(\beta)\right) \gt 0,$$ so that each $p_n \gt q_n$. If $n \gt m$, then $p_n \gt q_n \gt q_m$, so $p_n \gt q_m$. Thus all $p_n$ are strictly greater than all $q_n$. In particular, since $p_6 = 2 \sqrt{3} \lt 4$, this means that all $p_n \lt 4$ and thus all $q_n \lt 4$. Similarly, since $q_6 = 3$, all $q_n \ge 3$ and thus all $p_n \gt 3$. By continuing with the last-written equality above, $$p_n – q_n = \frac{n \sin(\beta)}{\cos(\beta)} \left(1 – \cos(\beta)\right) = \frac{n \sin(\beta) (1 – \cos^2 (\beta)}{\cos(\beta)(1 + \cos(\beta))} = \frac{n \sin^3(\beta)}{\cos(\beta) (1 + \cos(\beta))} \lt 2 n \sin^3(\beta) = \frac{2 q_n^3}{n^2} \lt \frac{128}{n^2},$$ since $\cos(\beta) \gt 1/2$ and all $q_n \lt 4$. Thus the difference between $p_n$ and $q_n$ decreases to arbitrarily small values for large $n$.

Recall from the above that all $p_n \gt 3$, so that the sequence $(p_n)$ of circumscribed semi-perimeters is bounded below. Thus by Axiom 1 the sequence $(p_n)$ has a greatest lower bound $L_1$. In fact, since all $p_n$ are greater than all $q_n$, any $q_n$ is a lower bound of the entire sequence $(p_n)$, so that we may write, for any $n \ge 6$, $p_n \geq L_1 \geq q_n$. Also, all $q_n \lt 4$, so that the sequence $(q_n)$ of inscribed semi-perimeters is bounded above, and thus has a least upper bound $L_2$. And, as before, since any $p_n$ is an upper bound for the entire sequence $(q_n)$, it also follows that $p_n \geq L_2 \geq q_n$. Thus both $L_1$ and $L_2$ are “squeezed” between $p_n$ and $q_n$, which, for sufficiently large $n$, are arbitrarily close to each other (according to the last displayed equation above), so that $L_1$ must equal $L_2$. Since the sequences $(p_n)$ and $(q_n)$ include the special case $n = 3 \cdot 2^k$, and the limits for these special cases have the common value $\pi$, this means that the common limits for the full sequences $(p_n)$ and $(q_n)$ are also equal to $\pi$. This completes the proof of Theorem 1a.

For Theorem 1b, let $r_n$ and $s_n$ denote the areas of the circumscribed and inscribed regular polygons for a circle of radius one. By examining the figure above (see also the previous blog), it can be seen that $r_n = n \tan (180^\circ/n)$ and $s_n = n \sin (180^\circ/n) \cos(180^\circ/n)$. Let $\beta = 180^\circ/n$. Now note that the difference between the circumscribed and inscribed areas is $$r_n – s_n = n (\tan(\beta_n) – \sin(\beta)\cos(\beta)) = n \left(\frac{\sin(\beta}{\cos(\beta)} – \sin(\beta) \cos(\beta)\right) $$ $$= \frac{n \sin(\beta) (1 – \cos^2(\beta))}{\cos(\beta)} = \frac{n \sin^3(\beta)}{\cos(\beta)} \lt 2n \sin^3(\beta) \lt \frac{128}{n^2},$$ since the final inequality was established a few lines above. As before, it follows that the greatest lower bound of the circumscribed areas $r_n$ is exactly equal to the least upper bound of the inscribed areas $s_n$. Furthermore, since the sequence $(p_n)$ of semi-perimeters of the circumscribed polygons is *exactly the same sequence* as the sequence $(r_n)$ 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 of Theorem 1b.

**Other formulas and algorithms for Pi**

We note in conclusion that Archimedes’ scheme is just one of the many known formulas and algorithms for $\pi$. See for example this collection. One such formula 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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The question of finding rational approximations to real numbers was first explored by the Greek scholar Diophantus of Alexandra (c. 201-285 BCE), and continues to fascinate mathematicians today, in a field known as Diophantine approximations.

It is easy to see that any real number can be approximated to any desired accuracy by simply taking the sequence of approximations given by the decimal digits out to some point, divided by the appropriate power

Continue reading New paper proves 80-year-old approximation conjecture

]]>The question of finding rational approximations to real numbers was first explored by the Greek scholar Diophantus of Alexandra (c. 201-285 BCE), and continues to fascinate mathematicians today, in a field known as Diophantine approximations.

It is easy to see that any real number can be approximated to any desired accuracy by simply taking the sequence of approximations given by the decimal digits out to some point, divided by the appropriate power of ten. For example, we can approximate $\pi$ to within one part in $10^{10}$ by simply writing $$\pi \approx \frac{31415926535}{10000000000}.$$

Researchers in the field have investigated more economical representations of this sort, in the sense of approximating a positive real number to a given tolerance by a rational whose denominator is small. One commonly used technique is the continued fraction algorithm, also known as the extended Euclidean algorithm. This operates by iteratively subtracting the greatest integer from the number, then taking the reciprocal, and then repeating, appropriately collecting the results along the way. For example, the first few approximations for $\pi$ produced in this manner are: $$3, \; 22/7, \; 333/106, \; 355/113, \; 103993/33102, \; 104348/33215, \; 208341/66317, \; \cdots.$$ The last approximation listed gives $\pi$ to approximately ten-digit accuracy, the same as for the digit expansion fraction above. See also the graph, which plots the logarithm base 10 of the error as a function of the number of steps taken in the algorithm applied to $\pi$.

An important result in this arena is Dirichlet’s approximation theorem: For any real number $\alpha$ and integer $N \ge 1$, there exists a pair of integers $(p,q), 1 \leq q \leq N$, such that $|q \alpha – p| \leq 1/N$. An immediate corollary is that given any irrational $\alpha$, the inequality $$\left|\alpha – \frac{p}{q}\right| < \frac{1}{q^2}$$ is satisfied by infinitely many distinct integer pairs $(p,q)$. One might wonder whether the exponent 2 on the right-hand side can be increased, and still yield infinitely many distinct integer pairs. For irrational algebraic numbers, Roth’s theorem shows that the 2 cannot be improved: For every irrational algebraic $\alpha$ and $\epsilon > 0$, the inequality $$\left|\alpha – \frac{p}{q}\right| < \frac{1}{q^{2 + \epsilon}}$$ has only finitely many solutions in terms of relatively prime integers $(p,q)$. Conversely, if one can demonstrate, for a given irrational constant $\alpha$, that there are infinitely many distinct solution pairs $(p,q)$ satisfying the inequality with an exponent higher than 2, then, by Roth's theorem, $\alpha$ must be transcendental.

Ever since Dirichlet first published his approximation theorem in 1840, researchers have explored generalizations, such as by considering different sequences of tolerances. In particular, let $\psi(q)$ be an arbitrary function from positive integers to nonnegative real numbers. Then given $\alpha \in [0,1]$, one can ask whether there are infinitely many pairs of integers $p,q$ such that $$\left|\alpha – \frac{p}{q}\right| \le \frac{\psi(q)}{q}.$$ As it turns out, the question is particularly difficult if $\psi(q)$ is somewhat irregular — for some $\psi$ and $\alpha$ there may be no solutions whatsoever. Nonetheless, one can show that under rather general conditions on $\psi$, this inequality will have infinitely many solutions for *almost all* $\alpha \in [0,1]$, in the measure theory sense.

One key result of this sort is Khinchin’s theorem: Suppose $\psi(q)$ is a function from the positive integers to nonnegative reals with the property that $q \psi(q)$ is decreasing, and let $\lambda$ denote Lebesque measure. Let $A$ denote the set of real $\alpha \in [0,1]$ for which the inequality $$\left|\alpha – \frac{p}{q}\right| \le \frac{\psi(q)}{q}$$ has infinitely many integer solution pairs $(p,q)$ with $0 \leq p \leq q$. Then (a): $\sum_q \psi(q) < \infty$ implies $\lambda(A) = 0$, and (b): $\sum_q \psi(q) = \infty$ implies $\lambda(A) = 1$.

In 1941, Richard J. Duffin and Albert C. Schaeffer investigated Khinchin’s theorem to see if the condition $q \psi(q)$ decreasing could be relaxed. They found that it was more natural and promising to work with integer pairs $(p,q)$ that are relatively prime. Among other things, they were able to show, by applying the first Borel-Cantelli lemma, that if $\phi(q)$ is the Euler phi function, and $\psi(q)$ satisfies $$\sum_{q=1}^\infty \frac{\phi(q) \psi(q)}{q} < \infty,$$ then $\lambda(A) = 0$, where $A$ is as defined above except that the integer pair $(p,q)$ is relatively prime. This led them to conjecture, under the same assumptions, that if $$\sum_{q=1}^\infty \frac{\phi(q) \psi(q)}{q} = \infty,$$ then $\lambda(A) = 1$. This assertion (stated more precisely below) is now known as the Duffin-Schaeffer conjecture. While there have been numerous partial results, a full-fledged proof has eluded the mathematical research community for nearly 80 years.

The surprising news is that the Duffin-Schaeffer has now been proven. In particular, in a new paper, Dimitris Koukoulopoulos of the University of Montreal and James Maynard of the University of Oxford have established the following:

**Duffin-Schaeffer conjecture (now proven by Koukoulopoulos and Maynard)**: Let $\phi(q)$ denote Euler’s phi function, and suppose that $\psi(q)$ is a function from the positive integers to nonnegative reals with the property that $$\sum_{q=1}^\infty \frac{\phi(q) \psi(q)}{q} = \infty.$$ Let $\lambda$ denote Lebesque measure, and define $A$ as the set of real $\alpha \in [0,1]$ for which the inequality $$\left|\alpha – \frac{p}{q}\right| \le \frac{\psi(q)}{q}$$ has infinitely many distinct solutions with relatively prime integers $(p,q)$. Then $\lambda(A) = 1$.

Koukoulopoulos and Maynard adopted a novel approach for this problem: they recast the problem as a question about connections between points and lines in a graph. In particular, the authors created a graph out of the integer denominators, connecting the points representing denominators with a line if they share numerous prime factors. In this way, the graph’s structure encodes the overlap between irrational numbers approximated by each denominator. With this framework in place, Koukoulopoulos and Maynard were able to analyze the graph using known techniques from graph theory, which then yielded their result.

As Dimitris Koukoulopoulos explained, “The graph is a visual aid — it’s a very beautiful language in which to think about the problem.”

Reaction from the mathematical community has been swift and very complimentary. Jeffrey Vaaler of the University of Texas, Austin (who himself published some earlier results on the Duffin-Schaeffer conjecture) declared “It’s a beautiful piece of work.” He added, “They had what I’d say was a great deal of self-confidence, which was obviously justified, to go down the path they went down.”

For additional details, see this Quanta article by Kevin Hartnett, this Scientific American article by Leila Sloman, and, of course, the Koukoulopoulos-Maynard technical paper.

]]>As of the present date (August 2019), more than 4000 exoplanets have been discovered orbiting other stars, and by the time you read this even more will have been logged. Several hundred exoplanets were announced in a July 2019 paper (although these await independent confirmation). All of this is a remarkable advance, given that the first confirmed exoplanet discovery did not occur until 1992.

Most of the discoveries mentioned above are planets that are either too large or too close to their sun to possess liquid water, much

Continue reading How many habitable exoplanets are there, really?

]]>As of the present date (August 2019), more than 4000 exoplanets have been discovered orbiting other stars, and by the time you read this even more will have been logged. Several hundred exoplanets were announced in a July 2019 paper (although these await independent confirmation). All of this is a remarkable advance, given that the first confirmed exoplanet discovery did not occur until 1992.

Most of the discoveries mentioned above are planets that are either too large or too close to their sun to possess liquid water, much less complex carbon-based compounds (see this analysis), and thus there is no conceivable chance that they harbor life even vaguely analogous to that on Earth. Thus researchers have been on the lookout for exoplanets in the circumstellar habitable zone around a star, which is loosely defined as an exoplanet that has a temperature regime capable of supporting liquid water, given sufficient atmospheric pressure, based on its distance from its host star. See this Wikipedia page, which lists more than 40 such potentially habitable exoplanets.

Along this line, an August 2019 study estimated that there are between 5 billion and 10 billion exoplanets in the Milky Way that reside in the habitable zone about their respective stars.

Among other things, researchers have focused microwave antennas and other receptors at these exoplanets, on the off chance that something might be heard at one of these locations. So far, nothing…

The public is clearly excited and fascinated by such reports. After reading some of these press reports, one might think that we are on the verge of discovering Earth 2.0, complete with little green men and women (or that we already have discovered Earth 2.0, but that “elites” are hiding the fact…). But is this type of enthusiasm really warranted, either in scientific literature or in the public arena?

Unfortunately, there are many reasons to hold the champagne. To begin with, just because an exoplanet is in a “habitable zone” about its star certainly does not mean that it actually has water, much less biological organisms. Many other factors need to be considered.

For example, Harvard researcher Laura Kreidberg has noted that the recently discovered exoplanet K2-18b, which has generated considerable excitement because its atmosphere has been confirmed to contain water, has a diameter about 2.7 times the size of Earth, making it more similar to Neptune than to Earth. What’s more, the atmospheric pressure near the rocky surface of this planet is bound to be thousands of times higher than on Earth, and the resulting temperature may exceed 2800 Celsius or 5000 Fahrenheit. There is no possible way any complex carbon-based molecule such as DNA could survive under such conditions.

In fact, as a recent New Scientist article points out, most likely *none* of the current list of 4000 exoplanets is capable of hosting life. This is because life needs much more than a water-friendly temperature regime. For example, a leading scenario for the emergence of life on Earth crucially involves ultraviolet light with a certain moderate energy level to enable simple molecules to combine to form more complex compounds.

To that end, Marcos Jusino-Maldonado and Abel Méndez, of the University of Puerto Rico at Arecibo, have defined an “abiogenesis” criterion, meaning that sufficient UV light of an appropriate energy level for abiogenesis (the origin of life from nonliving molecules) would be available. When they applied their criterion to a list of 40 known exoplanets in the habitable zone, only eight of these matched their abiogenesis condition, and most of these eight are not likely to harbor life because they have a large radius (and thus are probably not rocky planets but instead are gas giants). Only the single planet named Kepler-452b, orbiting a star 1400 light years away, remained a viable candidate. Its radius is 1.63 times that of the Earth, and it marginally meets the abiogenesis and habitability criteria.

Another major problem is that most of the “habitable” planets identified so far are planets orbiting red dwarf stars. Red dwarf stars are in the fact the most abundant and longest-living stars. Some researchers have championed such stars as likely places to hunt for exoplanets harboring life.

But as an August 2019 Scientific American article points out, red dwarf stars are notorious for frequent flares with x-rays and high-energy UV radiation that almost certainly would sterilize any planet in the “habitable” zone. In other words, if an exoplanet is close enough to a red dwarf for the star’s feeble light to permit water to exist, then it is also dangerously close for lethal radiation from stellar flares. What’s more, high-energy stellar winds would very likely strip away any protective atmosphere that any such planet might possess or develop.

Bolstering this conclusion is an August 2019 study by a team of researchers led by Laura Kreidberg of Harvard and Daniel D. B. Koll of MIT. They examined the exoplanet LHS3844b using a new astronomical technique, and showed that it lacks any significant atmosphere, very likely because its host star (a red dwarf) has stripped it away. They conclude that “hot terrestrial planets orbiting small stars may not retain substantial atmosphere.”

Other studies have found even more restrictive conditions on true life-hosting exoplanets. For instance, a team of researchers led by Paul Byrne at North Carolina State University recently found that many exoplanets, even those that are not gas giants but instead have solid crusts, might well be “toffee planets,” with surface rocks that are hot enough to slowly stretch and deform like toffee candy — see this technical paper for details. Such planets most likely would not exhibit plate tectonics, as on Earth, and thus are unlikely to enjoy the benefits of plate tectonics.

Plate tectonics and the Earth’s underlying geophysical features are now thought to be crucial to life on Earth. Among other things, plate tectonics acts as a global thermostat, regulating CO2 levels in the atmosphere to yield a moderate, long-term temperature regime. In addition, one major hazard to life on Earth is streams of high-energy particles emanating from the Sun and elsewhere, which radiation is lethally hazardous to most life. But here on Earth, almost all of this cosmic radiation is deflected by Earth’s magnetic field, which is generated by the same movement of molten iron in the Earth’s core that is the dynamo behind plate tectonics [Ward2000]. This magnetic field also significantly reduces the loss of the atmosphere to outer space.

In addition to Earth being special, the Sun and Solar System are also unusual in many ways. For example, an October 2018 Scientific American article noted that in most of the recently discovered exoplanet systems, planets tend to be of the same size — if one planet, is, say, 1.5 times the radius of Earth, the other planets in the same system are likely to be of roughly this same size also. This is in stark contrast to our Solar System, which features tiny planets such as Mercury and huge planets such as Jupiter, with roughly 20 times the radius (and 8000 times the volume) of Earth. The existence of a large planet such as Jupiter is now thought to be crucial to clearing out debris from the inner planets in the Solar System’s early life, so that, as a result, Venus, Earth and Mars have been relatively undisturbed by asteroid collisions over the past 3.8 billion years or so, allowing life to form and develop, at least on Earth [Ward2000].

Additionally, our system’s position in the Milky Way is also quite favorable: at roughly 27,000 light-years from the galactic center, our Solar System strikes a good balance between being close enough to the center to have a critical concentration of heavier elements for complex chemistry, and yet not so close as to be bathed in sterilizing radiation — only about 7% of the galaxy is in a “galactic habitable zone” by these criteria [Gribbin2018]. Along this line, roughly 85% of stellar systems in the Milky Way are binary systems (with two or more stars). Exoplanets in such systems typically have very irregular orbital patterns, almost certainly destroying any hope for a stable, long-term, life-friendly temperature/radiation regime.

A 2012 study, published in the Royal Astronomical Society of Canada, after surveying numerous criteria and other studies, found that, contrary to popular opinion, the Sun is a very special star: “[I]f one picked a star at random within our galaxy, then there is a 99.99% chance that it will *not* have the same intrinsic characteristics as our Sun and (basic) Solar System.”

See this 2018 Scientific American article by John Gribbin for additional facts and discussion.

In previous blogs (see Blog A and Blog B), we discussed the nagging puzzle known as Fermi’s paradox: If the universe (or even just the Milky Way) is teeming with life, why do we not see evidence of even a single other technological civilization? After all, if such a civilization exists at all, very likely it is thousands or millions of years more advanced, and thus exploring and even communicating with habitable planets in the Milky Way would be a relatively simple and inexpensive undertaking, even for a small group of individuals.

Numerous solutions have been proposed to Fermi’s paradox, but almost all of them have devastating rejoinders. Arguments such as “ETs are under a strict global command not to disturb Earth,” or “ETs have lost interest in space research and exploration,” or “ETs are not interested in a primitive planet such as Earth,” or “ETs have moved on to more advanced communication technologies,” all collapse under the principle of diversity, a fundamental feature of evolution. In particular, it is hardly credible that in a vast, diverse ET society (and much less credible if there are numerous such societies) that not a single individual or group of individuals has ever attempted to contact Earth, using a means of communication that an emerging technological society such as ours could quickly and easily recognize. And note that once such a signal has been sent to Earth, it cannot be called back, according to known laws of physics.

Some (see this PBS show for instance) have claimed that since only 50 years or so have elapsed since radio/TV and radio telescope transmissions began on Earth, this means that only ETs within 50 light-years of Earth (if any such exist) would even know of our existence. But this is clearly groundless, because networks of lights have been visible on Earth for hundreds of years, other evidence of civilization has been visible for thousands of years, large animal species (including early hominins) have been visible for millions of years, and atmospheric signatures of life have been evident for billions of years.

Arguments that exploration and/or communication are technologically “too difficult” for an ET society immediately founder on the fact that human society is on the verge of launching such technologies today, and ET societies, as mentioned above, are almost certainly thousands or millions of years more advanced. As a single example, since we now have rapidly improving exoplanet detection and analysis facilities, as mentioned above, surely any ET society has a far superior facility that can observe Earth. Within a few decades it will be possible to launch “von Neumann probes” that land on distant planets or asteroids, construct extra copies of themselves (with the latest software beamed from the home planet), and then launch these probes to other stars, thus exploring the entire galaxy if desired [Nicholson2013]. Such probes could then beam details of their discoveries back to the home planet and, importantly, even initiate communication with promising planets. Along this line, gravitational lenses, which utilize a star’s gravitational field as an enormously magnifying telescope, could be used to view images of distant planets such as Earth and to initiate communication with these planets [Landis2016].

So why have we not seen any such probes or communications? There is no easy answer. See this Math Scholar blog for more discussion of proposed solutions and rejoinders to Fermi’s paradox.

One cogent solution to Fermi’s paradox is the following: Perhaps the reason the heavens are silent is that Earth is an extraordinarily unique home for intelligent life, according to the criteria mentioned above and perhaps even other criteria that we do not yet understand, so that the closest Earth 2.0, if it exists at all, is exceedingly distant from our Earth. If so, this means that Earth is far more singular than anyone dreamed even a few years ago, and human society has a far greater obligation not to destroy, overheat or otherwise foul our nest — our biosphere in general, and our race in particular, are of cosmic importance.

Just as significantly, we may have to rethink the Copernican principle, namely the notion that there is nothing particularly special about human society, Earth or our position in the universe, a principle that has guided scientific research for decades if not centuries. To the contrary, it is increasingly clear that the Earth *is* rather special — at the very least, there does not appear to be any equivalent to Earth, complete with an advanced technological civilization, within hundreds of light-years of Earth. If the Copernican principle is overturned, even partially, this will mark a very significant juncture in the history of science.

On the other hand, we could hear an announcement tomorrow that not only has life been detected elsewhere, but even intelligent life, with which we can communicate. That would also be an event of incalculable significance, certainly among the most important scientific discoveries of all time.

Such considerations underscore why research into exoplanets is so important and so exciting. However this turns out, we eagerly await the new experimental findings!

]]>Yet physicists have known for many years that the standard model cannot be the

Continue reading How fast is the universe expanding? New results deepen the controversy

]]>Yet physicists have known for many years that the standard model cannot be the final answer. Most notably, quantum theory on one hand and general relativity on the other are known to be mathematically incompatible. This has led to research in string theory and loop quantum gravity as potential frameworks to resolve this incompatibility. Other difficulties may exist as well.

So how can physics advance beyond the standard model? There is only so far that mathematical theories can be taken in the absence of solid experimental results. As Sabine Hossenfelder has emphasized, beautiful mathematics published in a vacuum of experimental data can actually lead physics astray.

In a previous Math Scholar article, we described several anomalies that have arisen in recent physics experiments, any of which may potentially be a spark that leads to new physics beyond the standard model. Here are three:

*The proton radius anomaly*: This stems from the fact that careful measurements of the radius of a proton’s radius when orbited by an electron yield a radius of approximately 0.877 femtometers (i.e., 0.877 x 10^{-15}meters), whereas separate measurements of the proton’s radius when it is coupled with a muon (“muonic hydrogen”) yield a radius of 0.84 femtometers. These measurements differ by significantly more than the error bars of the two sets of experiments. See this Quanta article for details.*The neutron lifetime anomaly*: This stems from the fact that “bottle” measurements of a neutron’s average lifetime yield 879.3 seconds, whereas “beam” measurements yield 888 seconds. The error bar of the bottle measurements is just 0.75 seconds, and that of the beam measurements is just 2.1 seconds, so again the two measurements appear to be further apart than can reasonably be explained as statistical error. See this Quanta article for additional details.*The Hubble constant anomaly*: The Hubble constant H_{0}is a measure of the rate of expansion of the universe. One method to determine H_{0}is based on the flat Lambda cold dark matter (Lambda-CDM) model of the universe, combined with careful measurements of the cosmic microwave background (CMB) data from the Planck satellite. The latest (2018) result from the Planck team yielded H_{0}= 67.4, plus or minus 0.5 (the units are kilometers per second per megaparsec). Another approach is to employ a more traditional astronomical technique, based on observations of Cepheid variable stars, combined with parallax measurements as a calibration. In 2016, a team of astronomers using the Wide Field Camera 3 (WFC3) of the Hubble Space Telescope obtained the value H_{0}= 73.24, plus or minus 1.74. Again, these two values differ by significantly more than the combined error bars of the two measurements.

For each of these anomalies, experimental teams on both sides have been attempting to reduce error bars and to explore the fundamental theory to see if there is any heretofore ignored possibility of error.

In the past few months (as of August 2019), several new experimental studies have been published on the Hubble constant. In March 2019, a research team working with the Hubble Space Telescope reported that based on observations of 70 long-period Cepheid variable stars in the Large Magellanic Cloud, they were able to refine their estimate to H_{0} = 74.03, plus or minus 1.42. Needless to say, this new result does not help to resolve the discrepancy — it moves in the other direction.

In July 2019, a group reported results from another experimental approach, known as the “Tip of the Red Giant Branch” (TRGB). Their approach, which is analogous to but independent from the approach taken with Cepheid variable stars, is to analyze a surge in helium burning near the end of a red giant star’s lifetime. Using this scheme, they reported H_{0} = 69.8, plus or minus 1.7. This is slightly more than the Planck team value (67.8), but not nearly enough to close the gap with the Cepheid approach.

A third group also announced results in July 2019. This project, called H_{0} Lenses in COSMOGRAIL’s Wellspring (HoLiCOW) [yes, that is the acronym], employs gravitational lensing, namely the phenomenon predicted by general relativity that light bends as it passes near an intervening star or galaxy (see graphic above). The specific approach of the HoLiCOW project is to measure light from a very distant quasar, which is lensed by a closer galaxy. When this happens, multiple time-delayed images of the galaxy appear at the edges of the intervening galaxy, when viewed by earth-bound astronomers. The HoLiCOW project’s latest result is H_{0} = 73.3, plus or minus 1.76.

Needless to say, researchers are perplexed by the latest reports: the Planck team (based on the Lambda-CDM model) reports H_{0} = 67.4 (plus or minus 0.5); the TRGB team reports 69.8 (plus or minus 1.7); the HoLiCOW team reports 73.3 (plus or minus 1.76); and the Cepheid team reports 74.03 (plus or minus 1.42). Obviously these results cannot all simultaneously be correct. For example, the HoLiCOW team’s figure (73.3) represents a 5.3 sigma discrepancy from the Planck figure (67.4). While each of these teams is hard at work scrutinizing their methods and refining their results, there is an unsettling possibility that one or more of the underlying physical theories are just plain wrong, at least on the length and time scales involved.

Key among these theories is the Lambda-CDM model of big bang cosmology. Yet physicists and cosmologists are loath to discard this model, because it explains so much so well:

- The cosmic microwave background radiation and its properties.
- The large-scale structure and distribution of galaxies.
- The present-day observed abundances of the light elements (hydrogen, deuterium, helium and lithium).
- The accelerating expansion of the universe, as observed in measurements of distant galaxies and supernovas.

As Lloyd Knox, a cosmologist at the University of California, Davis, explains,

The Lambda-CDM model has been amazingly successful. … If there’s a major overhaul of the model, it’s hard to see how it wouldn’t look like a conspiracy. Somehow this ‘wrong’ model got it all right.

Various modifications to the Lambda-CDM model have been proposed, but while some of these changes partially alleviate the Hubble constant discrepancy, others make it worse. None are taken very seriously in the community at the present time.

For additional details and discussion, see this Scientific American article (although note that the Scientific American article’s report on the HoLiCOW measurement does not agree with the HoLiCOW team’s latest technical paper) and this Quanta article.

In spite of the temptation to jump to conclusions, throwing out the standard model or big bang cosmology, considerable caution is in order. After all, as mentioned above, in most cases anomalies are eventually resolved, usually as some defect of the experimental process or as a faulty application of the theory.

A good example of an experimental defect is the 2011 announcement by Italian scientists that neutrinos emitted at CERN (near Geneva, Switzerland) had arrived at the Gran Sasso Lab (in the Italian Alps) 60 nanoseconds sooner than if they had traveled at the speed of light. If upheld, this finding would have constituted a violation of Einstein’s theory of relativity. As it turns out, the experimental team subsequently discovered that the discrepancy was due to a loose fiber optic cable that had introduced an error in the timing system.

A good example of misapplication of underlying theory is the solar neutrino anomaly, namely a discrepancy in the number of observed neutrinos emanating from the interior of the sun from what had been predicted (incorrectly, as it turned out) based on the standard model. In 1998, researchers discovered that the anomaly could be resolved if neutrinos have a very small but nonzero mass; then, by straightforward application of standard model, the flavor of neutrinos could change enroute from the sun to the earth, thus resolving the discrepancy. Takaaki Kajita and Arthur McDonald received the 2015 Nobel Prize in physics for this discovery.

In any event, sooner or later some experimental result may be found that fundamentally upsets currently accepted theoretical theories, either for a specific framework such as Lambda-CDM big bang cosmology, or even for the foundational standard model. Are any of the above-mentioned anomalies of this earth-shaking character? Only time will tell.

]]>In an advance that may presage a dramatic new era of pharmaceuticals and medicine, DeepMind (a subsidiary of Alphabet, Google’s parent company) recently applied their machine learning software to the challenging problem of protein folding, with remarkable success. In the wake of this success, DeepMind and other private companies are racing to further extend these capabilities and apply them to real-world biology and medicine.

The protein folding problemProtein folding is the name for the physical process in which a protein chain, defined by a linear sequence of amino

Continue reading Protein folding via machine learning may spawn medical advances

]]>In an advance that may presage a dramatic new era of pharmaceuticals and medicine, DeepMind (a subsidiary of Alphabet, Google’s parent company) recently applied their machine learning software to the challenging problem of protein folding, with remarkable success. In the wake of this success, DeepMind and other private companies are racing to further extend these capabilities and apply them to real-world biology and medicine.

Protein folding is the name for the physical process in which a protein chain, defined by a linear sequence of amino acids, assumes its equilibrium 3-dimensional structure, a process that in nature typically occurs within a few milliseconds. The equilibrium or “native” structure determines most of the protein’s biological properties. Protein enzymes, for instance, control chemical reactions at the molecular scale. Accurately predicting protein structures is a “holy grail” of modern biology.

As a single example, the protein Cas9 is used to snip DNA in CRISPR gene editing, but the potential of the CRISPR technique is limited by the fact that using Cas9 entails an increased risk of mutations. A better understanding of the structure of Cas9 is thus essential to further advances in gene editing technology.

There are at least 20,000 different proteins in human biology, since there are that many genes in the human genome, but the actual tally is much higher, possibly as many as several billion.

Computationally simulating the protein folding process, and determining the final equilibrium conformation, is a difficult and challenging problem — indeed it has long been listed as one of the grand challenges of the high-performance computing field. Since exhaustively trying all possible conformations for a protein given by an n-long amino acid chain is well-known to be computationally infeasible (Levinthal’s paradox), researchers have broken the folding process into numerous steps, each of which has spawned numerous algorithmic approaches for computation. One recent survey of the computational problem and underlying physics is given in this paper.

The challenge of protein folding has led some to develop new computer architectures specifically devoted to this task. Notable among these efforts is the “Anton” system, which was designed and constructed by a team of researchers led by David E. Shaw, founder of the D.E. Shaw hedge fund. A technical paper they wrote describing the Anton system and one of their landmark calculations was named a winner of the 2009 ACM Gordon Bell Prize.

As mentioned above, DeepMind is a research subsidiary of Alphabet (Google’s parent company) devoted to machine learning (ML) and artificial intelligence (AI). In 2016, the “AlphaGo” program developed by DeepMind defeated Lee Se-dol, a Korean Go master, by winning four games of a five-game tournament, surprising observers who had predicted that this would not be done for decades, if ever. A year later, DeepMind’s improved AlphaGo program defeated Ke Jie, a 19-year-old Chinese player thought to be the world’s best.

Then DeepMind tried a new approach — rather than feeding their program over 100,000 published games by human competitors, they merely programmed the system with the rules of Go and a relatively simple reward function, and had it play games against itself. After just three days and 4.9 million training games, the new “AlphaGo Zero” program had advanced to the point that it defeated the earlier program 100 games to zero. After 40 days of training, its measured skill level was as far ahead of Ke Jie as Ke Jie is ahead of a typical amateur. For additional details, see this Math Scholar blog, this Scientific American article, and this DeepMind article.

Other DeepMind programs, using a similar machine learning approach, have conquered Chess and the Japanese game shogi. For details, see DeepMind’s technical paper and this nicely written New York Times article. This and some other recent developments in the ML/AI arena are summarized in this Math Scholar blog.

DeepMind’s latest conquest is protein folding: In their first attempt, DeepMind’s team easily took top honors at the 13th Critical Assessment of Structure Prediction (CASP), an international competition of protein structure computer programs. For protein sequences for which no other information was known (43 of the 90 test problems), DeepMind’s AlphaFold program made the most accurate prediction among the 98 competitors in 25 cases. This was far better than the second-place entrant, which won only three of the 43 test cases. On average, AlphaFold was 15% more accurate than its closest competitors on the most rigorous tests.

DeepMind’s team developed AlphaFold by training a neural network program on a large dataset of other known proteins, thus enabling the program to more efficiently predict the distances between pairs of amino acids and the angles between the chemical bonds connecting them. After this they employed a more classical “gradient descent” approach to minimize the overall energy level. In other words, their approach combined sophisticated deep learning models with brute force computational resources. A nice summary of these techniques is given in this Exxact blog (see diagram to right) and in this DeepMind article.Most researchers in the field were very impressed. One researcher described these results as absolutely stunning. The Guardian predicted that these results would “usher in a new era of medical progress.”

Large pharmaceutical firms have not traditionally paid much attention to computational approaches, preferring more conventional experimental methods. But the costs of such laboratory work are rising rapidly — by one estimate, large pharma firms spend roughly $2.5 billion bringing a new drug to market, a cost figure that ultimately must be paid for by consumers and their medical insurance companies in the form of sky-high prices for prescription drugs. One major reason for these escalating costs is the embarrassing fact that only about 10% of drugs that enter clinical trails are eventually approved by governmental regulatory agencies. Clearly the pharmaceutical companies must increase this success rate.

And the challenge looming ahead is even more daunting — the 20,000 genes of the human genome can malfunction in at least 100,000 ways, and the total number of interactions between human biology proteins is in the millions, if not higher. As Chris Gibson, founder of Recursion Pharmaceuticals, explains, “If we want to understand the other 97 percent of human biology, we will have to acknowledge it is too complex for humans.”

In the wake of DeepMind’s achievement (December 2018), numerous commercial enterprises are pursuing computational protein folding using ML/AI-based strategies. Venture capital operations are certainly taking notice, having poured more than $1 billion into ML/AI-based startups in the pharmaceutical field during the past year. In addition to Recursion, mentioned above, some other new firms include Insitro, which was recently acquired by Gilead Sciences; Benevolent AI, which has teamed up with AstraZeneca; and a University of California-based team that has partnered with GlaxoSmithKline. Along this line, Juan Alvarez, an associate Vice President for computational chemistry at Merck, says that ML-based methods will be “critical” to the drug discovery and development process in the coming years. See this Bloomberg article for additional details.

So it appears that machine learning and artificial intelligence-based technology is destined to have a major impact in the pharmaceutical-biomedical world in the coming years. The reasons are not hard to find. As GlaxoSmithKline senior Vice President Tony Wood explains, “Where else would you accept a 1-in-10 success rate? … If we could double that to 20% it would be phenomenal.”

]]>In his new book Homo Deus, Israeli scholar Yuval Noah Harari has published one of the most thoughtful and far-reaching analyses of humanity’s present and future. Building on his earlier Sapiens, Harari argues that although humanity has made enormous progress across in the past few centuries, the future of our society, and even of our species, is uncertain.

Harari begins with a reprise of human history, from prehistoric times to the present. He then observes that although religious beliefs are much more nuanced and sophisticated than in the past, human society still relies heavily on the narratives

Continue reading Homo Deus: A brief history of tomorrow

]]>In his new book Homo Deus, Israeli scholar Yuval Noah Harari has published one of the most thoughtful and far-reaching analyses of humanity’s present and future. Building on his earlier Sapiens, Harari argues that although humanity has made enormous progress across in the past few centuries, the future of our society, and even of our species, is uncertain.

Harari begins with a reprise of human history, from prehistoric times to the present. He then observes that although religious beliefs are much more nuanced and sophisticated than in the past, human society still relies heavily on the narratives they teach. Moral laws are widely believed to have been revealed from on high — whether in the Torah (Jews), in the New Testament (Christians), or in the Vedas (Hindus).

Are we now beyond these philosophical systems? Harari points out that modern “religions” also have their set of “divine” laws. Communism, for instance, was thought by Karl Marx and other early proponents to be a “scientific” theory, based on natural laws. Even modern liberalism, dating from the Enlightenment onward, along with present-day secular humanism, are also based on a religious creed, namely the fundamental belief that each human is endowed by natural law with free agency and unalienable rights. As Jefferson wrote in the U.S. Declaration of Independence,

We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.

But just as modern science, biblical scholarship and philosophy have undercut some of the fundamental tenets of various religious movements, in a similar way modern science and technology are now drawing into question some of the fundamental tenets of liberalism and secular humanism. Harari argues that future technology, in particular artificial intelligence, is certain to be even more disruptive to these philosophical systems, and may draw into question even the definition of “human.” Indeed, humanity almost certainly will be transformed by this technology, and humans in our present-day sense may no longer exist 100 years from now.

Here are some of his observations:

*Liberalism’s very success may contain the seeds of its ruin.*Scientists and technologists, pushed on by the supposedly infallible wishes of consumers, are devoting more and more energies to the three central projects of liberalism: (a) human life preservation and extension, (b) universal bliss — the “pursuit of happiness”, and (c) the approach to superhuman levels of knowledge and power.*Attempting to realize these dreams will almost certainly unleash new post-humanist technologies that may spell the end of humanity as we know it.*Artificial intelligence is already an unstoppable force, and is certain to change every aspect of human life. As we offload more and more of the operation of the world to intelligent systems, and as we enhance ourselves with a variety of dimly envisioned new technologies, what will “human” mean?*Modern science and technology are undermining both the notion of a single, human “self” as well as the notion of of free agency.*One conclusion of modern psychology is that human consciousness is not a single entity, but, in Marvin Minsky’s memorable words, a society of mind. Similar studies make it clear that one’s mind makes a decision between a few milliseconds to a second or more*before*consciously announcing the decision. What’s more, since the brain is fundamentally a computer, operating on inputs possibly together with atomic-level quantum events, where is “free will”? Finally, given current and future technology, should not the human “self” include the array of intelligent assistants that the person utilizes?

What strikes this reviewer is that, if anything, Harari’s roadmap of human society steadily adopting intelligent systems and robots, and careening to a posthuman state, appears to be happening even faster than he anticipated (the book was first released in the U.S. in 2017). Consider:

- Many have already offloaded numerous tasks to their smartphones and personal computers, certainly including: (a) keeping a calendar of appointments and events, (b) managing messages and email, (c) managing personal and family finances, (d) managing physical activity and health, (e) keeping track of spouses and children, and even (f) finding potential romantic partners. Why do we willingly surrender so many highly personal details to intelligent assistants? Because the intelligent assistants are better at these tasks!
- Artificially intelligent, machine-learning-based schemes have already taken over much of market trading. For example, most conventional actively managed mutual funds do not exceed the returns of a simple buy-and-hold index fund strategy. Instead, the state-of-the-art action is with highly mathematical, big-data-based hedge funds.
- Over 200 million smartwatches have been sold (as of June 2019), with sophisticated features that track highly personal details, such as heart rates, the length and type of daily exercise, and how frequently one takes a break to stand and stretch. The latest Apple smartwatch, for instance, includes GPS mapping, wireless communication, heart rate monitoring, an electrocardiogram generator, and can automatically alert emergency services if it detects that the wearer has fallen and was not able to get up. Future editions are certain to greatly extend the list of available features.
- Social media services such as Facebook and Twitter are collecting enormous amounts of data on users’ personal characteristics and preferences. Indeed, ensuring privacy, avoiding the misuse of this data, and stopping the proliferation of fake news and doctored images has emerged as a major technical challenge and political issue confronting the technology industry.
- Artificial intelligence applications are advancing at a torrid pace. Once unthinkable AI advances, such as self-teaching Go-playing computer programs and self-driving autos and trucks, have already been fielded. In May 2019, researchers at Google and several medical centers announced that an AI-based CT scanner detected and identified lung cancer on a par with the best human specialists. Such developments are certain to come at a breakneck pace in the coming years and decades.
- The wave of technology-based unemployment that has beset blue-collar and clerical industries is certain to take aim at higher-skill occupations. The financial industry, the medical care industry and even creative industries such as music composition are certain to be revolutionized. Harari has echoed warnings of others that a major societal challenge will be how to handle tens of millions of persons whose skills are no longer economically valued, and who are not really trainable for new emerging high-tech vocations. Even among other workers, hundreds of millions will need to re-invent themselves, perhaps more than once, over the space of their careers.
- Several large organizations are actively pursuing research into longevity and potential anti-aging drugs. Google, for instance, has allocated hundreds of millions of dollars to fund an anti-aging spinout named Calico. Amazon CEO Jeff Bezos has joined other investors to fund Unity, a San Francisco startup developing drugs aiming at ridding the body of “senescent” cells. Along this line, researchers recently found that a “cocktail” of three drugs appeared to rejuvenate the human body, stopping or turning back certain biological clocks.

The final chapter of Harari’s book addresses what he calls the “data religion” — the observation that the collection, dissemination and analysis of data is emerging as more than just a technology development, but in fact the central governing principle of human (and posthuman) society, as we inevitably approach a singularity, to use a term coined by Vernor Vinge, Ray Kurzweil and others. As Harari summarizes this “religion,”

Dataists believe that humans can no longer cope with the immense flows of data, hence they cannot distil data into information, let alone into knowledge or wisdom. The work of processing data should therefore be entrusted to electronic algorithms, whose capacity far exceeds that of the human brain.

In this regard, Harari echoes predictions made by numerous other scientists and scholars. For example, mathematician Steven Strogatz, in a recent New York Times article, writes,

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.

Harari ends with three simple but deeply significant questions:

- Are organisms really just algorithms, and is life really just data processing?
- What’s more valuable — intelligence or consciousness?
- What will happen to society, politics and daily life when non-conscious but highly intelligent algorithms know us better than we know ourselves?

These are good questions. But if anything, it seems to this reviewer that Harari is soft-pedaling the changes that are to come, which are certain to impact every aspect of modern society, from its fundamental philosophical underpinnings and governmental systems to highly personal details of day-to-day living. Hold on to your hats!

]]>Four mathematicians, Michael Griffin of Brigham Young University, Ken Ono of Emory University (now at University of Virginia), Larry Rolen of Vanderbilt University and Don Zagier of the Max Planck Institute, have proven a significant result that is thought to be on the roadmap to a proof of the most celebrated of unsolved mathematical conjecture, namely the Riemann hypothesis. First, here is some background:

The Riemann hypothesisThe Riemann hypothesis was first posed by the German

Continue reading Mathematicians prove result tied to the Riemann hypothesis

]]>Four mathematicians, Michael Griffin of Brigham Young University, Ken Ono of Emory University (now at University of Virginia), Larry Rolen of Vanderbilt University and Don Zagier of the Max Planck Institute, have proven a significant result that is thought to be on the roadmap to a proof of the most celebrated of unsolved mathematical conjecture, namely the Riemann hypothesis. First, here is some background:

The Riemann hypothesis was first posed by the German mathematician Georg Friedrich Bernhard Riemann in 1859, in a paper where he observed that questions regarding the distribution of prime numbers were closely tied to a conjecture regarding the behavior of the “zeta function,” namely the beguilingly simple expression $$\zeta(s) \; = \; \sum_{n=1}^\infty \frac{1}{n^s} \; = \; \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots$$ Leonhard Euler had previously considered this series in the special case $s = 2$, in what was known as the Basel problem, namely to find an analytic expression for the sum $$\sum_{n=1}^\infty \frac{1}{n^2} \; = \; \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} \cdots \; = \; 1.6449340668482264365\ldots$$ Euler discovered, and then proved, that in fact this sum, which is $\zeta(2)$, is none other than $\pi^2/6$. Similarly, $\zeta(4) = \pi^4/90$, $\zeta(6) = \pi^6/945$, and similar results for all positive even integer arguments. Euler subsequently proved that $$\zeta(s) \; = \; \prod_{p \; {\rm prime}} \frac{1}{1 – p^{-s}} \; = \; \frac{1}{1 – 2^{-s}} \cdot \frac{1}{1 – 3^{-s}} \cdot \frac{1}{1 – 5^{-s}} \cdot \frac{1}{1 – 7^{-s}} \cdots,$$ which clearly indicates an intimate relationship between the zeta function and prime numbers. Riemann examined the zeta function not just for real $s$, but also for the complex case. The zeta function, as defined in the first formula above, only converges for $s$ with real part greater than one. But one can fairly easily show that $$\left(1 – \frac{1}{2^{s-1}}\right) \zeta(s) \; = \; \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^s} \; = \; \frac{1}{1^s} – \frac{1}{2^s} + \frac{1}{3^s} – \cdots,$$ which converges whenever $s$ has positive real part, and that $$\zeta(s) = 2^s \pi^{s-1} \sin\left(\frac{\pi s}{2}\right) \Gamma(1 – s) \zeta(1 – s),$$ which permits one to define the function whenever the argument has a non-positive real part.

The zeta function has a simple pole singularity at $s = 1$, and clearly $\zeta(s) = 0$ for negative even integers, since $\sin (\pi s / 2) = 0$ for such values (these are known as the “trivial” zeroes). But it has an infinite sequence of more “interesting” zeroes, the first five of which are shown here:

Index | Real part | Imaginary part |

1 | 0.5 | 14.134725141734693790… |

2 | 0.5 | 21.022039638771554992… |

3 | 0.5 | 25.010857580145688763… |

4 | 0.5 | 30.424876125859513210… |

5 | 0.5 | 32.935061587739189691… |

Note that the real part of these zeroes is always 1/2. Riemann’s famous hypothesis is that

If the Riemann hypothesis is true, many important results would hold. Here are just two:

*The Mobius function*: Define the Mobius function of a positive integer $n$ as: $1$ if $n$ is a square-free positive integer with an even number of prime factors; $-1$ if $n$ is a square-free positive integer with an odd number of prime factors; and $0$ if $n$ has a squared prime factor. Then the statement that $$\frac{1}{\zeta(s)} \; = \; \sum_{n=1}^\infty \frac{\mu(n)}{n^s}$$ is valid for every s with ${\rm Re}(s) \gt 1/2$, with the sum converging, is equivalent to the Riemann hypothesis.*The prime-counting function*: For real $x \gt 0$, let $\pi(x)$ denote the number of primes less than $x$, and let ${\rm Li}(x) = \int_2^x 1/\log(t) \, {\rm d}t$. The “prime number theorem,” proven by Hadamard and de la Vallee Poussin in 1896, asserts that $\pi(x) / {\rm Li}(x)$ tends to one for large $x$. But if one assumes the Riemann hypothesis, a stronger result can be proved, namely $$|\pi(x) – {\rm Li}(x)| \lt \frac{\sqrt{x} \log(x)}{8 \pi}$$ for all sufficiently large $x$ (in fact, for all $x \ge 2657$).

For details and some other examples, see this Wikipedia article.

In a remarkable new paper, published in the *Proceedings of the National Academy Of Sciences*, the four mathematicians have resurrected a line of reasoning, long thought to be dead, originally developed by Johan Jensen and George Polya. The proof relies on “Jensen polynomials,” which for an arbitrary real sequence $(\alpha(0), \alpha(1), \alpha(2), \ldots)$, integer degree $d$ and shift $n$ are defined as: $$J_\alpha^{d,n} (x) = \sum_{j=0}^d {d \choose j} \alpha(n+j) x^j.$$ We say that a polynomial with real coefficients is hyperbolic if all of its zeroes are real. Define $\Lambda( s) = \pi^{-s/2} \Gamma(s/2) \zeta(s) = \Lambda(1-s)$. Consider now the sequence of Taylor coefficients $(\gamma(n), n \geq 1)$ defined implicitly by $$(4z^2 – 1) \Lambda(z+1/2) \; = \; \sum_{n=0}^\infty \frac{\gamma(n) z^{2n}}{n!}.$$

Polya proved that the Riemann hypothesis is equivalent to the assertion that the Jensen polynomials associated with the sequence $(\gamma(n))$ are hyperbolic for all nonnegative integers $d$ and $n$.

What Griffin, Ono, Rolen and Zagier have shown is that for $d \geq 1$, the associated Jensen polynomials $J_\gamma^{d,n}$ are hyperbolic for *all sufficiently large* $n$. This is not the same as *for every* $n$, but it certainly is a remarkable advance. In addition, the four authors proved that for $1 \leq d \leq 8$, that the associated Jensen polynomials are indeed hyperbolic for all $n \geq 0$. Previous to this result, the best result was for $1 \leq d \leq 3$ and all $n \geq 0$.

Ken Ono emphasizes that he and the other authors did not invent any new techniques or new mathematical objects. Instead, the advantage of their proof is its simplicity (the paper is only eight pages long!). The idea for the paper was a “toy problem” that Ono presented for entertainment to Zagier during a recent conference celebrating Zagier’s 65th birthday. Ono thought that the problem was essentially intractable and did not expect Zagier to make much headway with it, but Zagier was enthused by the challenge and soon had sketched a solution. Together with the other authors, they fleshed out the solution and then extended it to a more general theory.

Kannan Soundararajan, a Stanford mathematician who has studied the Riemann Hypothesis, said “The result established here may be viewed as offering further evidence toward the Riemann Hypothesis, and in any case, it is a beautiful stand-alone theorem.”

The authors emphasize that their work definitely falls short of a full proof of the Riemann hypothesis. For all they know, the hypothesis may still turn out to be false, or that what remains in this or any other proposed proof outline is so difficult that it may defy efforts to prove for many years to come. But the result is definitely encouraging.

It should be mentioned that some other manuscripts have circulated with authors claiming proofs, at least a few of which are by mathematicians with very solid credentials. However, none of these has ever gained any traction, so the only safe conclusion is that the Riemann hypothesis remains unproven and may be as difficult as ever.

Will it still be unproven 100 years from now? Stay tuned (if any of us are still around in 2119).

]]>In a remarkable development with far-reaching consequences, researchers at the Cambridge Laboratory of Molecular Biology have used a computer program to rewrite the DNA of the well-known bacteria Escherichia coli (more commonly known as “E. coli”) to produce a functioning, reproducing species that is far more complex than any previous similar synthetic biology effort.

Venter’s 2010 projectThis effort has its roots in a project spearheaded by J. Craig Venter, the well-known maverick biomedical researcher known for the “shotgun” approach to genome sequencing pioneered by his team at

Continue reading Computational tools help create new living organism

]]>In a remarkable development with far-reaching consequences, researchers at the Cambridge Laboratory of Molecular Biology have used a computer program to rewrite the DNA of the well-known bacteria Escherichia coli (more commonly known as “E. coli”) to produce a functioning, reproducing species that is far more complex than any previous similar synthetic biology effort.

This effort has its roots in a project spearheaded by J. Craig Venter, the well-known maverick biomedical researcher known for the “shotgun” approach to genome sequencing pioneered by his team at Celera Corporation (later acquired by Quest Diagnostics). In May 2010, Venter announced success in his team’s effort to synthesize an organism whose DNA had 1,080,000 DNA base pairs, designed by a computer program. In a final step, Venter reported that the bacteria used this synthetic DNA to generate proteins in preference to those of its own genome.

For additional details, see this Science article and this New York Times report.

In the latest advance, researchers at the Cambridge Laboratory of Molecular Biology, led by molecular biologist Jason Chin, constructed a variation of the E. coli bacteria with a synthetically designed genome consisting of four million DNA base pairs, compared with just one million in Venter’s 2010 experiment.

One objective of this study, according to Chin, is to explore why all living organisms on Earth today encode genetic information in the same curious way, and whether this design could be any different yet still yield functioning biology.

In normal biology, each group of three nonoverlapping base pairs code to produce one of 20 amino acids or “stop.” Note that biology could have produced up to 64 amino acids (since 4^{3} = 64), but only 20 are realized in real organisms, because multiple three-base-pair “words” code to the same amino acid, which according to DNA pioneer Francis Crick, may be a “frozen accident” of very early biology. For example, the DNA words CTT, CTC, CTA, CTG, TTA and TTG each code to produce the amino acid leucine, and TCT, TCC, TCA, TCG, AGT and AGC each code to produce serine. In total, 61 of the 64 possible three-base-pair words code to produce amino acids, and three words, namely UAA, UAG and UGA, code “stop.” See this table for details on the DNA code.

Chin’s team decided to try altering the E. coli genome, on the computer, by removing some of the superfluous codons. For example, they replaced all instances of TCG, which codes to serine, with AGC, which also codes to serine. In total, their genome uses just four codons to produce serine instead of six, and uses only two stop codons instead of three. This required editing the E. coli genome in over 18,000 DNA base-pair locations.

The actual implementation of this altered genome was a challenge. The genome is much too long and complicated to insert it into a cell in one attempt. Instead, the researchers swapped in segments, step by step. When they were done, there were only four errors in the four-million-long genome, which were quickly corrected.

The bottom line is that the resulting species is viable and reproduces, but grows somewhat more slowly that standard E. coli and develops longer, rod-shaped cells. Otherwise the cells appear normal. The researchers named the species “Syn61” (see graphic above).

Chin notes that such “designer lifeforms” could be useful for certain biopharmaceutical applications, for example to make insulin for diabetes patients and drugs for cancer, multiple sclerosis and other conditions. Production lines for these drugs, which employ altered strains of bacteria, are often plagued with viruses, but with designer bacteria of the type that Chin’s team has developed, these production lines could be based on highly virus-resistant artificial bacterial species.

Thomas Ellis, Director of the Center for Synthetic Biology at Imperial College London, described the achievement in glowing terms: “No one’s done anything like it in terms of size or in terms of number of changes before.” Geneticist George Church of Harvard University agrees: “Chin’s success will embolden the rest of us working to make many organisms (industrial microbes, plants, animals, and human cells) resistant to all viruses by this recoding approach.”

There is even the possibility that research like this will ultimately reveal the origin of life, or, at least, the DNA structure of the last universal common ancestor of all present-day species. For example, what is the bare minimum functionality for a species that can make copies of itself?

However this turns out, an era of synthetic biology will soon be upon us. How will we deal with it? Chin acknowledges the challenges:

People have legitimate concerns. … There is a dual use to anything we invent. But what’s important is that we have a debate about what we should and shouldn’t do. And that these experiments are done in a well controlled way.

For additional details, see this New York Times article, this BBC report and this Guardian report. The Cambridge Laboratory’s Nature paper is here.

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