30 декабря 2017, 01:45

What "Off The Grid" Indicators Reveal About The True State Of The US Economy

By Nicholas Colas, from DataTrek Research It’s that time of quarter again; today we review our “Off the Grid” economic indicators. And they all look pretty good in terms of launching the American economy into 2018. Pickup truck sales and used car prices remain robust, and there’s some actual inflation in our Bacon Cheeseburger Index. One warning: “Bitcoin” is among the top Google search autofills for the phrase “I want to buy… We started our “Off the Grid” economic indicators in the aftermath of the Financial Crisis as a way to dig deeper into the longer-lasting effects of that event on the American consumer. It seemed to us that standard economic measures like unemployment or CPI inflation missed a lot about the state of the country. So we started gathering up a list of intuitive metrics that could fill those gaps. A few examples from these datasets over the years: #1 Participation in the Supplemental Nutrition Assistance Program (commonly called Food Stamps) went from 26 million Americans in November 2006 to a high of 48 million in late 2012. At that high water mark, 16% of the entire US population needed government support to put food on the table. And since participation in the program is based on income, this meant a substantial portion of the US population was living at/near/below the poverty line. The latest data is more upbeat: as of August 2017 (latest data available), there are 41 million people enrolled in the program. Some of this reduction comes as states return to pre-Crisis rules for program participation, and some comes from rising incomes that allow households to exit the program. Another positive: Google searches for “Food Stamps” are back to pre-Crisis levels after a blip higher in the wake of the hurricanes in Florida and Texas. #2 During the Financial Crisis and its aftermath, Americans bought large amounts of gold and silver coins as a hedge against instability in the banking system. In any given month from 2009 to 2013, the US Mint shipped over $100 million in gold coins and $75 – 100 million in silver coins to dealers for retail sale. Demand for gold and silver coins in the US is now a fraction of those levels, averaging just $15 million and $7 million, respectively, per month in the second half of 2017. Google search volume data confirms the decline in interest, with “Gold coin” queries lower than at any point since 2004 (the start of the time series). #3 Sales of large pickup trucks, most commonly purchased by small businesses, reached a low of 70,000 units a month in early 2009. In November 2017, they were 191,000. Just as important, sales of large pickup trucks have been stable since 2014, growing at mid-single rates even as overall vehicle sales have plateaued. That’s a positive sign – small businesses don’t buy pickups for show. These are work vehicles, and an investment in a new one means they see business conditions remaining strong in 2018. With those three examples, you get the idea: the US economy has not only recovered from the Financial Crisis, but in many ways is firing on all cylinders. Our other OTG indicators generally point to the same conclusion. Here they are: Despite many pundits predicting their decline, used car prices are holding up well. The Manheim Used Vehicle Index (real price data from thousands of auctions) is our data source here. Their November 2017 reading is up 7.8% from last year. And since new car buyers almost always trade in their existing vehicle to buy a new one, higher used car prices effectively lower the cost of purchasing a new vehicle.   New vehicle inventories at dealer lots are currently at a seasonally normal 71 days supply. The caveat here: the hurricanes in Florida and Texas destroyed hundreds of thousands of cars and trucks. Selling rates in Q4 were therefore higher than usual, and “Days supply” is based on current rates. But given a strong economy, we are not overly worried that sales will fall off a cliff in the New Year. The amount of cash spent by the average American on a daily basis is up to $98/day, a post-Crisis high. (Source: Gallup Organization) We’ve seen a lot of change in Google’s autofill suggestions for “I want to buy” and “I want to sell”. Recall that the search engine tries to complete your partially entered query with words commonly used by other users. In Q4 2017, the most common thing users finished “I want to buy” was “a timeshare” (an obviously discretionary purchase). It dethroned “House”, which has been the most common entry since Q2 2015. A word of warning: “Bitcoin” has never made it into the top 4 autofills for “I want to buy”. Until now. We measure visible consumer inflation with our “Bacon Cheeseburger Index (BCI)”, equal weights of ground beef, cheese and bacon price data from the Consumer Price Index data. Good news here for the US Federal Reserve: consumers should start feeling a little more inflation in 2018. After a long bout of cheaper inputs for America’s favorite meal (well, mine anyway), the BCI is up 1.4% year over year. A year ago at this time, it showed a -5.7% decline. Our “Take This Job and Shove It” indicator is also in very healthy territory. This is a measure of quits as a percentage of total separations from the monthly JOLTS data. In October 2017, 61.4% of workers who left their jobs did so with a resignation letter rather than a pink slip. That’s not quite as high as the record 62.2% in September 2016, but still quite strong. And since Quits/Total Separations is a good proxy for Consumer Confidence, that important economic barometer has a full head of steam as we enter 2018. * * * The bottom line here: with few exceptions (Food Stamps, notably), the US economy is in exceptionally strong shape as we enter 2018. Small business confidence is strong, and savers do not see the need to hedge their bank accounts with gold coins. Timeshare salespeople are busy. Inflation that consumers use to anchor their expectations is rising at a modest pace. All that may be “Off the Grid”, but it gives us confidence that the standard – and currently quite bullish - economic data is actually on the mark.  

29 декабря 2017, 17:00

Zacks.com highlights: Sprouts Farmers Market, Thor Industries, Hibbett Sports, Patrick Industries and PetMed Express

Zacks.com highlights: Sprouts Farmers Market, Thor Industries, Hibbett Sports, Patrick Industries and PetMed Express

29 декабря 2017, 16:53

Add These 4 High Earnings Yield Stocks to Your Portfolio

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29 декабря 2017, 14:18

Bet on 4 Stocks with Rising Cash Flows to Scoop Big Gains

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29 декабря 2017, 13:42

Bet on 5 Momentum Stocks Using Driehaus Strategy

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28 декабря 2017, 23:28

Our Top Infographics of 2017

We count down our favorite infographics and charts from the year, including many you may not have seen before. Catch some of our best work in this Top 17 list. The post Our Top Infographics of 2017 appeared first on Visual Capitalist.

28 декабря 2017, 16:19

Zacks.com highlights: Broadcom, Arkema SA, Nutrisystem, Gibraltar Industries and Willdan Group

Zacks.com highlights: Broadcom, Arkema SA, Nutrisystem, Gibraltar Industries and Willdan Group

28 декабря 2017, 16:17

Zacks.com highlights: Marriott International, Western Digital, Morgan Stanley, Southern and TransUnion

Zacks.com highlights: Marriott International, Western Digital, Morgan Stanley, Southern and TransUnion

28 декабря 2017, 15:08

5 DuPont-Tested Winning Stocks

Use the DuPont technique to pick solid profit-generating stocks.

28 декабря 2017, 15:00

How to Highlight Your Talents in a Job Interview Without Showing Off

Bill Eppridge/life/Getty Images In a perfect world, job interviewers would be able to gauge your talents with great accuracy, so that you wouldn’t need to engage in any form of boasting or self-promotion. But talents can be notoriously hard to judge, particularly in short-term interactions with other such as job interviews: there is only a 4% overlap between interview ratings and subsequent job performance ratings. One reason for this elusive nature of talent is that it is quite easy to fake, especially when people are deceived about their own talents. Indeed, fooling others is much easier when you have managed to fool yourself. Unsurprisingly, narcissists and psychopaths can interview really well. Another reason is that those in charge of evaluating talent are not as talented at it as they think, so they over-rely on their intuition and misinterpret key signals: e.g., extraversion for social skills, confidence for competence, and charisma for leadership potential. You and Your Team Series Communication How to Work with a Bad Listener Rebecca Knight 8 Ways to Get a Difficult Conversation Back on Track Monique Valcour Stop Trying to Sound Smart When You’re Writing Liane Davey So what should you do if you are interested in communicating your talents to others without coming across as a show-off or as being somewhat deluded? Here are four simple suggestions to consider: Briefly quantify your most relevant experience: While past behavior doesn’t always predict future behavior— particularly if the context changes—people will want to be reassured that you “have done it before.” Fortunately, this is one of the first questions you will be asked when your talents are being evaluated, so you wont be accused of showing off for simply answering what you are asked. That said, it is essential that you keep it brief. We are living in an age of short attention spans. The risk here is that you get so immersed in your answer that people stop paying attention to what you are saying and make assumptions about your lack of brevity. This could work against you: people who talk too much about themselves are often perceived as self-centered, arrogant, or even narcissistic. It is also likely that if you big up your achievements too much while not being sufficiently specific or concrete, your audience will assume you are exaggerating. It is therefore best to quantify any relevant experience and get to the point quickly: e.g., “I have 18 years of global experience in X area/industry”, “I managed a team of 75 people”, “We grew our business unit by 150%”, or “I have led a division that was responsible for 60% of company revenues”. And if you don’t know what to report, pick what’s most relevant for the role you are being considered for. Speak about your passion rather than your skills: For some reason, it is OK to be enthusiastic about your interests and passions, but not so much about your talents. For example, you can’t expect others to celebrate the fact that you are really impressed with your sense of humor or your (self-rated) leadership skills. Try telling people “I am one of the funniest people in the world” and they will immediately assume you are not—probably rightly so. The same goes for telling them that you are a “great leader” or a “disruptive thinker.” However, if you report that you really enjoy managing teams, thinking of unconventional ways of doing things, or that you always “try to see the funny side of things,” it may at least suggest to others that you could have some talent in those areas. Focus on your potential: Recent research suggests that, when it comes to judging others, we are more interested in their future than their past. In “The Picture of Dorian Gray”, Oscar Wilde famously stated that he prefers women with a past and men with a future, but it seems that in the real world there is not much gender differentiation on this matter. Regardless of their gender, we are generally more interested in people’s futures than their past, not least because the past is already written and cannot be affected by our decisions. Furthermore, inferring the future requires real skills and effort, so it’s a much higher-stakes activity. Thus you can help others speculate about your own future—and place a bet on your talents—by describing the key qualities of your potential. Psychological reviews indicate that there are three main areas to discuss: learning ability, drive, and people skills. Fortunately, talking about these qualities will not be regarded as a blatant attempt to show off, and employers themselves are aware of how important these traits are, for they largely decide whom to employ and promote based on these qualities. The key, however, is not to make generic self-promotional statements — e.g., “I’m a fast learner”, “I’m super driven”, or “I have great people-skills” — but to demonstrate these things with concrete examples (back to point 1) and during the interview. For example, if you have good people skills, you will avoid interrupting the interviewers, speaking for too long, or showing off. And if you want others to believe that you have good learning ability, then talk about concrete difficult problems you have solved or niche expertise you have acquired. Note that while these examples highlight past accomplishments they will invite people to infer your potential and therefore make inferences about your future performance. Turn your fans into advocates: Ultimately, your reputation is made of what others (not you) think of you. It is therefore obvious that you are better off being promoted by others than promoting yourself. While references—such as recommendation letters—are a poor predictor of future performance, they can still play a very important role in determining your success. The right things said by the right person to the right person will matter more than any objective indicator of talent. You should therefore treasure your supporters as much as you can. Moreover, if you can turn your mentors (and fans) into advocates, you will not have to work as hard to promote yourself. Your reputation will always have a bright and a dark side. Turn the custodians of your bright side reputation into your brand ambassadors and opportunities will flourish, especially if your dark side is not public. A final point to remember is that there are strong cultural differences in people’s tolerance to self-promotion, and social etiquette more broadly. Even in countries that are largely similar, the same behaviors will be celebrated in one place, but punished in another. For example, what may come across as normal self-presentation in the U.S. would be considered crass bragging in the U.K., whereas the British way of showing off—fake self-deprication—will often be interpreted as anxious insecurity in the U.S. However, even when the signals or apparent signs may change, in pretty much every culture people will look for the same thing—namely, for people who appear to actually have some talent.

28 декабря 2017, 08:23

Further points on how to understand modern India (from the comments)

Good post. There are a few other topics that can serve as useful handles to “understand” India. 1. Study the folk history of the popular Indian pilgrimage sites – For a lot of people, Hinduism is associated with abstruse metaphysics, mysticism, Vedanta, and Yoga. And this obsession with the high falutin theoretical stuff, means that […] The post Further points on how to understand modern India (from the comments) appeared first on Marginal REVOLUTION.

27 декабря 2017, 16:59

5 Top Stocks with Solid Net Profit Margin

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26 декабря 2017, 17:43

4 Impressive Liquid Stocks for Marvelous Returns

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26 декабря 2017, 17:41

5 Top Stocks with Attractive Sales Growth to Buy Right Away

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26 декабря 2017, 17:39

Bet on These 5 Low Leverage Stocks to Ensure Safe Returns

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26 декабря 2017, 17:22

Bet on These 5 Stocks With Exciting Interest Coverage Ratio

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26 декабря 2017, 17:21

Add Value to Your Investment With These 4 Low P/CF Stocks

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25 декабря 2017, 23:30

The Blockchain and Us (2017)

 In 2008, Satoshi Nakamoto invented bitcoin and the blockchain. For the first time in history, his invention made it possible to send money around the globe without banks, governments or any other intermediaries. The concept of the blockchain isn’t very intuitive. But still, many people believe... [[ This is a content summary only. Visit http://FinanceArmageddon.blogspot.com or http://lindseywilliams101.blogspot.com for full links, other content, and more! ]]

24 декабря 2017, 00:28

The Biggest Math Story of 2017

Mathematics is very unlike any other discipline, and 2017 demonstrated it in spades! Imagine, if you will, what it takes to prove out a hypothesis or conjecture in science. Years of careful and often costly experimentation or observation produces results that, should any tiny detail lead to results that might overturn part or all of the hypothesis and its predictions through new observations and experiments, will lead to the hypothesis being tested more acutely or perhaps continue to be used after having its limitations considered (the best case scenario), shelved, or unceremoniously dumped into the rubbish bin of failed scientific propositions. Starts With a Bang's Ethan Siegel recently lamented that scientific proof is a myth: You've heard of our greatest scientific theories: the theory of evolution, the Big Bang theory, the theory of gravity. You've also heard of the concept of a proof, and the claims that certain pieces of evidence prove the validities of these theories. Fossils, genetic inheritance, and DNA prove the theory of evolution. The Hubble expansion of the Universe, the evolution of stars, galaxies, and heavy elements, and the existence of the cosmic microwave background prove the Big Bang theory. And falling objects, GPS clocks, planetary motion, and the deflection of starlight prove the theory of gravity. Except that's a complete lie. While they provide very strong evidence for those theories, they aren't proof. In fact, when it comes to science, proving anything is an impossibility. The reason that absolute scientific proof is an impossibility is because reality is, in reality, very complicated. Our limitations as real people living in the real world constrains our ability to see, test or even to imagine every possible detail of existence that might affect how reality works. As a result, every genuine scientific theory and hypothesis can potentially be shown to be false, which is a major part of what separates science from pseudoscience. There is only one serious human discipline that doesn't have that limitation: mathematics. One of the most fun descriptions that gets into what mathematical proofs can do that scientific proofs cannot that we saw this year was presented in Thomas Oléron Evans and Hannah Fry's book The Indisputable Existence of Santa Claus: The Mathematics of Christmas: The scientific method takes a theory - in our case that Santa is real - and sets about trying to prove that it is false. Although this may seem a little counterintuitive on the surface, it actually does make a lot of sense. If you go out looking for evidence that Santa doesn't exist and don't find any ... well then, that is pretty revealing. The harder you try, and fail, to show that Santa cannot exist, the more support youy have for your theory that he must. Eventually, when enough evidence has been gathered that all points in the same direction, your original theory is accepted as fact. Mathematical proof is different. In mathematics, proving something "beyond all reasonable doubt" isn't good enough. You have to prove it beyond all unreasonable doubt as well. Mathematicians aren't happy unless they have demonstrated the truth of a theory absolutely, irrefutably, irrevocably, categorically, indubitably, unequivocally, and indisputably. In mathematics, proof really means proof, and once something is mathematically true, it is true forever. Unlike, say, the theory of gravity - hey, Newton? So with the differences between the nature of scientific and mathematical proofs in mind, let's get to the biggest math stories of 2017 where, because we're practical people who live in the real world, we've focused upon the stories where the outcome of maths can make a practical difference to people's lives. Have you ever walked around with an open-topped cup of coffee? If so, you probably have run into the problem of having the contents of your mug slosh and spill out as you attempted to walk with it, which wastes both your precious coffee and, if it gets on your hands, potentially causes burns requiring first aid. Fortunately, 2017 is the year in which mathematics solved the problem of sloshing coffee! Americans drink an average of 3.1 cups of coffee per day; for many people, the popular beverage is a morning necessity. When carrying a liquid, common sense says to walk slowly and refrain from overfilling the container. But when commuters rush out the door with coffee in hand, chances are their hastiness causes some of the hot liquid to slosh out of the cup. The resulting spills, messes, and mild burns undoubtedly counteract coffee's savory benefits. Sloshing occurs when a vessel of liquid—coffee in a mug, water in a bucket, liquid natural gas in a tanker, etc.—oscillates horizontally around a fixed position near a resonant frequency; this motion occurs when the containers are carried or moved. While nearly all transport containers have rigid handles, a bucket with a pivoted handle allows rotation around a central axis and greatly reduces the chances of spilling. Although this is not necessarily a realistic on-the-go solution for most beverages, the mitigation or elimination of sloshing is certainly desirable. In a recent article published in SIAM Review, Hilary and John Ockendon use surprisingly simple mathematics to develop a model for sloshing. Their model comprises a mug on a smooth horizontal table that oscillates in a single direction via a spring connection. "We chose the mathematically simplest model with which to understand the basic mechanics of pendulum action on sloshing problems," J. Ockendon said. But that's not the best part of the story! That part comes from where the idea for the study originated, which illustrates Evans and Fry's point of the extremes to which mathematicians will go that scientists do not: The authors derive their inspiration from an Ig Nobel prize-winning paper describing a basic mechanical model that investigates the results of walking backwards while carrying a cup of coffee.... The authors evaluate this scenario rather than the more realistic but complicated use of a mug as a cradle that moves like a simple pendulum. To further simplify their model, they assume that the mug in question is rectangular and engaged in two-dimensional motion, i.e., motion perpendicular to the direction of the spring's action is absent. Because the coffee is initially at rest, the flow is always irrotational. "Our model considers sloshing in a tank suspended from a pivot that oscillates horizontally at a frequency close to the lowest sloshing frequency of the liquid in the tank," Ockendon said. "Together we have written several papers on classical sloshing over the last 40 years, but only recently were we stimulated by these observations to consider the pendulum effect." The Ockendons focused on rectangular containers in and indicate that their mathematical study may be extended to include cylindrical cups in the future, but rest assured, the mathematical work won't stop there until cups of every possible geometry have been considered! Still, while the maths that might lead to preventing sloshing and spilling may be terribly important for coffee drinkers, it's not the biggest math story of the year, so we much continue our search! In the sport of basketball, the practical ability to skillfully place a round ball through an elevated circular ring with netting attached to it is one that can determine whether a player earns millions of dollars a season as a professional athlete or does not. Sometimes however, the limits of skilled players seem to stretch as they appear have an easier time in consistently shooting baskets than at other times, where they seem to have what's called a "hot hand" as they exceed their usual level of performance over an extended period of time during a game. Scientists and statisticians have studied this apparent phenomenon over the years and have chalked it all up to randomness, where statistically speaking, it is something that periodically happens when a player's performance is much closer to the long end of the tails of a normal distribution describing their performance than it is to their mean performance, where the apparent "hot hand" is little more than a cognitive illusion as people who see it are really being fooled by randomness. That consensus view was challenged in a paper by Joshua Miller and Adam Sanjurjo, who instead argue that earlier finding was based on a misreading of the math of probabilities. Our surprising finding is that this appealing intuition is incorrect. For example, imagine flipping a coin 100 times and then collecting all the flips in which the preceding three flips are heads. While one would intuitively expect that the percentage of heads on these flips would be 50 percent, instead, it's less. Here's why. Suppose a researcher looks at the data from a sequence of 100 coin flips, collects all the flips for which the previous three flips are heads and inspects one of these flips. To visualize this, imagine the researcher taking these collected flips, putting them in a bucket and choosing one at random. The chance the chosen flip is a heads – equal to the percentage of heads in the bucket – we claim is less than 50 percent. If flip 42 were heads, then flips 39, 40, 41 and 42 would be HHHH. This would mean that flip 43 would also follow three heads, and the researcher could have chosen flip 43 rather than flip 42 (but didn't). If flip 42 were tails, then flips 39 through 42 would be HHHT, and the researcher would be restricted from choosing flip 43 (or 44, or 45). This implies that in the world in which flip 42 is tails (HHHT) flip 42 is more likely to be chosen as there are (on average) fewer eligible flips in the sequence from which to choose than in the world in which flip 42 is heads (HHHH). This reasoning holds for any flip the researcher might choose from the bucket (unless it happens to be the final flip of the sequence). The world HHHT, in which the researcher has fewer eligible flips besides the chosen flip, restricts his choice more than world HHHH, and makes him more likely to choose the flip that he chose. This makes world HHHT more likely, and consequentially makes tails more likely than heads on the chosen flip. In other words, selecting which part of the data to analyze based on information regarding where streaks are located within the data, restricts your choice, and changes the odds. Similar mathematical reasoning applies for the statistics behind the counterintuitive phenomenon described by the Monty Hall problem. It's a really cool insight, although one that we're afraid has limited potential for practical application, which is why we cannot call this the biggest math story of the year. The field of mathematics is notorious for developing conjectures that defy proof for centuries. 2017 saw the delivery of a formal proof of the Kepler Conjecture, which identifies the maximum density by which spherical objects of equal size can be packed together within a given space, which was first proposed by Johannes Kepler in 1611. In the real world, the results can be seen anywhere spherically-shaped objects are packed together, such as oranges packed into a rectangular crate, which if optimally packed according to the Kepler Conjecture, will mean that a little over 74% of the available space will be filled by orange, while the rest of the space would be empty. 306 years later, a team of 19 researchers led by Thomas Hales appears to have finally cracked it and published a formal proof that can be confirmed by mathematician referees. Or rather, by their computers, because Hales' team's proof is so sufficently complex that modern computing technology is the only way that humans have to verify the findings. In 2003, Hales anticipated that it would take 20-person years of labor for computers to verify every step of the proof in launching the project that finally delivered the proof 14 calendar years later. Not every mathematical conjecture involving discrete geometry endures for centuries however. Some only last for decades, as was the case in Zilin Jiang's and Alexandr Polyanskii's proof of László Fejes Tóth’s zone conjecture, which says that if a unit sphere is completely covered by several zones, their combined width is at least equal to the irrational mathematical constant pi. At first glance, these kinds of proofs may not seem to to have terribly practical applications, but effective solutions to discrete geometry problems like these do have real world impact. Discrete geometry studies the combinatorial properties of points, lines, circles, polygons and other geometric objects. What is the largest number of equally sized balls that can fit around another ball of the same size? What is the densest way to pack equally sized circles in a plane, or balls in a containing space? These questions and others are addressed by discrete geometry. Solutions to problems like these have practical applications. Thus, the dense packing problem has helped optimize coding and correct mistakes in data transmission. A further example is the four-color theorem, which says that four colors suffice to plot any map on a sphere so that no two adjacent regions have the same color. It has prompted mathematicians to introduce concepts important for graph theory, which is crucial for many of the recent developments in chemistry, biology and computer science, as well as logistics systems. And also secure, garble-free, long distance (including interplanetary) communications, to name an up-and-coming application that would be an outcome for doing this kind of math! Perhaps the biggest mathematical breakthrough honored in 2017 was the discovery by Maryanthe Malliaris and Saharon Shelah that two different variants of infinity, long thought to be different in nature, are actually equal in size. In a breakthrough that disproves decades of conventional wisdom, two mathematicians have shown that two different variants of infinity are actually the same size. The advance touches on one of the most famous and intractable problems in mathematics: whether there exist infinities between the infinite size of the natural numbers and the larger infinite size of the real numbers. The problem was first identified over a century ago. At the time, mathematicians knew that “the real numbers are bigger than the natural numbers, but not how much bigger. Is it the next biggest size, or is there a size in between?” said Maryanthe Malliaris of the University of Chicago, co-author of the new work along with Saharon Shelah of the Hebrew University of Jerusalem and Rutgers University. In their new work, Malliaris and Shelah resolve a related 70-year-old question about whether one infinity (call it p) is smaller than another infinity (call it t). They proved the two are in fact equal, much to the surprise of mathematicians.... Most mathematicians had expected that p was less than t, and that a proof of that inequality would be impossible within the framework of set theory. Malliaris and Shelah proved that the two infinities are equal. Their work also revealed that the relationship between p and t has much more depth to it than mathematicians had realized. Surprise is the right word, because Malliaris' and Shelah's result is very counterintuitional. And incredibly cool. But alas, not the biggest math story of 2017! There is a class of mathematical problems named after Diophantus of Alexandria that are, unsurprisingly, known as Diophantine equations, whose components are made up of only sums, products, and powers in which all the constants are integers, and where the only solutions of interest are expressed as either integers or as rational numbers. If you think back to when you might have taken a class in algebra and recall those really wicked polynomial equations that you encountered or had to factor, that's the kind of problem that we're talking about. Wicked being the operative word, because there's a really difficult Diaphantine equation that mathematicians have been working to solve for over four decades called the "cursed curve", where they've been seeking to prove that the equation only has a limited number of rational solutions. In November 2017, a team of mathematicians succeeded. Last month a team of mathematicians — Jennifer Balakrishnan, Netan Dogra, J. Steffen Müller, Jan Tuitman and Jan Vonk — identified the rational solutions for a famously difficult Diophantine equation known as the “cursed curve.” The curve’s importance in mathematics stems from a question raised by the influential mathematician Jean-Pierre Serre in 1972. Mathematicians have made steady progress on Serre’s question over the last 40-plus years, but it involves an equation they just couldn’t handle — the cursed curve. (To give you a sense of how complicated these Diophantine equations can get, it’s worth just stating the equation for the cursed curve: y4 + 5x4 − 6x2y2 + 6x3z + 26x2yz + 10xy2z − 10y3z − 32x2z2 − 40xyz2 + 24y2z2 + 32xz3 − 16yz3 = 0.) In 2002 the mathematician Steven Galbraith identified seven rational solutions to the cursed curve, but a harder and more important task remained: to prove that those seven are the only ones (or to find the rest if there are in fact more). The authors of the new work followed Kim’s general approach. They constructed a specific geometric object that intersects the graph of the cursed curve at exactly the points associated to rational solutions. “Minhyong does very foundational theoretical work in his papers. We’re translating the objects in Kim’s work into structures we can turn into computer code and explicitly calculate,” said Balakrishnan, a mathematician at Boston University. The process proved that those seven rational solutions are indeed the only ones. "Kim's general approach" in this case refers to the work of the University of Oxford's Minhyong Kim, who has been working to apply concepts derived from the science of physics to the solution of difficult mathematical problems. The proof for the seven solutions of the cursed curve is an exciting development for number theory, where the intersection of physics and mathematics brings us up to the biggest math story of the year. We began this article with a discussion of the main difference between the standards of scientific proof and mathematical proof. Nowhere in 2017 is that difference more on display than in the biggest math story of the year, in which mathematicians have demonstrated that the famed Navier-Stokes equations that describe the flow of fluids in the real world, break down under "certain extreme conditions". The Navier-Stokes equations capture in a few succinct terms one of the most ubiquitous features of the physical world: the flow of fluids. The equations, which date to the 1820s, are today used to model everything from ocean currents to turbulence in the wake of an airplane to the flow of blood in the heart. While physicists consider the equations to be as reliable as a hammer, mathematicians eye them warily. To a mathematician, it means little that the equations appear to work. They want proof that the equations are unfailing: that no matter the fluid, and no matter how far into the future you forecast its flow, the mathematics of the equations will still hold. Such a guarantee has proved elusive. The first person (or team) to prove that the Navier-Stokes equations will always work — or to provide an example where they don’t — stands to win one of seven Millennium Prize Problems endowed by the Clay Mathematics Institute, along with the associated $1 million reward. Mathematicians have developed many ways of trying to solve the problem. New work posted online in September raises serious questions about whether one of the main approaches pursued over the years will succeed. The paper, by Tristan Buckmaster and Vlad Vicol of Princeton University, is the first result to find that under certain assumptions, the Navier-Stokes equations provide inconsistent descriptions of the physical world. By "inconsistent descriptions of the physical world", Buckmaster's and Vicol's work is pointing to the situation where, when given exactly the same fluid and starting conditions, instead of providing a single, unique solution for what the flow of fluids will be at a particular point of time in the future, the Navier-Stokes equations will instead provide two or more non-unique solutions, where they cannot accurately predict the future state of the resulting fluid flow. That puts the physics associated with the Navier-Stokes equations into the situation where physicists and engineers must consider the limitations of where they can be shown to be valid in using them in their applications. Since Navier-Stokes is considered to be the "gold standard of the mathematical description of a fluid flow", that finding could potentially disrupt a whole lot of apple carts in multiple sciences and engineering disciplines whose applications might approach those limits. That disruptive potential is why the story of the mathematical proof of the ability of the Navier-Stokes equations to describe unique solutions for all fluid flows under all conditions counts as being the biggest math story of 2017! Previously on Political CalculationsThe Biggest Math Story of the Year is how we've traditionally marked the end of our posting year since 2014. Here are links to our previous editions, along with our coverage of other math stories during 2017: The Biggest Math Story of the Year (2014) The Biggest Math Story of 2015 The Biggest Math Story of 2016 The Biggest Math Story of 2017 Connecting the Lonely Dots The Map of Mathematics Illegal Numbers in America Big Data Gone Bad for Fighting Crime Think Outside of the Box Composition of Functions Babylonian Trig Powers of Ten Computing All the Prime Numbers from Here to Infinity How to Use Math to Optimally Divide Up Resources Have a Merry Christmas, and we'll see you again in the New Year!

23 декабря 2017, 18:15

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