darenw shares a report: Freeman J. Dyson, a mathematical prodigy who left his mark on subatomic physics before turning to messier subjects like Earth's environmental future and the morality of war, died on Friday at a hospital near Princeton, N.J. He was 96. His daughter Mia Dyson confirmed the death. As a young graduate student at Cornell in 1949, Dr. Dyson wrote a landmark paper -- worthy, some colleagues thought, of a Nobel Prize -- that deepened the understanding of how light interacts with matter to produce the palpable world. The theory the paper advanced, called quantum electrodynamics, or QED, ranks among the great achievements of modern science. But it was as a writer and technological visionary that he gained public renown.

He imagined exploring the solar system with spaceships propelled by nuclear explosions and establishing distant colonies nourished by genetically engineered plants. "Life begins at 55, the age at which I published my first book," he wrote in "From Eros to Gaia," one of the collections of his writings that appeared while he was a professor of physics at the Institute for Advanced Study in Princeton -- an august position for someone who finished school without a Ph.D. The lack of a doctorate was a badge of honor, he said. With his slew of honorary degrees and a fellowship in the Royal Society, people called him Dr. Dyson anyway. Further reading: Slashdot's interview with Freeman Dyson (2013).

The New York Times: They asked Katherine Johnson for the moon, and she gave it to them. Wielding little more than a pencil, a slide rule and one of the finest mathematical minds in the country, Mrs. Johnson, whose death at 101 was announced on Monday by NASA, calculated the precise trajectories that would let Apollo 11 land on the moon in 1969 and, after Neil Armstrong's history-making moonwalk, let it return to Earth. A single error, she well knew, could have dire consequences for craft and crew. Her impeccable calculations had already helped plot the successful flight of Alan B. Shepard Jr., who became the first American in space when his Mercury spacecraft went aloft in 1961. The next year, she likewise helped make it possible for John Glenn, in the Mercury vessel Friendship 7, to become the first American to orbit the Earth. Yet throughout Mrs. Johnson's 33 years in NASA's Flight Research Division -- the office from which the American space program sprang -- and for decades afterward, almost no one knew her name.

Mrs. Johnson was one of several hundred rigorously educated, supremely capable yet largely unheralded women who, well before the modern feminist movement, worked as NASA mathematicians. But it was not only her sex that kept her long marginalized and long unsung: Katherine Coleman Goble Johnson, a West Virginia native who began her scientific career in the age of Jim Crow, was also African-American. In old age, Mrs. Johnson became the most celebrated of the small cadre of black women -- perhaps three dozen -- who at midcentury served as mathematicians for the space agency and its predecessor, the National Advisory Committee for Aeronautics. Their story was told in the 2016 Hollywood film "Hidden Figures," based on Margot Lee Shetterly's nonfiction book of the same title, published that year. The movie starred Taraji P. Henson as Mrs. Johnson, the film's central figure. It also starred Octavia Spencer and Janelle Monae as her real-life colleagues Dorothy Vaughan and Mary Jackson.

Long-time Slashdot reader sandbagger shared this report from the Hugo award-winning science fiction fanzine File 770: North Bellmore, New York fan Elyse Rosenstein, 69, died suddenly on February 20th. She had been undergoing rehabilitation after suffering a broken leg. At the time of her death, she was a retired secondary school science teacher. With Joyce Yasner, Joan Winston, Linda Deneroff and Devra Langsam, she organized the very first Star Trek convention, held in New York City in 1972. The convention was not only the very first media convention, it was also the biggest science fiction convention to date by a considerable margin...

At the time, Star Trek fans were often looked down on by many science fiction fans, who were more into books and magazines than TV shows. The pair hoped that a convention specifically geared towards Star Trek would do a lot to bring fans together. The rest, as they say, is fan history....

Elyse Rosenstein had a BS in physics and math, and an MS in physics, and taught science for more than two decades. She was a member of the New York Academy of Sciences and the Long Island Physics Teachers Association...

She was nicknamed "The Screaming Yellow Zonker" by Isaac Asimov.

An anonymous reader quotes a report from Observer: If you like to cook but not to shop or plan your own meals, and if you weren't too hungry, and if you didn't like cooking for too many friends, then Blue Apron -- the startup delivering precisely measured, prepackaged amounts of just enough salmon, green beans, butter and lemon for one meal, no leftovers -- was for you. Exactly who it was that was both upwardly mobile to pay for this service while also having a barren kitchen, nobody really knew -- but by the divine math of Silicon Valley gamblers, your existence made this an idea worth several billion dollars and potent enough to "disrupt" the grocery business. People actually believed this. Or they did until Jeff Bezos and Amazon went shopping and bought out Whole Foods. Or until HelloFresh launched. Or until Blue Apron spent millions on packaging and shipping, as well as marketing, literally gifting away boxes of neatly assorted ingredients to millennials who never ordered another box. All this conspired to, one-by-one, wreck Blue Apron's IPO, crater stock prices to all-time list lows, kick founders out of company leadership and now, at last, the seemingly undeniable, ultimate doom of the company.

After losing another $23.7 million in the last three months of 2019, Blue Apron is laying off 240 workers and shutting down the shop at its Arlington, Texas warehouse location. Blue Apron will keep, for now, its California and New York assembly-and-distribution shops, while leaders ponder peddling what's left at a paltry $50 million price tag. Meanwhile, customers continue to desert Blue Apron, down to 351,000 in the last quarter of 2019, from 557,000 the year before. Selling off Blue Apron that low would mean a loss in the neighborhood of $143 million for Blue Apron's capital investors, including Fidelity and Goldman Sachs. That hurts, but as usual, retail investors took the worst hit. Stock-market playing rubes, who bought in when Blue Apron went public at $11 a share, have lost more than 80% on their investment -- and that represents a recovery. Shares were trading for $3.60 at the close on Wednesday, up from 2018 when Blue Apron was worth less than a dollar. There's no other analysis than this: Blue Apron was one of the biggest-ever Silicon Valley catastrophes, a mix of hubris, unrealistic expectations, a misunderstanding of how people exist in the world -- and, Amazon.

An anonymous reader shares a problem that Amazon asks in its interviews: A cable of 80 meters is hanging from the top of two poles that are both 50 meters off the ground. What is the distance between the two poles (to one decimal point) if the center cable is (a) 20 meters off the ground and (b) 10 meters off the ground?

Andrew Yang, tech entrepreneur and founder of Venture for America, will end his campaign for president after a disappointing showing in the New Hampshire primary. The Washington Post reports: "I am a numbers guy," Yang said in an interview before addressing supporters at Manchester's Puritan Backroom. "In most of these [upcoming] states, I'm not going to be at a threshold where I get delegates, which makes sticking around not necessarily helpful or productive in terms of furthering the goals of this campaign. If I become persuaded that there's a particular candidate that gives us a superior chance of beating Donald Trump, and I think it's important to make that opinion known, then I would consider it for sure," Yang said. He also said he would be open to becoming another candidate's running mate or joining a presidential Cabinet.

In his stump speech, Yang warned of the societal and economic changes automation would continue to bring to the United States. He proposed countering it by implementing universal basic income in the form of a $1,000-a-month "Freedom Dividend" for U.S. citizens. His sometimes bleak message on the campaign trail was contrasted with his upbeat, irreverent style of campaigning: Yang once crowd-surfed at a candidate forum and sometimes challenged other celebrities to pickup basketball games. He half-danced onto just about every stage to the '90s Mark Morrison R&B hit "Return of the Mack" and spawned a loyal following of supporters who dubbed themselves the "Yang Gang." They often showed up at his events wearing trademark "math" hats, a nod both to his self-described emphasis on facts and research and to the geek culture that surrounded his candidacy. "This is the nerdiest campaign in history," Yang told The Washington Post last year. Yang was also the first presidential candidate to use campaign funds for a pilot program meant to resemble his universal basic income proposal. "He told CNN on Monday that the concept of a freedom dividend was 'not going anywhere,' and emphasized on Tuesday that he had forced a new idea into Democratic politics," reports The Washington Post. "He made that point with math."

"Now, 66 percent of Democrats support a universal basic income," Yang said. "It's got 72 percent of young people, aged 18 to 34."

By exploiting randomness, three mathematicians have proved an elegant law that underlies the chaotic motion of turbulent systems. From a report: Picture a calm river. Now picture a torrent of white water. What is the difference between the two? To mathematicians and physicists it's this: The smooth river flows in one direction, while the torrent flows in many different directions at once. Physical systems with this kind of haphazard motion are called turbulent. The fact that their motion unfolds in so many different ways at once makes them difficult to study mathematically. Generations of mathematicians will likely come and go before researchers are able to describe a roaring river in exact mathematical statements. But a new proof finds that while certain turbulent systems appear unruly, they actually conform to a simple universal law. The work is one of the most rigorous descriptions of turbulence ever to emerge from mathematics. And it arises from a novel set of methods that are themselves changing how researchers study this heretofore untamable phenomenon.

"It may well be the most promising approach to turbulence," said Vladimir Sverak, a mathematician at the University of Minnesota and an expert in the study of turbulence. The new work provides a way of describing patterns in moving liquids. These patterns are evident in the rapid temperature variations between nearby points in the ocean and the frenetic, stylized way that white and black paint mix together. In 1959, an Australian mathematician named George Batchelor predicted that these patterns follow an exact, regimented order. The new proof validates the truth of "Batchelor's law," as the prediction came to be known. "We see Batchelor's law all over the place," said Jacob Bedrossian, a mathematician at the University of Maryland, College Park and co-author of the proof with Alex Blumenthal and Samuel Punshon-Smith. "By proving this law, we get a better understanding of just how universal it is."

An anonymous reader quotes a report from Ars Technica: The Trump administration has for several years been working to weaken federal vehicle fuel-efficiency standards. To justify these changes, regulatory agencies argued that more stringent standards would both cost consumers more and reduce road safety. A draft version of the new final rule, however, seems to directly contradict those lines of reasoning. The draft of the Safer Affordable Fuel-Efficient (SAFE) Vehicles rule has not been released publicly, but Sen. Thomas Carper (D-Del.) has seen it. In a letter (PDF) to the White House, Carper says not only is the rule "replete with numerous questionable legal, procedural, and technical assertions," as well as "apparent typographical and other errors," but it also completely fails to provide the safety or economic benefits initially claimed.

"Remarkably, the costs of the Trump administration's draft final rule exceed its benefits to Americans" relative to the current standards. The senator writes: "While the draft final rule finds that the per vehicle purchase price would be reduced relative to the Obama rules by $977 (EPA greenhouse gas standards)/$1,083 (DOT's fuel economy standards), the draft final rule also projects that the increased gasoline consumers would have to use to operate the less fuel-efficient vehicles would ad $1,461 (EPA greenhouse gas standards)/$1,423 (DOT fuel economy standards) to these costs. Adding hundreds of dollars to the cost of each vehicle would seem to be the opposite of the more "affordable" vehicles the SAFE rule promised." Further, Carper notes, the estimate of lives potentially saved over a nearly 50-year time period by upgrading to new cars does not take into account the lives potentially lost to illness and disease attributable to increased pollution from less efficient cars. And of course, Carper notes, lower fuel-economy standards that result in consumers buying and using more gas, means burning more fossil fuels at a time when we should be doing the opposite. "My office's review of the draft final rule indicates that it utterly fails to provide any demonstrable safety, environmental, or economic benefit to consumers or the country," Carper concludes. "It should be abandoned. At a minimum, I seek your commitment that you will not allow the finalization of this extreme and unlawful environmental rollback in any form that even remotely resembles" the current draft.

Slashdot reader JoshuaZ writes:
In a major breakthrough in quantum computing it was shown that MIP* equals RE. MIP* is the set of problems that can be efficiently demonstrated to a classical computer interacting with multiple quantum computers with any amount of shared entanglement between the quantum computers. RE is the set of problems which are recursive; this is essentially all problems which can be computed.

This result comes through years of deep development of understanding interactive protocols, where one entity, a verifier, has much less computing power than another set of entities, provers, who wish to convince the verifier of the truth of a claim. In 1990, a major result was that a classical computer with a polynomial amount of time could be convince of any claim in PSPACE by interacting with an arbitrarily powerful classical computer. Here PSPACE is the set of problems solvable by a classical computer with a polynomial amount of space. Subsequent results showed that if one allowed a verifier able to interact with multiple provers, the verifier could be convinced of a solution of any problem in NEXPTIME, a class conjectured to be much larger than PSPACE. For a while, it was believed that in the quantum case, the set of problems might actually be smaller, since multiple quantum computers might be able to use their shared entangled qubits to "cheat" the verifier. However, this has turned out not just to not be the case, but the exact opposite: MIP* is not only large, it is about as large as a computable class can naturally be.

This result while a very big deal from a theoretical standpoint is unlikely to have any immediate applications since it supposes quantum computers with arbitrarily large amounts of computational power and infinite amounts of entanglement.

The paper in question is a 165 tour de force which includes incidentally showing that the The Connes embedding conjecture, a 50 year old major conjecture from the theory of operator algebras, is false.

The university underemployed more than 1,000 students -- primarily undergraduates in computer science and engineering -- in order to avoid paying union benefits, UAW Local 2865 says. From a report: The University of California at Berkeley owes student workers $5 million in back pay, a third-party arbitrator ruled on Monday, teaching assistants at the university say. More than 1,000 students -- primarily undergraduates in Berkeley's electrical engineering and computer science department -- are eligible for compensation, the United Auto Workers (UAW) Local 2865, which represents 19,000 student workers in the University of California system, told Motherboard. In some cases, individual students will receive around $7,500 per term, the union says. "This victory means that the university cannot get away with a transparent erosion of labor rights guaranteed under our contract," Nathan Kenshur, head steward of UAW Local 2865 and a third-year undergraduate math major at Berkeley, told Motherboard.

Thanks to their union contract, students working 10 hours a week or more at Berkeley are entitled to a full waiver of their in-state tuition fees, $150 in campus fees each semester, and childcare benefits. (Graduate students also receive free healthcare.) But in recent years, Berkeley has avoided paying for these benefits, according to UAW Local 2865. Instead, the university has hired hundreds of students as teaching assistants with appointments of less than 10 hours a week. On Monday, an arbitrator agreed upon by the UAW and the university ruled that Berkeley had intentionally avoided paying its student employees' benefits by hiring part-time workers. It ordered the university to pay the full tuition amount for students who worked these appointments between fall 2017 and today, a press release from the union says.

According to physicist Jason Steffen, letting slower passengers board airplanes first actually results in a more efficient process and less time before takeoff. An anonymous reader shares a report from Ars Technica: Back in 2011, Jason Steffen, now a physicist at the University of Nevada, Las Vegas, became intrigued by the problem and applied the same optimization routine used to solve the famous traveling salesman problem to airline boarding strategies. Steffen fully expected that boarding from the back to the front would be the most efficient strategy and was surprised when his results showed that strategy was actually the least efficient. The most efficient, aka the "Steffen method," has the passengers board in a series of waves. "Adjacent passengers in line will be seated two rows apart from each other," Steffen wrote at The Conversation in 2014. "The first wave of passengers would be, in order, 30A, 28A, 26A, 24A, and so on, starting from the back."

Field tests bore out the results, showing that Steffen's method was almost twice as fast as boarding back-to-front or rotating blocks of rows and 20-30 percent faster than random boarding. The key is parallelism, according to Steffen: the ideal scenario is having more than one person sitting down at the same time. "The more parallel you can make the boarding process, the faster it will go," he told Ars. "It's not about structuring things as much as it is about finding the best way to facilitate multiple people sitting down at the same time." Steffen used a standard agent-based model using particles to represent individual agents. This latest study takes a different approach, modeling the boarding process using Lorentzian geometry -- the mathematical foundation of Einstein's general theory of relativity. Co-author Sveinung Erland of Western Norway University and colleagues from Latvia and Israel exploited the well-known connection between microscopic dynamics of interacting particles and macroscopic properties and applied it to the boarding process. In this case, the microscopic interacting particles are the passengers waiting in line to board, and the macroscopic property is how long it takes all the passengers to settle into their assigned seats. The paper has been published in the journal Physical Review E.

Sabine Hossenfelder, research fellow at the Frankfurt Institute for Advanced Studies, writes: What we have here in the foundation of physics is a plain failure of the scientific method. All these wrong predictions should have taught physicists that just because they can write down equations for something does not mean this math is a scientifically promising hypothesis. String theory, supersymmetry, multiverses. There's math for it, alright. Pretty math, even. But that doesn't mean this math describes reality. Physicists need new methods. Better methods. Methods that are appropriate to the present century. And please spare me the complaints that I supposedly do not have anything better to suggest, because that is a false accusation. I have said many times that looking at the history of physics teaches us that resolving inconsistencies has been a reliable path to breakthroughs, so that's what we should focus on. I may be on the wrong track with this, of course.

Why don't physicists have a hard look at their history and learn from their failure? Because the existing scientific system does not encourage learning. Physicists today can happily make career by writing papers about things no one has ever observed, and never will observe. This continues to go on because there is nothing and no one that can stop it. You may want to put this down as a minor worry because -- $40 billion dollar collider aside -- who really cares about the foundations of physics? Maybe all these string theorists have been wasting tax-money for decades, alright, but in the large scheme of things it's not all that much money. I grant you that much. Theorists are not expensive. But even if you don't care what's up with strings and multiverses, you should worry about what is happening here. The foundations of physics are the canary in the coal mine. It's an old discipline and the first to run into this problem. But the same problem will sooner or later surface in other disciplines if experiments become increasingly expensive and recruit large fractions of the scientific community. Indeed, we see this beginning to happen in medicine and in ecology, too.

An anonymous reader quotes Ars Technica: The next big thing in 3D printing just might be so-called "4D materials" which employ the same manufacturing techniques, but are designed to deform over time in response to changes in the environment, like humidity and temperature. They're also sometimes known as active origami or shape-morphing systems. MIT scientists successfully created flat structures that can transform into much more complicated structures than had previously been achieved, including a human face. They published their results last fall in the Proceedings of the National Academy of Sciences...

MIT mechanical engineer Wim van Rees, a co-author of the PNAS paper, devised a theoretical method to turn a thin flat sheet into more complex shapes, like spheres, domes, or a human face. "My goal was to start with a complex 3-D shape that we want to achieve, like a human face, and then ask, 'How do we program a material so it gets there?'" he said. "That's a problem of inverse design..." van Rees and his colleagues decided to use a mesh-like lattice structure instead of the continuous sheet modeled in the initial simulations. They made the lattice out of a rubbery material that expands when the temperature increases. The gaps in the lattice make it easier for the material to adapt to especially large changes in its surface area. The MIT team used an image of [19th century mathematician Carl Friedrich] Gauss to create a virtual map of how much the flat surface would have to bend to reconfigure into a face. Then they devised an algorithm to translate that into the right pattern of ribs in the lattice.

They designed the ribs to grow at different rates across the mesh sheet, each one able to bend sufficiently to take on the shape of a nose or an eye socket. The printed lattice was cured in a hot oven, and then cooled to room temperature in a saltwater bath.

And voila! It morphed into a human face.
"The team also made a lattice containing conductive liquid metal that transformed into an active antenna, with a resonance frequency that changes as it deforms."

A reader shares a report from SciTechDaily: MIT mathematicians and engineers have developed a mathematical model that predicts how stable a knot is, based on several key properties, including the number of crossings involved and the direction in which the rope segments twist as the knot is pulled tight. "These subtle differences between knots critically determine whether a knot is strong or not," says Jorn Dunkel, associate professor of mathematics at MIT. "With this model, you should be able to look at two knots that are almost identical, and be able to say which is the better one." "Empirical knowledge refined over centuries has crystallized out what the best knots are," adds Mathias Kolle, the Rockwell International Career Development Associate Professor at MIT. "And now the model shows why."
[...]
In comparing the diagrams of knots of various strengths, the researchers were able to identify general "counting rules," or characteristics that determine a knot's stability. Basically, a knot is stronger if it has more strand crossings, as well as more "twist fluctuations" -- changes in the direction of rotation from one strand segment to another. For instance, if a fiber segment is rotated to the left at one crossing and rotated to the right at a neighboring crossing as a knot is pulled tight, this creates a twist fluctuation and thus opposing friction, which adds stability to a knot. If, however, the segment is rotated in the same direction at two neighboring crossing, there is no twist fluctuation, and the strand is more likely to rotate and slip, producing a weaker knot. They also found that a knot can be made stronger if it has more "circulations," which they define as a region in a knot where two parallel strands loop against each other in opposite directions, like a circular flow.

By taking into account these simple counting rules, the team was able to explain why a reef knot, for instance, is stronger than a granny knot. While the two are almost identical, the reef knot has a higher number of twist fluctuations, making it a more stable configuration. Likewise, the zeppelin knot, because of its slightly higher circulations and twist fluctuations, is stronger, though possibly harder to untie, than the Alpine butterfly -- a knot that is commonly used in climbing. The findings have been published in the journal Science.

A computer made from strands of DNA in a test tube can calculate the square root of numbers up to 900. New Scientist reports: Chunlei Guo at the University of Rochester in New York state and colleagues developed a computer that uses 32 strands of DNA to store and process information. It can calculate the square root of square numbers 1, 4, 9, 16, 25 and so on up to 900. The DNA computer uses a process known as hybridization, which occurs when two strands of DNA attach together to form double-stranded DNA.

To start, the team encodes a number onto the DNA using a combination of ten building blocks. Each combination represents a different number up to 900, and is attached to a fluorescence marker. The team then controls hybridization in such a way that it changes the overall fluorescent signal so that it corresponds to the square root of the original number. The number can then be deduced from the color. The DNA computer could help to develop more complex computing circuits, says Guo. Guo believes DNA computers may one day replace traditional computers for complex computations. The findings have been published in the journal Small.

Long-time Slashdot reader theodp writes: A Microsoft blog post notes the company has lined up K-12 educators to sing the praises of Minecraft Education Edition at the Future of Education Technology Conference, where it'll also be pitching Microsoft Education in general. A 2019 Recap of Minecraft: Education Edition (and an accompanying video) highlight Microsoft's success in getting teachers to use Minecraft to teach subjects across the K-12 curriculum, not just Hour of Code tutorials. Microsoft's ambitions for Minecraft were tipped in a 2015 press release, which included the lofty claim that "Minecraft has the power to transform learning on a global scale...."

There are some teacher walkthrough videos available for review, like the unlisted one for Math Bed Wars! , a Common Core-aligned Minecraft-based lesson that teaches multiplication commutativity ("Students build arrays to show commutative properties of multiplication while constructing defenses as part of a Minecraft mini-game"). The lesson plan for Math Bed Wars! warns that children who fail to get enough hands-on Minecraft play time aren't likely to get much of a math education:

"While there is not much actually doing of math in the section of the lesson plan, it is by far the most important. It is in the game play where they get its meaning, and deeper thinking happens. For example, they will start thinking how to use math to build strategically. However, the most important part is what it does for the students' engagement across math. So please give them at least 30 minutes of game play, even if you have to break up the lesson into two days."

Is it okay for schools to make children play Microsoft Minecraft if the kids want to learn math and other subjects?

Schools are increasingly adopting a "one-to-one" policy of giving each child a digital device -- often an iPad -- and most students in the U.S. now use digital learning tools in school. There's near-universal enthusiasm for technology on the part of educators. Unfortunately, the evidence is equivocal at best. Some studies have found positive effects, at least from moderate amounts of computer use, especially in math. But much of the data shows a negative impact. It looks like the most vulnerable students can be harmed the most by a heavy dose of technology -- or, at best, not helped. Why are these devices so unhelpful for learning? Various explanations have been offered. When students read text from a screen, they absorb less information than when they read it on paper, for example. But there are deeper reasons, too. Unless we pay attention to these, we risk embedding a deeper digital divide.

Guillaume Lample and Francois Charton, at Facebook AI Research in Paris, say they have developed an algorithm that can calculate integrals and solve differential equations. MIT Technology Review reports: Neural networks have become hugely accomplished at pattern-recognition tasks such as face and object recognition, certain kinds of natural language processing, and even playing games like chess, Go, and Space Invaders. But despite much effort, nobody has been able to train them to do symbolic reasoning tasks such as those involved in mathematics. The best that neural networks have achieved is the addition and multiplication of whole numbers. For neural networks and humans alike, one of the difficulties with advanced mathematical expressions is the shorthand they rely on. For example, the expression x^3 is a shorthand way of writing x multiplied by x multiplied by x. In this example, "multiplication" is shorthand for repeated addition, which is itself shorthand for the total value of two quantities combined.

Enter Lample and Charton, who have come up with an elegant way to unpack mathematical shorthand into its fundamental units. They then teach a neural network to recognize the patterns of mathematical manipulation that are equivalent to integration and differentiation. Finally, they let the neural network loose on expressions it has never seen and compare the results with the answers derived by conventional solvers like Mathematica and Matlab. The first part of this process is to break down mathematical expressions into their component parts. Lample and Charton do this by representing expressions as tree-like structures. The leaves on these trees are numbers, constants, and variables like x; the internal nodes are operators like addition, multiplication, differentiate-with-respect-to, and so on. [...] Trees are equal when they are mathematically equivalent. For example, 2 + 3 = 5 = 12 - 7 = 1 x 5 are all equivalent; therefore their trees are equivalent too. These trees can also be written as sequences, taking each node consecutively. In this form, they are ripe for processing by a neural network approach called seq2seq.

The next stage is the training process, and this requires a huge database of examples to learn from. Lample and Charton create this database by randomly assembling mathematical expressions from a library of binary operators such as addition, multiplication, and so on; unary operators such as cos, sin, and exp; and a set of variables, integers, and constants, such as [pi] and e. They also limit the number of internal nodes to keep the equations from becoming too big. [...] Finally, Lample and Charton put their neural network through its paces by feeding it 5,000 expressions it has never seen before and comparing the results it produces in 500 cases with those from commercially available solvers, such as Maple, Matlab, and Mathematica. The comparisons between these and the neural-network approach are revealing. "On all tasks, we observe that our model significantly outperforms Mathematica," say the researchers. "On function integration, our model obtains close to 100% accuracy, while Mathematica barely reaches 85%." And the Maple and Matlab packages perform less well than Mathematica on average. The paper, called "Deep Learning For Symbolic Mathematics," can be found on arXiv.

Cade Metz from The New York Times pays tribute to Tony Brooker, the mathematician and computer scientist who designed the programming language for the world's first commercial computer. Brooker died on Nov. 20 at the age of 94. From the report: Mr. Brooker had been immersed in early computer research at the University of Cambridge when one day, on his way home from a mountain-climbing trip in North Wales, he stopped at the University of Manchester to tour its computer lab, which was among the first of its kind. Dropping in unannounced, he introduced himself to Alan Turing, a founding father of the computer age, who at the time was the lab's deputy director. When Mr. Brooker described his own research at the University of Cambridge, he later recalled, Mr. Turing said, "Well, we can always employ someone like you." Soon they were colleagues.

Mr. Brooker joined the Manchester lab in October 1951, just after it installed a new machine called the Ferranti Mark 1. His job, he told the British Library in an interview in 2010, was to make the Mark 1 "usable." Mr. Turing had written a user's manual, but it was far from intuitive. To program the machine, engineers had to write in binary code -- patterns made up of 0s and 1s -- and they had to write them backward, from right to left, because this was the way the hardware read them. It was "extremely neat and very clever but pretty meaningless and very unfriendly," Mr. Brooker said. In the months that followed, Mr. Brooker wrote a language he called Autocode, based on ordinary numbers and letters. It allowed anyone to program the machine -- not just the limited group of trained engineers who understood the hardware. This marked the beginning of what were later called "high-level" programming languages -- languages that provide increasingly simple and intuitive ways of giving commands to computers, from the IBM mainframes of the 1960s to the PCs of the 1980s to the iPhones of today.

Mathematicians regard the Collatz conjecture as a quagmire and warn each other to stay away. But now Terence Tao has made more progress than anyone in decades. From a report: 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."

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.