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Archive for December 11th, 2019

Mathematician Proves Huge Result on “Dangerous” Problem

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Kevin Hartnett writes in Quanta:

Experienced mathematicians warn up-and-comers to stay away from the Collatz conjecture. It’s a siren song, they say: Fall under its trance and you may never do meaningful work again.

The Collatz conjecture is quite possibly the simplest unsolved problem in mathematics — which is exactly what makes it so treacherously alluring.

“This is a really dangerous problem. People become obsessed with it and it really is impossible,” said Jeffrey Lagarias, a mathematician at the University of Michigan and an expert on the Collatz conjecture.

Earlier this year one of the top mathematicians in the world dared to confront the problem — and came away with one of the most significant results on the Collatz conjecture in decades.

On September 8, Terence Tao posted a proof showing that — at the very least — the Collatz conjecture is “almost” true for “almost” all numbers. While Tao’s result is not a full proof of the conjecture, it is a major advance on a problem that doesn’t give up its secrets easily.

“I wasn’t expecting to solve this problem completely,” said Tao, a mathematician at the University of California, Los Angeles. “But what I did was more than I expected.”

The Collatz Conundrum

Lothar Collatz likely posed the eponymous conjecture in the 1930s. The problem sounds like a party trick. Pick a number, any number. If it’s odd, multiply it by 3 and add 1. If it’s even, divide it by 2. Now you have a new number. Apply the same rules to the new number. The conjecture is about what happens as you keep repeating the process.

Intuition might suggest that the number you start with affects the number you end up with. Maybe some numbers eventually spiral all the way down to 1. Maybe others go marching off to infinity.

But Collatz predicted that’s not the case. He conjectured that if you start with a positive whole number and run this process long enough, all starting values will lead to 1. And once you hit 1, the rules of the Collatz conjecture confine you to a loop: 1, 4, 2, 1, 4, 2, 1, on and on forever.

Over the years, many problem solvers have been drawn to the beguiling simplicity of the Collatz conjecture, or the “3+ 1 problem,” as it’s also known. Mathematicians have tested quintillions of examples (that’s 18 zeros) without finding a single exception to Collatz’s prediction. You can even try a few examples yourself with any of the many “Collatz calculators” online. The internet is awash in unfounded amateur proofs that claim to have resolved the problem one way or the other.

Try it for yourself here.

“You just need to know multiplying by 3 and dividing by 2 and you can start playing around with it right away. It’s very tempting to try,” said Marc Chamberland, a mathematician at Grinnell College who produced a popular YouTube video on the problem called “The Simplest Impossible Problem.”

But legitimate proofs are rare.

In the 1970s, mathematicians showed that almost all Collatz sequences — the list of numbers you get as you repeat the process — eventually reach a number that’s smaller than where you started — weak evidence, but evidence nonetheless, that almost all Collatz sequences incline toward 1. From 1994 until Tao’s result this year, Ivan Korec held the record for showing just how much smaller these numbers get. Other results have similarly picked at the problem without coming close to addressing the core concern.

“We really don’t understand the Collatz question well at all, so there hasn’t been much significant work on it,” said Kannan Soundararajan, a mathematician at Stanford University who has worked on the conjecture.

The futility of these efforts has led many mathematicians to conclude that the conjecture is simply beyond the reach of current understanding — and that they’re better off spending their research time elsewhere.

“Collatz is a notoriously difficult problem — so much so that mathematicians tend to preface every discussion of it with a warning not to waste time working on it,” said Joshua Cooper of the University of South Carolina in an email.

An Unexpected Tip

Lagarias first became intrigued by the conjecture as a student at least 40 years ago. For decades he has served as the unofficial curator of all things Collatz. He’s amassed a library of papers related to the problem, and in 2010 he published some of them as a book titled The Ultimate Challenge: The 3x + 1 Problem.

“Now I know lots more about the problem, and I’d say it’s still impossible,” Lagarias said.

Tao doesn’t normally spend time on impossible problems. In 2006 he won the Fields Medal, math’s highest honor, and he is widely regarded as one of the top mathematicians of his generation. He’s used to solving problems, not chasing pipe dreams.

“It’s actually an occupational hazard when you’re a mathematician,” he said. “You could get obsessed with these big famous problems that are way beyond anyone’s ability to touch, and you can waste a lot of time.”

But Tao doesn’t entirely resist the great temptations of his field. Every year, he tries his luck for a day or two on one of math’s famous unsolved problems. Over the years, he’s made a few attempts at solving the Collatz conjecture, to no avail.

Then this past August an anonymous reader left a comment on Tao’s blog. The commenter suggested trying to solve the Collatz conjecture for “almost all” numbers, rather than trying to solve it completely.

“I didn’t reply, but it did get me thinking about the problem again,” Tao said.

And what he realized was that the Collatz conjecture was similar, in a way, to the types of equations — called partial differential equations — that have featured in some of the most significant results of his career.

Inputs and Outputs

Partial differential equations, or PDEs, can be used to model many of the most fundamental physical processes in the universe, like the evolution of a fluid or the ripple of gravity through space-time. They arise in situations where the future position of a system — like the state of a pond five seconds after you’ve thrown a rock into it — depends on contributions from two or more factors, like the water’s viscosity and velocity.

Complicated PDEs wouldn’t seem to have much to do with a simple question about arithmetic like the Collatz conjecture.

But Tao realized there was something similar about them. With a PDE, you plug in some values, get other values out, and repeat the process — all to understand that future state of the system. For any given PDE, mathematicians want to know if some starting values eventually lead to infinite values as an output or whether an equation always yields finite values, regardless of the values you start with.

For Tao, this goal had the same flavor as investigating whether you always eventually get the same number (1) from the Collatz process no matter what number you feed in. As a result, he recognized that techniques for studying PDEs could apply to the Collatz conjecture.

One particularly useful technique involves a statistical way of studying the long-term behavior of a small number of starting values (like a small number of initial configurations of the water in a pond) and extrapolating from there to the long-term behavior of all possible starting configurations of the pond.

In the context of the Collatz conjecture, imagine starting with a large sample of numbers. Your goal is to study how these numbers behave when you apply the Collatz process. If close to 100% of the numbers in the sample end up either exactly at 1 or very close to 1, you might conclude that almost all numbers behave the same way.

But for the conclusion to be valid, you’d have to construct your sample very carefully. The challenge is akin to generating a sample of voters in a presidential poll. To extrapolate accurately from the poll to the population as a whole, you’d need to weight the sample with the correct proportion of Republicans and Democrats, women and men, and so on.

Numbers have their own “demographic” characteristics. There are odd and even numbers, of course, and numbers that are multiples of 3, and numbers that differ from each other in even subtler ways. When you construct a sample of numbers, you can weight it toward containing certain kinds of numbers and not others — and the better you choose your weights, the more accurately you’ll be able to draw conclusions about numbers as a whole.

Weighty Choices

Tao’s challenge was much harder than just figuring out how to create an initial sample of numbers with the proper weights. At each step in the Collatz process, the numbers you’re working with change. One obvious change is that almost all numbers in the sample get smaller.

Another, maybe less obvious change is that  . . .

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Written by LeisureGuy

11 December 2019 at 1:58 pm

Posted in Math

Benefits of flaxseed for inflammation — and for high blood pressure

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Maybe my getting my blood pressure back to normal without medication is not totally due to cutting salt and drinking hibiscus tea.

Written by LeisureGuy

11 December 2019 at 1:18 pm

The ‘Russia Hoax’ Is a Hoax

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Adam Serwer writes in the Atlantic:

If you are following mainstream news outlets, you know that in 2016, Donald Trump benefited from a Russian hacking and disinformation campaign designed to help him get elected, even as he sought permission from the Russian government to build a hotel in Moscow. You know that he deflected blame from Russia for that campaign, even as he sought to benefit from it politically. You know that shortly after the election, Trump told Russian officials in the Oval Office that he didn’t mind their efforts on his behalf, inviting further interference. And you know that while those acts may not have amounted to criminal conspiracy, the president’s insistence that there was “no collusion” flies in the face of established facts.

If you are ensconced in the pro-Trump-propaganda universe of Fox News and its spawn, you know something different. You know that the Russia investigation was a “hoax” developed by the “deep state” and the media, an attempt by a fifth column within the FBI to engage in a “coup,” a conspiracy, a frame job, “nothing less than the attempted overthrow of the U.S. government.” Any evidence of wrongdoing by the president, in this universe, has been manufactured by Trump’s shadowy and powerful enemies—George Soros, liberals in the FBI, Barack Obama.

The belief that Trump is the victim of a vast and ongoing conspiracy is a crucial element of the president’s enduring appeal to his supporters. If the allegations against the president are all completely false, then his supporters can continue to back him with a clear conscience, because anything and everything negative they hear about the president must be false. The consistency of that message is more important than the actual details, which frequently end up contradicting complex explanations for the president’s innocence that are often incongruous with each other, such as the insistence that Robert Mueller’s investigation was a “total exoneration” of the president, but also “total bullshit.”

The Department of Justice inspector general’s probe into the origins of the Russia investigation, which was released Monday, found no evidence that any of the Trump conspiracy theories surrounding the origin of the investigation are true. The investigation was not launched on Obama’s orders, it was not an effort by pro–Hillary Clinton FBI agents to prevent Trump from getting elected, and it was not predicated on the existence of opposition research gathered by the former British intelligence officer Christopher Steele. The president’s defenders have taken to referring to the entire investigation as “the Russia hoax,” insisting that the entire investigation was an effort by “persons within the FBI and Barack Obama’s Justice Department” who “worked improperly to help elect Clinton and defeat Donald Trump in the 2016 presidential election.” But the IG report shows that the “Russia hoax” defense is itself a hoax, and a highly successful one, aimed at reassuring Trump supporters who might otherwise be troubled by the president’s behavior.

The inconsistencies and contradictions of the “Russia hoax” narrative appear not to trouble the president’s supporters. Rather, as George Orwell wrote in 1944, “For quite long periods, at any rate, people can remain undisturbed by obvious lies, either because they simply forget what is said from day to day or because they are under such a constant propaganda bombardment that they become anaesthetized to the whole business.” The numbness to every new Trump revelation, no matter how shocking, is in part a product of the president’s success in fatiguing anyone who might be interested in what the facts are.

The IG report knocked down the various claims that Trump and his allies have made, one by one. The report confirmed that the Russia investigation originated, as has been previously reported, with the Trump campaign adviser George Papadopoulos bragging to an Australian diplomat about Russia possessing “dirt” on Hillary Clinton, which the IG determined “was sufficient to predicate the investigation.” The widespread conservative belief that the investigation began because of the dubious claims in the Steele dossier was false. “Steele’s reports played no role” in the opening of the Russia investigation, the report found, because FBI officials were not “aware of Steele’s election reporting until weeks later.”

Republicans’ claim that the investigation began because the FBI misled the Foreign Intelligence Surveillance Court to obtain permission to surveil the former Trump campaign aide Carter Page was false. The IG also “did not find any records” that Joseph Mifsud, the professor who told Papadopoulos the Russians had obtained “dirt” on Clinton, was an FBI informant sent to entrap him. The former FBI agent Peter Strzok and the former FBI attorney Lisa Page, who shared anti-Trump sentiments over text and have become key villains in the Trumpist narrative of a “coup,” never had the power to do what has been attributed to them. The IG report notes that Page “did not play a role in the decision to open” the Russia investigation, and that Strzok was “was not the sole, or even the highest-level, decision maker as to any of those matters.”

The IG report also determined that “the FBI had an authorized purpose when it opened [the Russia investigation] to obtain information about, or protect against, a national security threat or federal crime, even though the investigation also had the potential to impact constitutionally protected activity.” Moreover, the IG found “no evidence” that “political bias or improper motivation influenced the decisions” to investigate Trump advisers with ties to Russia.

There is, in short, no “deep state” anti-Trump conspiracy, no network of perfidious liberals in the FBI seeking to take down Trump. There is, however, voluminous evidence of reprehensible behavior by the president, first taking advantage of a foreign attack on the 2016 election for personal and political profit, seeking to obstruct the investigation into that interference, and then falsely concocting an elaborate conspiracy theory to avoid accountability for his actions.

Nevertheless, there are important systemic problems with the FBI and the way that the U.S. government approves invasive surveillance techniques on American citizens. The report notes that . . .

Continue reading.

Written by LeisureGuy

11 December 2019 at 1:15 pm

The massive triumph of the rich, illustrated by stunning new data

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Class warfare is already being waged, and the wealthy are (a) aware of it and (b) participating actively and (c) winning. Greg Sargent writes in the Washington Post:

The triumph of the rich, which is one of the defining stories of our time, is generally described as largely the reflection of two factors. The first, of course, is the explosion of income among top earners, in which a tiny minority has vacuumed up a ballooning share of the gains from the past few decades of economic growth.

The second factor — which will be key to the 2020 presidential race — has been the hidden decline in the progressivity of the tax code at the top, in which the wealthiest earners have over those same decades seen their effective tax rates steadily fall.

Put those two factors together, and they tell a story about soaring U.S. inequality that is in some ways even more dramatic than each is on its own.

A new analysis prepared at my request by Gabriel Zucman — the French economist and “wealth detective” who has become famous for tracing the hidden wealth of the super-rich — illustrates that dual story in a freshly compelling way.

The top-line finding: Among the bottom 50 percent of earners, average real annual income even after taxes and transfers has edged up a meager $8,000 since 1970, rising from just over $19,000 to just over $27,000 in 2018.

By contrast, among the top 1 percent of earners, average income even after taxes and transfers has tripled since 1970, rising by more than $800,000, from just over $300,000 to over $1 million in 2018.

Among the top 0.1 percent, average after-tax-and-transfer income has increased fivefold, from just over $1 million in 1970 to over $5 million in 2018. And among the top .01 percent, it has increased nearly sevenfold, from just over $3.5 million to over $24 million.

I’m emphasizing the phrase “after taxes and transfers” because this is at the core of Zucman’s new analysis. The idea is to show the combined impact of both the explosion of pretax income at the top and the decline in the effective tax rate paid by those same earners — in one result.

The declining progressivity of the tax code is the subject of “The Triumph of Injustice,” a great new book by Zucman and fellow Berkeley economist Emmanuel Saez. It charts the slow strangulation of that progressivity at the top.

As they demonstrate, the effective tax rate (federal, state, local and other taxes) paid by top earners has steadily declined since the 1950s and 1960s, when the tax code really was quite progressive, to a point where the highest income groups pay barely more, percentage wise, than the bottom.

Indeed, in 2018, the top 400 earners for the first time paid a lower effective overall tax rate than working-class Americans. There are many reasons for this radical decline in progressivity, including domestic and international tax avoidance, the whittling away of the estate and corporate taxes, and the repeated downsizing of top marginal rates.

I asked Zucman to calculate the average totals in raw dollar amounts that each income group has taken home over the decades — after taxes and transfers. This includes federal, state and payroll taxes of numerous kinds, and government spending on transfer programs, including Social Security, Medicare, Medicaid, and disability and veteran benefits, among other things.

Zucman is able to do this because he and Saez created a database to do the new book’s calculations, and this database includes as much income, tax and transfer information at all levels of government as they were able to assemble.

Here are Zucman’s results:

As noted, real after-tax-and-transfer income edged up just a bit among the bottom 50 percent while rocketing upward to an extraordinary degree among the top-earning groups.

Meanwhile, among middle-class Americans in the 50 percent to 90 percent section of the income scale (sometimes called the “middle 40”), average income went from just over $44,000 in 1970 to just over $75,000 in 2018.

“After government transfers, the income of the working and middle class has grown a bit faster than before, but still their income growth rate has been very low,” Zucman told me. “For the working class, their after-taxes-and-transfers income has barely increased.”

And among those in the group from the top 10 percent to 1 percent, average income went from just over $90,000 in 1970 to over $185,000 in 2018. For this group, income doubled — they showed real gains. But the bottom very decidedly did not, and the middle 40’s gains were comparatively modest.

Here’s another way to look at what happened at the turn of each decade:

All this offers another way of looking at a phenomenon that political philosopher Samuel Moyn has labeled the “victory of the rich.”

These data help debunk frequent arguments against making the tax code far more progressive. Hedge-funder Leon Cooperman recently penned a whiny letter to Sen. Elizabeth Warren (D-Mass.) — demonstrating what Paul Krugman described as an odd billionaire “self-pitying” streak — arguing that the superwealthy already pay a lot and that the current tax code is more progressive than you think, once you factor in transfers.

It’s true that the wealthy often pay a lot. But the fact that after-tax income has pulled into the stratosphere at the very top, even as effective tax rates have fallen dramatically, shows that the current tax bill of the rich does not, in and of itself, constitute any kind of meaningful argument about the fairness of today’s after-tax distribution.

What’s more, the above also shows how limited the impact of transfer programs has been on soaring inequality over the decades.

“People have this idea that government redistribution has upset some of the rise in inequality, but essentially that’s not the case,” Zucman told me.

Finally, Zucman’s analysis could give weight to progressive arguments for a deep structural economic overhaul. Even Joe Biden’s plan, which would raise corporate tax rates and return top marginal income tax rates to where they were before President Trump’s tax cuts, is surprisingly progressive, a sign of how much this debate has shifted.

Meanwhile, Warren has proposed a tax on extreme wealth. And Sen. Bernie Sanders (I-Vt.) has offered an even more ambitious wealth tax, plus a proposal to boost top income tax rates far more than any other rival.

The larger context for this debate is that real-world take-home income has positively exploded at the very top, even as it has barely budged for the bottom half — the sum total of two separate but intertwined stories that have unfolded over the decades.

“You have two trends reinforcing each other,” Zucman told me. “There has been the rise in . . .

Continue reading.

Written by LeisureGuy

11 December 2019 at 10:43 am

Bamboo shoot redux

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Here’s another bamboo shoot, peeled and with the bottom cut off and halved. The bottom is already growing the internal hollow structure of mature bamboo, but the top portion lacks the cavities. I will chop and sauté this, including leaves (why not?), along with some other veggies, mushrooms, tempeh, and cooked barley.

Written by LeisureGuy

11 December 2019 at 9:33 am

The Arctic may have crossed key threshold, emitting billions of tons of carbon into the air, in a long-dreaded climate feedback

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Andrew Freedman reports some bad tidings in the Washington Post:

The Arctic is undergoing a profound, rapid and unmitigated shift into a new climate state, one that is greener, features far less ice and emits greenhouse gas emissions from melting permafrost, according to a major new federal assessment of the region released Tuesday.

The consequences of these climate shifts will be felt far outside the Arctic in the form of altered weather patterns, increased greenhouse gas emissions and rising sea levels from the melting Greenland ice sheet and mountain glaciers.

The findings are contained in the 2019 Arctic Report Card, a major federal assessment of climate change trends and impacts throughout the region. The study paints an ominous picture of a region lurching to an entirely new and unfamiliar environment.

Especially noteworthy is the report’s conclusion that the Arctic already may have become a net emitter of planet-warming carbon emissions due to thawing permafrost, which would only accelerate global warming. Permafrost is the carbon-rich frozen soil that covers 24 percent of the Northern Hemisphere’s land mass, encompassing vast stretches of territory across Alaska, Canada, Siberia and Greenland.

There has been concern throughout the scientific community that the approximately 1,460 billion to 1,600 billion metric tons of organic carbon stored in frozen Arctic soils, almost twice the amount of greenhouse gases as what is contained in the atmosphere, could be released as the permafrost melts.

Warming temperatures allow microbes within the soil to convert permafrost carbon into the greenhouse gases — carbon dioxide and methane — which can be released into the air and accelerate warming. Ted Schuur, a researcher at Northern Arizona University and author of the permafrost chapter, said the report “takes on a new stand on the issue” based on other published work, including a study in Nature Climate Change in November.

Taking advantage of the new studies — one on regional carbon emissions from permafrost in Alaska during the warm season, and another on winter season emissions in the Arctic compared to how much carbon is absorbed by vegetation during the growing season — the report concludes permafrost ecosystems could be releasing as much as 1.1 billion to 2.2 billion tons of carbon dioxide per year. This is almost as much as the annual emissions of Japan and Russia in 2018, respectively.

“These observations signify that the feedback to accelerating climate change may already be underway,” the report concludes.

“Each of the studies has some parts of the story. Together they really paint the picture of — we’ve turned this corner for Arctic carbon,” Schuur said. “Together they complement each other nicely and really in my mind are a smoking gun for this change already taking place.” . . .

Continue reading.

Meanwhile, conservatives and fossil-fuel companies deny that this is happening. I hope that works, but so far denial has proved ineffective at stopping (or even slowing) the trend..

Written by LeisureGuy

11 December 2019 at 8:42 am

First commercial aircraft to use motors rather than engines takes flight

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Ian Bailey reports in the Globe & Mail:

The chief executive officer of B.C.-based Harbour Air Seaplanes took to the air at the controls of a float plane equipped with an electric engine Tuesday, a move that puts the company in the global race to develop electric flight and reduce emissions from passenger aircraft.

Greg McDougall, the sole occupant of the aircraft, flew the six-seater de Havilland about 16 kilometres in eight minutes, taking off from the company’s terminal on the Fraser River south of Vancouver International Airport.

It was the first full-fledged flight for the DHC-2 de Havilland Beaver, following a previous test in which it was flown about 60 to 90 metres. The conventional internal combustion engine of the de Havilland has been replaced by a 750-horsepower electric engine developed by magniX, a company based in Redmond, Wash.

“We had no way of knowing really exactly how the aircraft would actually perform until we actually flew it, ” Mr. McDougall told a news conference following the flight. “That was the first real time it had flown.”

While the plane has a cutting-edge electric propulsion system, Mr. McDougall noted “the airframe that the motor is attached to is actually one year younger than me, so 62 years old.”

Harbour Air, which carries about 500,000 passengers a year on routes to and from locations such as Victoria, Vancouver Island and the Gulf Islands, is now in a race to perfect electric flight where competitors include Airbus, Sweden’s Heart Aerospace and the Scottish carrier Loganair.

Mr. McDougall, who founded Harbour Air in 1982 with a pair of small seaplanes, said electrification of the transportation industry is a given, prompting the 11 months of work his company has done on the concept.

“Obviously aviation is one of the tougher ones to do, but it’s going to happen,” he said. “We need to be at the forefront of it.”

In a statement responding to the Harbour Air test flight, Transport Canada said the federal government is “steadfast” in its belief that transportation electrification is a key part of Canada’s transition to a low-carbon economy.

The federal department says it offers regulatory and certification advice to all industry partners, including Harbour Air, “seeking to develop innovative aviation ideas,” but does not have specific standards for electric-powered aircraft. . .

Continue reading.

Amy Smart also has a report in the Vancouver Sun:

As Greg McDougall prepared to fly the world’s first all-electric commercial aircraft Tuesday morning, he said “nervous” wasn’t quite the word to describe how he was feeling.

The fact that the Harbour Air CEO would be the first person to take the modified de Havilland Beaver on a full test flight didn’t faze him, nor did knowledge of a charging glitch the night before.

McDougall had gone for a dinner break Monday evening while a crew of designers and engineers stared at their computers with furrowed brows, and he returned later to find them smiling and laughing, crisis averted.

“The emotion isn’t necessarily excitement, it’s more sort of anticipation and focus,” he said.

With the sun hanging low over the Fraser River in Richmond, McDougall shifted the throttle into gear and took off. After landing, he said it felt just like flying any other plane, only with more kick.

“For me, that flight was just like flying a Beaver but it was a Beaver on electric steroids,” he said, adding he had to throttle back in order to delay the takeoff to be in line with about a dozen cameras.

“It wanted to fly. With the tailwind it was going to leap off the water.”

The brief but successful test flight marked a significant win for Harbour Air and partner magniX, which designed the electric motor, in the race to electrify commercial aviation fleets.

Dozens of companies are working on electric planes, including Boeing and Airbus. Israeli company Eviation unveiled a nine-seat, all-electric plane named “Alice” at the Paris Air Show in June, which also happens to be a magniX project.

Roei Ganzarski, CEO of Seattle-based engineering firm magniX, described the test flight as the beginning of a revolution in aviation.

In 1903, the Wright brothers made history with the first successful flight and, in 1939, the Heinkel jet launched the jet age, he said.

“Since 1939, we’ve pretty much stayed stable. Today that team made history,” Ganzarski said, gesturing toward the design team.

Harbour Air announced in March that it had partnered with magniX with the goal of becoming the world’s first all-electric airline.

The 62-year-old Beaver was outfitted with a 750-horsepower electric motor, which gives it capacity to fly about 160 kilometres before needing a recharge.

Weight, altitude and storage remain the biggest barriers to flying electric. A mid-sized passenger plane weighs 100 times as much as a mid-sized car and the battery technology hasn’t quite adjusted to the aviation market.

Fuel also remains about 40 to 50 times more power dense than batteries, Ganzarski said. But the team expects innovation in the battery industry to continue in the same way for aviation as it has for electric cars. The key will be developing batteries that are more compact at the same time that they are more powerful.

The test flight used lithium-ion batteries because they are the most “tried and true,” but there are already others on the market that are more powerful, McDougall said.

“The evolution of lithium batteries is constant and there are literally billions of dollars being poured into that technology as we speak,” he said. . .

Continue reading.

 

Written by LeisureGuy

11 December 2019 at 8:35 am

The Drunken Goat and the G.B. Kent BK4, with the Above the Tie R

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This is the last of my natural-bristle brushes. The brush in the cover photo of the Guide is a Kent BK8, but I found I preferred the BK4 size. Mickey Lee’s now-defunct Soapworks made a number of excellent shaving soaps, and The Drunken Goat is one of my favorites. Too bad it’s no longer available; it has an excellent fragrance.

The razor head is the Above the Tie R1, but mine is stamped simply “R” since it is from 2014, before the R2 arrived. Here it’s on a UFO handle, and it did an extremely nice job.

A splash of The Drunken Goat aftershave, and the day is launched.

Written by LeisureGuy

11 December 2019 at 8:19 am

Posted in Shaving

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