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Home»February»Discover Dialogue: Mathematician Peter Woit Discover Dialogue: Mathematician Peter Woit"No one has a plausible idea about how string theory can explain anything."By Susan Kruglinski|Monday, February 20, 2006 RELATED TAGS: STRING THEORYPeter Woit is a lecturer in Columbia University's mathematics department. He launched a blog, called Not Even Wrong, two years ago to contest the underpinnings of string theory. It was a hit and has become a lively forum for bickering scientists. You have a Ph.D. in physics. Why did you end up in a math department? W: Well, one reason actually had to do with string theory. After I received my Ph.D. in theoretical physics, it became clear that if you wanted to keep working in theoretical physics, especially in the mathematical end of theoretical physics, you would pretty much have to do string theory. And I really wasn't very interested in that, so I thought the math department would be a better idea. What year was this? W: I got my Ph.D. in '84, which is right around the time the string theory fad started. One effect of it was if you were doing something else involving mathematics and physics, people just don't want to hear about it, they were just not interested. If you started looking for a job, you found out that nobody really wants what you're doing. Why are you so interested in the problems with string theory? W: In the mid-eighties, my reaction to it was it didn't seem that promising to me and there were all these other smart people doing it. I thought, within a couple of years either it will get somewhere, in which case they'll all want to work on it, or it won't go anywhere and they'll give up. It seemed like a perfectly reasonable thing for people to be doing. And as the years went on—and we're now 21 years past this—it became more and more disturbing that this had taken on a very different character than just a few years of people working on a very speculative idea. It had reached this kind of critical mass and totally had taken over the field. I think much of it has really gotten to the point where it's not even a legitimate science anymore. How is it not a legitimate science? W: At this point they really don't even have a plausible idea about how to ever make a prediction out of this, or how to use this in order to really explain anything about the world. So there's an ongoing discussion now almost at the level of philosophy of science: Is this even a science? I think I am not the only one who thinks this has gone past the point where it's not even really a science anymore. And yet they are clearly working with math, which is scientific. How do you describe what they are doing? W: The science writer John Horgan has a nice line about this. He calls it science fiction in mathematical form. They are certainly using mathematics, and they are building models and writing down equations for them, but the models they are working with just aren't connected to the real world. There isn't even any plausible way you could imagine that they are going to be able to connect that to the real world and to use these models to explain some experiment we are seeing. Even string theorists admit that it is not really a theory. What is it? W: The best way to say it is what people have now is really an approximation to a theory. The kinds of equations that they have now are the kinds of equations you would get in an approximation scheme to some underlying theory, but nobody knows what the underlying theory is. What about the positive things that have come out of string theory? W: It has had a very good effect on mathematics. It's gone through several different stages. In the mid-eighties, it went through something called conformal field theory, and some truly great mathematics was done because of that. And more recently there has been a lot of work in what's called topological string theory and this has led to a lot of fantastic mathematics recently, and a bit earlier on there was something called mirror symmetry which give you a completely new information about higher dimensional spaces. So from the point of view of mathematics, it's been a big success. Also, over the last ten years, there has been some very interesting work using string theory to understand the theory of the strong interactions. So if string theory is so useful, what are your issues? W: I think what most bothers me about it, the problem with it, is the way it has driven out other sorts of research. The way in which it has been pursued has made it virtually impossible to work on other things in the field. Why is string theory such a phenomenon? W: One of the big factors is that the field is a victim of its own success. The standard model, which was in place by around 1973, has been absurdly successful. There are literally zero experimental results that disagree with this model. Normally, one thing that has kept physics from becoming overly speculative or going off into the wrong direction is that sooner or later an experimentalist comes along and shows you that that was the wrong direction. That just hasn't happened. Another aspect has to do with Edward Witten, who is the most amazing figure in the field. He legitimately is an incredible genius. And throughout the early eighties he was doing stuff that was ten times more interesting than anyone else in the field. No one had ever seen anyone like him. He got interested in this in 1984, and he was pushing the idea very strongly. So I think it was a combination of by far the most influential person in the field pushing the idea very strongly, combined with the fact that there weren't any other good ideas around, and there wasn't anything from experiments telling us which way to go. By now you have several generations of physicists—this has been going on for more than 20 years—who have spent their whole career doing this. People don't like to give up on something they have their lives invested in and try something else. It's not human nature. Doesn't string theory fit a physics need? W: Certainly the reason people originally got interested in it was that it held out hopes of unifying the standard model in particle physics and general relativity, the theory of gravitation. And I think there are still some people who believe in that promise. The other thing to say about string theory is that nobody really quite knows what it is. It's still a very mysterious business. It really is more of a set of hopes that some things that people already understand are an approximation of some deeper theory, although nobody knows what it is. Why is your blog called Not Even Wrong? W: It's a famous quote from Wolfgang Pauli, a well-known physicist from the earlier part of the century. I certainly wasn't the first one to use it or the first one even to apply it to string theory. Pauli supposedly was asked about some paper, and he just described it as, "That paper is not even wrong." Meaning that it is just so undefined that you can't even tell if it is wrong or not. It's appropriate for string theory. It's so ill-formulated that you really can't tell whether it's wrong or not. Sometimes people use it as a term of abuse, as in, "That's so stupid, it's not even wrong." But it also has an implication that something is not well-defined enough that you can even decide whether it is wrong or not. What inspired you to start a blog? W: Since I'm responsible for the computer system in the department, I'd played a little bit with the software. I thought, I've got a lot of things I want to say and this may be a good forum with which to do this. So I guess it was about a year and a half ago that I started it. I was immediately surprised at how much attention it started getting. Had you felt you were not getting much reaction for earlier articles on this subject? W: I wrote something and initially tried to get Physics Today to publish it, but they wouldn't, and then I put it up on a physics web archive, and I got quite an amazing reaction to it. I probably heard from at least 50 to 100 physicists who said, "We've always thought this. It's great that someone is saying this." And I heard from only 2 or 3 people who said, "You don't know what you're talking about." So there was a huge positive reaction. How many hits do you get on your blog? W: Very quickly it was a few hundred, and then a thousand, then a couple thousand, and now its up to about 5,000 per day. Do you know of people who have changed their mind about what they were pursuing in physics because of your blog? W: People have told me that my blog has had an effect on students who were trying to decide whether to go into string theory or not. Brian Greene, one of the most prominent advocates of string theory, is in your department. How has your high profile as an anti-string theorist affected your relationship? W: We get along fine. He's a very reasonable guy. We disagree about string theory to some extent, but unlike a lot of other string theorists, Brian is certainly someone who is willing to publicly admit that string theory is something that may very well be wrong. Have you influenced each other at all? W: The few times I ever thought there was something to string theory was after hearing Brian talk. He's certainly a very convincing speaker. I haven't talked to him much about it, so I have no idea whether I've succeeded in making him more skeptical about the prospects of string theory or not. This is not something you sit down and talk about? W: No, we haven't discussed it very much at all. It seems people's attitude toward string theory has changed recently. Is that true? W: I think definitely up until last year or so it was very rare to see anything skeptical about string theory. I think within the last year this idea has really gotten out there, has become much more commonplace, that there quite possibly is a problem with all of this. Any idea why the climate has changed? W: Well, one thing is, in string theory itself, things are really not going well. Regularly, every year or two, there would be some new idea they would come out with, so the field was kind of bubbling along. I think what's happened in the last two or three years, there really haven't been any new ideas to try. The other problem is that there's been a split within the string theory community. There are some who have basically decided that whatever this theory is, it has infinitely complex possible solutions [known as the string theory landscape]. As for the dream that there's going to be one solution of string theory and it's going to be the real world, I think a lot of them have given up on that. So they're trying to pursue this idea that string theory really is an infinitely complex thing. I think a lot of other string theorists are well aware that if you go down that road you really can't predict anything and you're in danger of leaving what is normal science. Do you think there was a backlash because string theory was pushed into the public spotlight too early or too hard? W: I think it was certainly oversold. I think if you talk to most string theorists, they actually see themselves as suffering from this overselling of the theory. They're actually not happy that they have to contend with that. Can you compare this time in physics to another time in physics? Is this an exciting time? W: I'm afraid I don't think it is an exciting time. It's a very difficult and challenging time. I think it really is an unparalleled situation in the history of physics. There is this new accelerator that will come online in Geneva at CERN, the LHC, Large Hadron Collider. And there are a lot of good reasons to believe that that accelerator will finally have enough energy to start to get some interesting data. And so I think what's going on among theorists is people are just kind of waiting and hanging in there for a couple of years and hoping that this will really finally return the field to a much healthier state, where there will be new experimental results coming in which will start telling us what direction to go in. So I think when you do hear people saying that this is an exciting time, often what they are referring to is that it's exciting that two or three years from now we're going to finally start getting some new data which will turn things around. How do you respond to critics who say you are unqualified to discuss string theory? W: I have a PhD from Princeton in particle theory and have been thinking about this subject for twenty years. The only criticism of what I've been doing over the years that I would actually agree with is that I should have spent less time thinking about string theory and complaining about it and learning about it just to criticize it, instead of devoting time to more positive things that I should be pursuing. To you, in a perfect world, what would today's field of physics be like? How would you imagine it would be functioning at its best capacity? W: Broadly, I think what the field needs is for people to acknowledge that this particular speculative idea doesn't really work and there aren't any obvious good ideas out there. I think it would actually be healthier for theoretical physics these days to take a look at how mathematicians operate, because mathematics has always been a less faddish subject. In mathematics there is much more of a culture where people spread out and devote their lives to thinking hard about something that interests them. There has always been much more of a culture in physics that you want to work on something where you can get results and produce a paper a few months from now. And when the problems are very hard and no one knows what to do, I think people need to be willing to dig in and spend years thinking about something different than what other people are thinking. And there really isn't the kind of institutional support within the physics community for this kind of behavior, whereas there is in mathematics. I think it would be a lot healthier if physicists would acknowledge that they are in a different situation than we've been in historically because of this lack of experiment, and under the circumstances people do need to behave differently. Is part of it that, often, mathematicians are not searching for a larger meaning? Do you think that because physicists are looking for larger meaning, it's throwing off their objectivity? W: The thing that keeps mathematicians honest is the notion of mathematical rigor and mathematical proof. Mathematicians have this very strong culture where everything you say has to be very precise and you have to be able to rigorously prove it. Whereas physicists have never worried too much about that because even if you are doing something that is logically a bit inconsistent, it doesn't really matter because it is all going to get sorted out by the experiment in the end. I think physicists need to spend a lot more time being clear about exactly what's working and what isn't. We have to think about what really is beautiful -- what really is a powerful, beautiful new idea and what isn't. Do you think physicists have lost the idea of what is beautiful and powerful because they have gotten carried away? W: I see what's happening in the field is that they are getting backed into a corner. They've got this underlying speculative idea that they don't want to give up on. If an idea doesn't work, you can always make it try to work by making it more complicated, so a lot of what is going on is that they're being forced into more complicated and more ugly things, to the extent that some of them are even trying to make a virtue of this. They're saying, well, the world really is this incredibly complicated, ugly, Rube Goldberg kind of place. That's just the way it is.You might also likeDiscover Interview: The Math Behind the Physics Behind the UniverseThe Extremely Long Odds Against the Destruction of EarthNothingness of Space Could Illuminate the Theory of EverythingThe Large Hadron Collider Will Finally Start Smashing in SeptemberTesting String TheoryDiscover Interview: Roger Penrose Says Physics Is Wrong, From String Theory to Quantum MechanicsGravity LeaksPhysicists Are Close to Uncovering the Fundamental Rules of Reality3 Ideas That Are Pushing the Edge of Science

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Search this site: Plus Blog Mathematical theatre at the Science Museum: X&Y What is the shape of the Universe? Is it finite or infinite? Does it have an edge? In their new show X&Y Marcus du Sautoy and Victoria Gould use mathematics and the theatre to navigate the known and unknown reaches of our world. Through a series of surreal episodes, X and Y, trapped in a Universe they don't understand and confronted for the first time with another human being, tackle some of the biggest philosophical and scientific questions on the books: where did the Universe come from, does time have an end, is there something on the other side, do we have free will, can we ever prove anything about our Universe for sure or is there always room for another surprise? Marcus and Victoria met while working on A disappearing number, Complicite's multi award-winning play about mathematics. X&Y has developed from that collaboration and pursues many of the questions at the heart of A disappearing number. X&Y is on at the Science Museum in London 10 - 16 October 2013. Click here to book tickets. You can read about A disappearing number, an interview with Victoria Gould and several articles by Marcus du Sautoy on Plus. The Magic Cube: Get puzzling! If you like the Rubik's cube then you might love the Magic Cube. Rather than having colours on the little square faces it has number on it. So your task is not only to put the large faces together in the right way, but also to figure out what this right way is. Which numbers should occur together on the same face and in what order? Jonathan Kinlay, the inventor of the Magic Cube, has estimated that there are 140 x 1021 different configurations of the Magic Cube. That's 140 followed by 21 zeroes and 3000 more configurations than on an ordinary Rubik's cube. To celebrate the launch of the Magic Cube, Kinlay's company Innovation Factory is running a competition to see who can solve the cube first. To start it off they will be shipping a version the puzzle directly to 100 of the world's leading quantitative experts, a list that includes people at MIT, Microsoft and Goldman Sachs. You can join too by nominating yourself (or someone else). Innovation Factory will accept up to 20 nominees (in addition to those that have already been picked). The competition will launch in September and run for 60 days. To nominate someone please send an email to [email protected], giving the name and email, mailing address of the nominee and a brief explanation of why you think they should be included in the competition. If you don't get accepted, don't worry — the Magic Cube will go on sale after the competition has ended. The winner will receive lots of glory and a metal version of the Magic Cube precision-machined from solid aluminium, and they will be featured on the Innovation Factory website. As a warm-up you can read about the ordinary Rubik's cube on Plus. The paper galaxy

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Past recipients have included: Energy Access Foundation, an organization that increases access to clean and renewable energy through rural energy enterprises Aqua Para La Vida, an organization that works in rural Nicaragua to build safe drinking water and sanitation systems Trees, Water, People, a group dedicated to helping communities protect, conserve and manage natural resources IDEA WILD, a group dedicated to helping preserve the earth’s biodiversity Union of Concerned Scientists, a group dedicated to improving the environment Marine Conservation Biology Institute (MCBI), a group dedicated to advancing the science of marine conservation biology Sustainable Ecosystems Institute (SEI), a group that uses science-based, cooperative solutions to maintain natural ecosystems and the human communities that depend on them Conversation Law Foundation, an organization working to solve significant environmental challenges facing New England Sustainability Mini-ReviewsBiodiversity Mini-ReviewsDeep Ocean Mini-ReviewsEnergy Mini-ReviewsNon-Government Organizations (NGOs) Mini-Reviews Rise of the NGO Fifty years ago, about four dozen non-governmental organizations (NGOs) had a voice at United Nations conferences. Since then, these groups have proliferated both in number and influence to the point they are now major players in the environmental arena. In 1992, representatives of 13,000 NGOs convened in Rio de Janeiro for a UN-sponsored Earth Summit-the largest ever gathering of environmentalists and heads of state. The Summit yielded the Convention on Biological Diversity-an important treaty, ratified by 178 nations, which calls for the assessment and protection of threatened plants and animals. “That conference would not have taken place if NGOs hadn’t pushed for it,” comments William Moomaw, an environmental policy specialist at Tufts University. “Without NGOs”, he adds, “there wouldn’t be a Kyoto Protocol aimed at curbing the release of greenhouse gases. Society needs a voice that’s not always provided by governments or the corporate sector, which is why NGOs have flourished.” Today there are more than 20,000 NGOs. These organizations have become a “third force” on the world stage, taking “their place at the table of business and governments,” according to Harvard biologist E.O. Wilson. When Wilson advised conservation groups in the 1970s, their role was “basically that of beggars and evangelists,” raising awareness about problems in the hopes that someone would follow through. By the 1990s, groups such as the Nature Conservancy and Conservation International had the clout to buy up large tracts of land. Actions by both private organizations and governments have brought about 9 percent of the world’s land mass and 1 percent of its waters under some form of protection. Over the decades, NGOs like the World Wildlife Fund and Natural Resources Defense Council have grown from shoestring operations into empires with a million or more members and multi-million dollar budgets. Along with the change in size has come a different attitude. “People got tired of the gloom and doom approach,” says Bud Ris, executive director of the Union of Concerned Scientists from 1984 through 2003. “They wanted to hear about solutions.” Getting results today often entails a very different approach. “In the early days, the process was adversarial because that was the only way to get anyone’s attention. But some companies have finally gotten the message,” Moomaw says, leading to unlikely alliances between environmental groups and their former rivals in industry. The Conservation Law Foundation, for example, is now forging partnerships with companies that, a decade ago, they might have sued. Minireviews are written by Steve Nadis, a writer based in Cambridge, Massachusetts who has worked for the Union of Concerned Scientists and the World Resources Institute. His articles have appeared in Nature, Scientific American, the Atlantic Monthly and other magazines. He was a 1997/98 Knight Science Journalism Fellow at MIT. Protecting the Amazon About one third of the world’s tropical rainforests lie in the Amazon Basin, an ecologically-rich region that is home to more than a fifth of the known plant and animal species. But the Amazon is disappearing fast: About 15 percent of its original forests are gone, and deforestation continues at a frightening clip. Ten thousand square miles of rainforest-more than twice the size of the state of Connecticut-were lost from July 2001 through June 2002, according to a satellite image survey carried out by the Brazilian government. Drastic measures are needed to save this region of unparalleled biological diversity, and the Amazon Region Protected Areas (ARPA) initiative, which was launched in September 2002, is aggressively confronting the challenge. Under ARPA, the Brazilian government has pledged to set aside at least 10 percent of its Amazon land within a decade, thereby safeguarding over 190,000 square miles-an area larger than the entire U.S. National Park System. The government asked the World Wildlife Fund (WWF), the world’s largest privately-funded conservation group, to help devise the conservation plan. WWF and other participating organizations proposed a plan that included strictly-protected parks and managed reserves where indigenous people will be allowed to hunt, fish and farm on a limited basis. WWF and its partners are now identifying new Amazon regions that should be considered for federal protection. Tumucumaque National Park, which was created in 2003, is the biggest single addition to ARPA. At 15,000 square miles, Tumucumaque is the world’s largest tropical rainforest reserve-six times bigger than the Florida Everglades. No major roads have yet been cut into the pristine forest where jaguars, sloths, harpy eagles and other creatures are thought to roam. Information furnished by local indigenous tribes equipped with Global Positioning System handsets was combined with aerial photos to create the most detailed map of an Amazon region ever produced. Later in 2003, the 2,600 square mile Chandless State Park, which provides habitat for rare spider monkeys and endangered species such as the jaguar and Goeldi’s tamarin, was added to ARPA. Within a half year of the project's inauguration, more than 20,000 square miles of Brazilian land came under protection. Advocates of the initiative hope momentum is gathering. “Nothing like ARPA has ever been attempted before,” says WWF vice president Guillermo Castilleja. “WWF, together with the other partners in this program, share a vision to make ARPA the most successful large-scale forest conservation effort in history.” NGO Minireviews are written by Steve Nadis, a writer based in Cambridge, Massachusetts who has worked for the Union of Concerned Scientists and the World Resources Institute. His articles have appeared in Nature, Scientific American, the Atlantic Monthly and other magazines. He was a 1997/98 Knight Science Journalism Fellow at MIT. Protecting the Cardamoms For decades, the Cardamom Mountains in southwestern Cambodia remained off limits to most people. The dense forested mountains-scented with the sweet spice for which they’re named-had been the last stronghold of the brutal Khmer Rouge regime. Ironically, the presence of the armed guerillas, along with the minefields and booby traps they laid, helped make the Cardamoms a largely untouched wilderness area. The Khmer Rouge movement collapsed in the late 1990s, leaving its mountain sanctuary up for grabs. The cash-poor Cambodian government sold five logging concessions to timber companies for parcels in the central mountains. Several roads had been cut into the heart of the range, paving the way for farmers, hunters, and settlers. That’s when Conservation International (CI) stepped in. The Washington-based conservation giant, which is helping to protect more than 100 million acres worldwide, helped finance a Cardamom wildlife survey in 2000 conducted by Flora and Fauna International, a British NGO. The survey confirmed ecologists’ suspicions that the region is a biological treasure trove. Covering just six percent of Cambodia, the Cardamoms are home to most of the country’s large mammals and also shelter about half of its birds, reptiles and amphibians. Threatened species found there include the Indochinese tiger, the Asian elephant, the Malaysian sun bear, the pileated gibbon and the Siamese crocodile. Even rarer species such as the Javan rhinoceros and the khiting vor-a bizarre half-sheep, half-antelope creature thought to exist though never seen before-are rumored to inhabit the mountain slopes. Convinced of the area’s importance, CI struck a deal with the Cambodian government in 2001 that prohibited logging in the mountains while their permanent status was being determined. A year later, Cambodia announced the creation of a million-acre protected forest in the central Cardamoms. The new park abuts two existing sanctuaries, adding up to a combined preserve of 2.44 million acres. It is the largest and most pristine wildlife refuges in the Southeast Asia mainland, covering nearly one-fourth of the total mountain range. CI’s work is not yet done. It now hopes to expand conservation corridors that would link the Cardamoms with the coast, thereby securing key elephant habitats. CI is also paying 50 forest rangers, at a cost of about $250,000 per year, to patrol the park and prevent poaching and illegal logging. The organization knows from experience that creating a wilderness preserve is just the first step; a continued presence is required to make it last. Room to Roam: Yellowstone To Yukon National Parks, even spacious ones like Yellowstone Park (the largest in the United States), are not big enough to accommodate predators like grizzly bears and wolves. Radio-collar tracking studies have shown that a single wolf can rove over a 60,000 square mile area. Other research indicates that the 400 or so grizzly bears thought to occupy Yellowstone need an area 10 times bigger than the park’s confines to thrive. Roads and other obstacles further limit the bears’ range. Small, isolated populations can become inbred; the loss of genetic hardiness brings a greater susceptibility to disease and famine. A new breed of conservationist believes the traditional “patchwork” approach of cordoning off scattered wildlife enclaves is far from optimal. “Little island parks,” claims Canadian environmental lawyer Harvey Locke, can become “islands of extinction.” A decade ago, Locke and other conservationists conceived of the Yellowstone to Yukon (Y2Y) plan for creating a continuous, 2,000 mile seam of protected or otherwise managed land, half a million square miles in all, that stretches from Yellowstone to the Yukon Territories. This “bright green thread” championed by Y2Y proponents will tie wilderness areas together by means of wildlife corridors at least 30 miles wide. Y2Y spokesperson Jeff Gailus calls the effort “the biggest, boldest conservation initiative in history.” He and his colleagues are taking a long-term perspective, thinking in terms of a 50- to 100-year process. In the meantime, they’re trying to encourage the U.S. and Canadian governments-as well as environmental groups, big and small-to preserve additional land or, failing that, to make sure land management practices conform to the needs of wildlife. It’s an ambitious task, admits Gailus, “but we’re not starting from scratch.” The Y2Y plan incorporates 11 existing national parks, along with dozens of other parks and protected areas. And the team is steadily expanding the base: The government of British Columbia recently protected 24,000 square miles in the northeastern section of the province as part of an agreement with native Americans, wildlife advocates and industry representatives. In a similar arrangement, Canada has begun to protect nearly 40,000 square miles in the Mackenzie Valley. Meanwhile, conservationists are trying to establish wildlife-friendly links between the Glacier-Waterton National Park complex and Banff and Jasper to the north. With a staff of just 10 people operating out of offices in Canmore, Alberta and Missoula, Montana, the Y2Y organization cannot begin to manage the vast sweep of land under consideration. The success of Y2Y rests instead on collaborations with 180 other participating groups. “People aren’t working alone anymore to protect their own river, mountain, or valley,” says Gailus. “We’re all working together toward a greater vision, which makes it even more rewarding.” The Last Great Places: The Nature Conservancy Palmyra Atoll-a string of islands in the center of the Pacific, 1,000 miles south of Hawaii-had, by some miracle, emerged from the 20th century virtually intact, without being ruined, or colonized by humans. Still, plans were afoot to “develop” this tropical paradise, a U.S. territory. One proposal called for building a resort and casino; another scenario envisioned a spent nuclear fuel repository. The Nature Conservancy (TNC), unenthused about either prospect, came up with a simple strategy for saving Palmyra. In November 2000, they bought the whole package-680 acres of land and 15,500 acres of coral reefs and lagoons-for $30 million. Some might consider that an unusual approach to conservation, but it’s standard practice for TNC, which has already protected more than 116 million acres of land and waters worldwide. As part of the largest private conservation campaign ever initiated, the group has recently pledged to spend $1 billion to save 200 of the “Last Great Places on Earth,” Palmyra being one of them. In 2001, the U.S. Fish and Wildlife Service established the Palmyra Atoll National Wildlife Refuge to protect the atoll and its surrounding waters. Palmyra was a coveted addition to the national refuge system because it contains one of the last undeveloped atolls in the entire Pacific and some of the most spectacular corals found anywhere, as well as hosting diverse marine species. More than 130 different hard corals grow there-three times more species than are found in the Caribbean Sea or in all the Hawaiian Islands put together. The islands also provide a habitat for more than one million nesting seabirds-their only sanctuary within 450,000 square miles of ocean. Palmyra supports one of the world’s largest colonies of red-footed boobies, second only to the Galapagos Islands. Migratory birds, such as the bristle-thighed curlew, make their first rest stop there while passing through from Alaska. Seabirds are by no means the only visitors to Palmyra. Pilot whales, bottled-nosed dolphins, tiger sharks, manta rays, sea turtles, giant clams, parrot fish, lumphead wrasses and groupers are also well represented. Through its purchase, TNC is trying to ensure that the atoll remains a wildlife haven for the indefinite future. In so doing, various commercial plans for the site, including the construction of an offshore bank, manufacturing center, fish processing plant, and missile launch site have been thwarted. Although TNC is in favor of progress, they believe that sometimes means keeping things the way they are. Call for Calm in the Seas: Natural Resources Defense Council In the past half-century, the oceans have become at least 10 times noisier due to the growing din from shipping, mining, drilling, military sonar systems and other human activities. The problem of underwater noise pollution is not widely known to terrestrial creatures like humans. But to ocean denizens such as whales and dolphins-who rely on sound to communicate with one another, navigate, avoid peril and find food-a vastly noisier world would be hard to ignore. Marine mammals certainly have their champions in environmental circles to the extent that the “save the whales” rallying cry has long been a cliche. But on the noise pollution front, the Washington-based Natural Resources Defense Council (NRDC), has taken the lead. NRDC’s efforts in this area are largely due to the initiative of one man, senior attorney Joel Reynolds. In 1994, Reynolds heard about clandestine experiments the U.S. Navy was conducting off the California coast. After questioning military officials and scientists, and wading through volumes of documents during a nine-month investigation, Reynolds learned about a Navy technique for detecting enemy submarines through the use of “low-frequency active” (LFA) sonar. The Navy had tested the technology repeatedly without studying how it affected marine life, despite its own calculations showing the sonar emitted sound levels of 140 decibels-comparable to a space-shuttle launch-300 miles from the source. The Navy also failed to obtain permits mandated by the Marine Mammal Protection and Endangered Species Acts. In response to pressure from Reynolds, the Navy agreed to study the effects of LFA on marine life. The agency later admitted that one of its mid-frequency sonar systems contributed to the March 2000 deaths of at least eight Cuvier’s beaked whales that had beached themselves in the Bahamas. Reynolds and NRDC won a major lawsuit in October 2003, when a federal court ruled that the Navy could only test its LFA system in a limited area of the North Pacific, with additional restrictions imposed to protect migratory species. But the battle is far from over. The Navy has appealed its loss-a move NRDC has vowed to fight. The successful lawsuit is only a first step, Reynolds explains, as it only applies to the U.S. Navy-not to other countries who are developing similar systems-and only pertains to low-frequency sonar. Since the problem of undersea noise is international in scope, NRDC has recently launched an international program to combat it-a measure that may offer some relief to whales and dolphins whose movements are not constrained by political boundaries. Protecting Fisheries: Conservation Law Foundation The waters off New England once teemed with millions of cod-a term that encompasses 10 families and more than 200 species of fish worldwide. The abundance of cod lured Pilgrims to New England in the early 1600s and kept those settlers from starving, helping to convert the region into an economic powerhouse. For centuries, the sea’s bounty appeared limitless, but all that changed in the last century with the advent of technology that enabled people to remove fish from the seas faster than they could reproduce. By the 1990s, populations of cod, haddock, flounder and other “groundfish” dropped to the lowest levels ever recorded, signaling the collapse of famed fisheries in the Gulf of Maine and Georges Bank. A Boston-based advocacy group, Conservation Law Foundation (CLF), is working to mitigate this environmental disaster. CLF, a group founded by lawyers, is not afraid to use the arm of the law to achieve its ends. The organization began investigating the state of fisheries in the late 1980s in response to pleas from fishermen for help in stemming the decline in fish stocks. CLF filed a lawsuit in 1991 after concluding that a management plan endorsed by the National Marine Fisheries Service (NMFS) did little to address overfishing. The settlement led to a more stringent policy and the unprecedented closures of fisheries. In 2000, CLF was the lead agency in another suit against NMFS for failing to prevent overfishing and for sanctioning inadequate fishery rebuilding plans. Two fishing groups intervened on behalf of the conservation coalition. CLF scored a big victory, as did the cause of marine conservation, when the federal court ruled in its favor a year later. CLF is now waiting to see whether NMFS’s new management plan complies with the Sustainable Fisheries Act. “If the right plan is approved, the outlook is very good for many, though not all, groundfish species,” says Priscilla Brooks, who heads CLF’s Marine Resources Project. Populations of haddock and yellowtail flounder have almost fully rebounded, although cod stocks remain low as overfishing persists. “Our goal is to reduce pressure on cod and hope it responds positively,” says Brooks. “Once we can get stocks to come back up, the trick then will be to establish sustainable fishing practices so we don’t drive them back down again.” Her group vows to closely monitor the status of overtaxed groundfish, which constitute a New England treasure. CLF realizes that vigilance alone will not save the day, but without a watchful eye, we may soon bid farewell to cod and other prized species. An Urban Success Story: Charles River Watershed Association Once regarded as a glorified sewer separating Boston from Cambridge, the Charles River is on the mend. According to the Charles River Watershed Association (CRWA), which monitors the river daily, the waterway is clean enough for swimming, except after heavy downpours when some sewage seeps in through storm drains. Within a few years, the Charles could be “fishable and swimmable” every summer day, regardless of storms-at which point the river of “Dirty Water” fame will become a thing of the past. “This is an urban success story,” claims Kari Dolan, a researcher with the National Wildlife Federation. Much of the credit for this remarkable turnaround goes to CRWA, one of the nation’s first watershed organizations, which was formed in 1965 to address concerns over the river’s declining state. In 1995, the Environmental Protection Agency (EPA) gave the river a “D” for water quality, but since 1998, the Charles has received “B’s”-largely because CRWA has identified pollution sources by measuring bacteria and chemical toxins at various sites, while the EPA has shut down those sources. It has been a fruitful partnership, with CRWA providing the research that guides the cleanup process and EPA supplying the enforcement muscle. Although the Charles has come a long way, there’s still much to be done. Going from a “B” to an “A” in EPA ratings will be harder than going from “D” to “B”, mainly due to all the contaminants lodged in river sediments. Removing them would require dredging-a costly job. For CRWA, the cleanup itself is just half the story. The group is equally focused on preserving the river flow through strategies that keep rainwater in the Charles River watershed, a 300-square-mile basin, rather than discharging it through sewers into Boston Harbor. Their agenda goes beyond fishing and swimming, setting the broader goal of restoring the watershed through innovative water conservation and recovery techniques. CRWA’s efforts are clearly paying off. There are other measures of success than EPA report cards, one being the increased presence of blue herons, osprey, marsh hawks, turtles and other forms of wildlife along the river and its banks. “The Charles was written off for 50 years, but most Bostonians don’t write it off anymore,” says the group’s executive director Robert Zimmerman. “It’s important to put nature on display in urban areas so people realize the environment is not confined to places like Yellowstone.” Restoring Wetlands: The Wetlands Institute, The Trustees of Reservations Since the 1800s, the 273-mile-long Illinois River has been dammed, channelized and otherwise transformed, serving at various times as a canal between the Mississippi River and the Great Lakes, as well as a sewer for Chicago. To redress more than a century of abuse, The Wetlands Initiative (TWI), a Chicago-based non-profit group, recently began restoring 2,600 acres of Illinois River floodplain near Hennepin, Illinois-the first step in TWI's plan to revive broad stretches of wetlands alongside the river and its tributaries. If TWI realizes its goals, the project will be the largest wetlands restoration in the history of Illinois, a state that has already lost 90 percent of its wetlands. Work at Hennepin got started in 2001 after TWI purchased the floodplain site that once supported two lakes (Hennepin and Hopper) as well as wetland, prairie and fen communities. Since the 1920s, however, the land had been pumped dry to permit corn and soybean cultivation. TWI turned off the pumps, allowing precipitation and groundwater to refill the lake beds. In response, the adjacent wetlands and marshes quickly sprang back to life. Frogs, birds and plants returned after nearly a century’s absence. The sounds of Western chorus frogs, American toads and spring peepers were also heard. Muskrats and beavers began reshaping the landscape, and state-threatened bird species, such as the pied-bill grebe and black tern, magically appeared. Meanwhile, a more modest effort is underway in Hingham, Massachusetts to resuscitate a 15-acre salt marsh that spans a narrow peninsula called World’s End. It’s not a big undertaking as restoration ventures go, but salt marshes are endangered ecosystems, and every marsh that can be revived can yield environmental benefits. The Trustees of Reservations (TTOR), the non-profit land conservation organization which owns the land, hope to inspire similar projects along the East Coast. A once-thriving salt marsh was drained for agriculture in the early 1600s by the first European settlers, who installed two dikes to keep saltwater out of the area. In 2003, TTOR took steps to reverse some of the centuries-long damage inflicted by humans. The first priority, restoring tidal flows into and out of the former marsh, was accomplished by installing four-foot by eight-foot concrete culverts in the middle of each dike. Since then, dense stands of phragmites, an invasive reed, have retreated, making way for native salt marsh grasses. A popular destination for hikers and picnickers, World’s End now offers unique opportunities for witnessing a wetland restoration in progress, says TTOR ecologist Andy Walsh. “People can come here to see a coastal zone in transition-a salt marsh coming back to life.” Defending Bats and Insects: Bat Conservation International, Xerces Society In 1992, a spelunker named Steve Smith searched for artifacts in an abandoned mine in Iron Mountain, Michigan, just before the mine was scheduled to be sealed from the public for good. Instead of uncovering artifacts, Smith found thousands of bats and realized they would soon be entombed. Smith alerted Bat Conservation International (BCI) in Austin, Texas. Merlin Tuttle, who started the group in 1982, immediately flew to Michigan where he saw signs of a million brown bats hibernating in the mine. Tuttle persuaded officials from the town and mining company to jettison their plans for covering the mine’s entrances with backfill. Instead, steel gates-with bars spaced widely enough to let bats fly through, while keeping people out-were installed at the entrances. That decision saved a million bats from certain death and protected their home, prompting BCI to look at other mines throughout the country. In 1993, BCI and the U.S. Bureau of Land Management started the North American Bats and Mines Project to prevent the loss of bats due to the closure of abandoned mines. The program is important because more than half of North America’s 46 bat species find sanctuary in mines, after having been driven from traditional roosts in caves and forests. Nearly 2,000 bat-friendly gates have been installed to date, at a cost of roughly $5,000 apiece. BCI has been successful in getting mining companies to share the expenses. The Xerces Society, like BCI, is focused on preserving noncharismatic creatures like insects. “We’re equal opportunity, so long as it doesn’t have a backbone,” says the group’s executive director Scott Hoffman Black. Xerces petitioned the U.S. Fish and Wildlife Service a few years ago when it learned that fewer than 200 butterflies called Carson wandering skippers were left in this country, confined to two sites in California and Nevada. In 2000, the agency protected the skipper under the Endangered Species Act, which provided a legal mandate for preserving its habitat. Xerces, which has a staff of just six, is now trying to save an orange, black, and white checkered butterfly called the Taylor’s checkerspot, whose prairie habitat in the Pacific Northwest has shrunk by more than 99 percent and whose population now numbers in the hundreds. The group is conducting surveys to see where the butterfly lives, while also working with land trusts-the Nature Conservancy as well as smaller, local organizations-to safeguard critical sites. “It’s a combination of advocacy, land management, and science,” says Xerces staffer Matthew Shepherd, “wrapped up in one small butterfly with a two-inch wingspan.” NEB Nature Conservancy: Trees, Water & People Trees, Water & People (TWP) is dedicated to improving people’s lives by helping communities to protect, conserve and manage the natural resources upon which their long-term well being depends. To that end, TWP has developed ongoing watershed protection, sustainable agriculture, reforestation and appropriate technology programs in cooperation with communities throughout Central America. Throughout Central America, native forests are being felled at an alarming rate. If current deforestation trends are not slowed, many scientists agree that the region could be completely void of native forests in the next century. Such massive deforestation causes severe soil erosion, water degradation, loss of wildlife habitat and therefore precious biodiversity. While it is essential to protect natural resources and prevent deforestation, the forest is a resource essential to the survival of human populations. Millions of families throughout Central America rely on forests and land to provide fuel to cook meals, wood to build homes, water to drink and space to live and grow crops. Programs designed to help these communities meet their current needs without endangering long-term ecosystem health are essential. Since establishment in 1998, Trees, Water & People has helped hundreds of low-income communities to balance the needs of human populations with the long-term health of the ecosystem that supports them. In less than six years, TWP and community volunteers have planted nearly 1,000,000 trees, helping to prevent soil erosion, protect native forests and ensure the future of regional biodiversity. TWP staff and local volunteers have also provided sustainable agriculture and watershed protection training to more than 50,000 community members. Finally, in order to compliment their reforestation and watershed protection programs, TWP also works with families to introduce improved cooking stoves that reduce the demand for fuel wood by approximately 70%. These stoves are a huge improvement over traditional open fire cooking stoves still used by most of Central America’s rural population. Not only do they save forests and the life they support, but by removing all smoke from the home via a chimney, they protect families from life-endangering respiratory ailments caused by indoor air pollution. If you are interested in learning more about Trees, Water & People’s work in Central America, please contact: Trees, Water & People 633 Remington Street http://www.treeswaterpeople.org [email protected] LeadershipNews and Press ReleasesCertificationsEnvironmental CommitmentBusiness Development OpportunitiesInternational Ordering & SupportCareersContact UsResearch at NEB Home ›About NEB ›Environmental Commitment ›Global Conservation ›Non-Government Organziations (NGOs) Mini-Reviews

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On-Line Recreation Permits Mapping Products Jobs/USAJOBS Land Use Planning/NEPA Our Offices/Centers Eastern States Operations Center Nat'l Training Center BLM > Arizona > On-Line Recreation Permits > Paria Canyon Wilderness Area/Vermilion Cliffs National Monument > Paria Canyon Permit Area > Prehistory & History Print Page Vermilion Cliffs National MonumentParia Canyon/Vermilion Cliffs WildernessParia Canyon Permit Area - Prehistory & History People of Paria Canyon - Those Who Came BeforeThe Paria River flows intermittently from its headwaters in Bryce Canyon National Park through Utah and Arizona to deposit silt and sediment in the Colorado River. The river basin has been used by various cultures for thousands of years.Some archaeologists believe Paria Canyon was inhabited for at least 10,000 years before the first Europeans arrived in the 1770s. The nature of its geography probably made Paria Canyon a travel route between what is now southern Utah and northern Arizona.From about A.D. 200 to A.D. 1200, the Anasazi occupied the region and may have had small farms and granaries in some stretches of the canyon. Paiute people later occupied and traversed much of the area 600 years before the first Europeans entered the mouth of the canyon. The word "Paria" is Paiute, and may mean water that tastes salty.Most archaeological sites located in Paria Canyon are petroglyphs or rock art sites. Petroglyphs are images or symbols carved, incised, or pecked into the rocks while pictographs are painted on the rock. No habitations or large villages have been found in the canyon, leading researchers to conclude that the canyon was primarily used as a travel route in prehistoric times.More Recent ResidentsTwo missionaries, Fathers Dominguez and Escalante, were the first documented European visitors in the region in 1776. They attempted to establish a route from Santa Fe, New Mexico, to Monterey, California. On their return route to Santa Fe, after failing to reach California, the expedition camped several days at the mouth of Paria Canyon, which would become the site of Lees Ferry 100 years later. They succeeded in exiting the canyon through what is now Dominguez Pass, which is located high on the northeast rim of the lower canyon. (Dominguez-Escalante Expedition Site) What to ExpectPlan and PreparePack It In, Pack It OutCamping ResponsiblyThe Wilderness Experience InformationCampgroundsClimate and WeatherDriving DirectionsGuidesMaps & GuidebooksShuttle ProvidersTrailheadsPhoto GalleryFAQs Natural WondersPrehistory and HistoryGeologyVegetationWildlife PermitsHow to Obtain a PermitObtain a Permit Note: If you are unable to access the "Obtain" link above, please check your security settings.Click "Tools" on the menu bar, then click "internet options", the "advanced" tab, scroll down to the Security Section and check the box "Use TLS 1.0." Click "OK" and try to access it again. Lees Ferry was established in 1871 by John D. Lee, who was the first settler and operator of the ferry. Lee's diary mentioned how rugged Paria Canyon was in the late 1800s. Crampton and Rusho write that Lee, during his eight-day trip through Paria Canyon, struggled for two days and one night without stopping because a safe place to camp out of flash flood danger could not be found."We concluded to drive down the creek (Paria), which took us some eight days of toil, fatigue, and labour through brush, water, ice, and quicksand -- without seeing the sun for 48 hours."John D. LeeUtah State Historical SocietyIn 1870, the small settlement of Pahreah, which was located northwest of the modern-day Paria Contact Station, included 47 families, a church and a post office. This frontier settlement, like many in the West, was frequented by Native Americans, pioneers and the occasional outlaw. John Wesley Powell, the first director of the United States Geological Survey, in surveying the region, used the spelling Paria, which is the name found on modern topographical maps.Jacob Hamblin, envoy for the Church of Jesus Christ of Latter Day Saints, reported his observations about Pahreah on March 27, 1870."The settlement was progressing with a guardhouse and a small corral, where men can cook and lodge safely with 20-25 horses." (Feduska 6)Pahreah TownsiteUtah State Historical SocietyBy that time, Pahreah also had vegetable farms, fruit and nut orchards, and cattle.Unfortunately, severe flooding during the 1880s brought alkaline soil and entrenched arroyos and by 1889, only eight families remained at the settlement. By the 1930s the town of Pahreah had vanished. Near the abandoned settlement, now a ghost town being slowly swept away by the river, a western movie set was built. Here famous characters like Calamity Jane, Buffalo Bill and others came to life. Zane Grey, a famous western writer during the 1920s, had some of his novels filmed in the area, including Revelation, Heritage of the Desert, and A Biography of Buffalo Jones. Today, the ghost town and movie set can be visited by traveling 30-miles east from Kanab along U.S. Highway 89.The Arizona Strip, a portion of Arizona geographically isolated from the rest of the state by the Colorado River, has always been a difficult area to access.According to Crampton and Rusho (1992), Zane Grey described Paria Creek in 1907 thus:"Dawn opened my eyes to what seemed the strangest and most wonderful place in the world. Paria Creek watered this secluded and desert bound spot."Zane GreyJane FosterCrampton and Rusho (1992) also wrote that south of Paria Canyon in House Rock Valley, two men named Uncle Jim Owens and Buffalo Jones, established a Buffalo Ranch in the early 1900s."The original intent of the ranch was to produce hybrid offspring from buffalo and cattle called cattalo."AZ Game & Fish DepartmentThe attempt failed, but today the buffalo herd is managed by the Arizona Game and Fish Department. The Buffalo Ranch is 22-miles south of U.S. Highway 89A, and can be reached by USFS Road 8910.Water DevelopmentWater is the essence of life in the West. According to James J. Ligner of the U. S. Geological Survey, the gaging station on the Colorado River at the mouth of the Paria River is:"The most important station in the United States." (Reilly, 1997).This remote gaging station was important in the development of the Colorado River Compact of 1922, an agreement among western states to divide the Colorado River into an upper basin, which is located north of its confluence with the Paria River, and a lower basin south of the confluence. The compact allocates the water from the Colorado River to individual states. Vermilion Cliffs National Monument Monument Manager: Kevin Wright345 E. Riverside Drive St. George, UT 84790-6714 (435) 688-3200 Hours: 7:45 a.m.-5:00 p.m. Monday through Friday 10:00 a.m.-3:00 p.m. Saturday Closed Sunday

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close Do not display this message againClose window Math Matters ­ Maplesoft PRODUCTS Maple Home : Math Matters - A Brief Look at How Mathematics has Influenced Modern Life. --- Make Selection --- Water Supply & Distribution Petroleum & Petrochemicals High Performance Materials Health Technologies Agricultural Mechanization Laser & Fiber Optics In 2003, the National Academy of Engineering (USA) published A Century of Innovation: Twenty Engineering Achievements that Transformed Our Lives. This book celebrates the top twenty technological advances of the twentieth century that fundamentally changed society. These advances have influenced where and how we live, what we eat, what we do for work or leisure, and even how we think about our world and the universe. Mathematics has played a major role in bringing these innovations to reality. Many mathematical theories and models of real world problems have helped scientists and engineers grapple with seemingly impossible tasks. Today, mathematical techniques reach even further into our society. In addition to making technology more efficient and effective, mathematical techniques help organizations deal with financial, manufacturing, and even marketing issues. This poster is a tribute to the National Academy of Engineering as well as the men and women who have focused their brilliance to transform the modern world. The poster is a mosaic of the ways mathematics helps us utilize and benefit from these great technological achievements. Some achievements will be familiar. Some will be a surprise. All, hopefully, will encourage you to investigate these topics further. 1 George Constable and Bob Somerville, A Century of Innovation: Twenty Engineering Achievements That Transformed Our Lives (Washington: National Academies Press, 2003). The content on this poster does not necessarily reflect the position or views of the National Academy of Engineering, and no official endorsement should be inferred.

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Search this site: Plus Blog Mathematical theatre at the Science Museum: X&Y What is the shape of the Universe? Is it finite or infinite? Does it have an edge? In their new show X&Y Marcus du Sautoy and Victoria Gould use mathematics and the theatre to navigate the known and unknown reaches of our world. Through a series of surreal episodes, X and Y, trapped in a Universe they don't understand and confronted for the first time with another human being, tackle some of the biggest philosophical and scientific questions on the books: where did the Universe come from, does time have an end, is there something on the other side, do we have free will, can we ever prove anything about our Universe for sure or is there always room for another surprise? Marcus and Victoria met while working on A disappearing number, Complicite's multi award-winning play about mathematics. X&Y has developed from that collaboration and pursues many of the questions at the heart of A disappearing number. X&Y is on at the Science Museum in London 10 - 16 October 2013. Click here to book tickets. You can read about A disappearing number, an interview with Victoria Gould and several articles by Marcus du Sautoy on Plus. The Magic Cube: Get puzzling! If you like the Rubik's cube then you might love the Magic Cube. Rather than having colours on the little square faces it has number on it. So your task is not only to put the large faces together in the right way, but also to figure out what this right way is. Which numbers should occur together on the same face and in what order? Jonathan Kinlay, the inventor of the Magic Cube, has estimated that there are 140 x 1021 different configurations of the Magic Cube. That's 140 followed by 21 zeroes and 3000 more configurations than on an ordinary Rubik's cube. To celebrate the launch of the Magic Cube, Kinlay's company Innovation Factory is running a competition to see who can solve the cube first. To start it off they will be shipping a version the puzzle directly to 100 of the world's leading quantitative experts, a list that includes people at MIT, Microsoft and Goldman Sachs. You can join too by nominating yourself (or someone else). Innovation Factory will accept up to 20 nominees (in addition to those that have already been picked). The competition will launch in September and run for 60 days. To nominate someone please send an email to [email protected], giving the name and email, mailing address of the nominee and a brief explanation of why you think they should be included in the competition. If you don't get accepted, don't worry — the Magic Cube will go on sale after the competition has ended. The winner will receive lots of glory and a metal version of the Magic Cube precision-machined from solid aluminium, and they will be featured on the Innovation Factory website. As a warm-up you can read about the ordinary Rubik's cube on Plus. The paper galaxy

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2x2 Mixed Factorial Design Background information you need to know to understand the 2x2 mixed analysis is covered in the PsychWorld commentary "Within-Subjects Designs" and "2x2 Between Subjects Designs". The mixed factorial design is, in fact, a combination of these two. It is a factorial design that includes both between and within subjects variables. One special type of mixed design, that is particularly common and powerful, is the pre-post-control design. This is a design in which all subjects are given a pre-test and a post-test, and these two together serve as a within-subjects factor (test). Participants are also divided into two groups. One group is the focus of the experiment (i.e., experimental group) and one group is a base line (i.e., control) group. So, for example, if we are interested in examining the effects of a new type of cognitive therapy on depression, we would give a depression pre-test to a group of persons diagnosed as clinically depressed and randomly assign them into two groups (traditional and cognitive therapy). After the patients were treated according to their assigned condition for some period of time, let’s say a month, they would be given a measure of depression again (post-test). This design would consist of one within subject variable (test), with two levels (pre and post), and one between subjects variable (therapy), with two levels (traditional and cognitive) (Figure 1). Pre-Test Post-Test Cognitive Therapy Cognitive Pre Cognitive Post Traditional Therapy Traditional Pre Traditional Post Figure 1. Example of Pre-Post-Control Design When a researchers uses the pre-post-control design he or she is usually looking for an interaction such that one cell in particular stands out, and that is the experimental group’s post test score. Ideally the pre-test scores will be equivalent. It is the post-test score difference between the experimental and control group that is important (see Figure 2). Figure 2. Hypothetical Means for Experiment in Figure 1 Therefore, in terms of post-hoc tests the most important comparison is between the post-test mean for the experimental group and the post-test mean for the control group (see Figure 3). Figure 3. Comparison of Post-Test Means Also, it is typical for the experimenter to expect a change in the experimental group from pre to post, but not in the control group, which would make the important post-hoc comparisons between pre- and post-test for the experimental groups and between pre- and post-test for the control group (see Figure 4). Figure 4. Comparison of Pre vs. Post Test Means for Both Groups Of course, the pre-post-control design is not the only type of mixed design. Another common type of mixed design (and within-subjects design in general) is one that includes a change over time, so that one independent variable consists of multiple measures of one group of people over time. So, for example, we might be interested in comparing the interest of males vs. females in math and science over some time period during development. More specifically, we could give a group of school children a measure of interest in math and science at age 10 and then give the same group of students the same measure of interest at age 18. Our design then would look like Figure 5, and one set of possible means would look like the means in Figure 6, which would represent an interaction. Age 10 Age 18 Males Males-Age 10 Males-Age 18 Females Females-Age 10 Females-Age 18 Figure 5. Mixed Design with Time as a Within-Subjects Factor Figure 6. Hypothetical Means for the Experiment in Figure 5 The two-way mixed analysis of variance is the most complex type of design/analysis that is covered in the PsychConnections.com modules. The VirtualStatistician and experimental psych modules cover the inferential tests listed below. Although, of course, there are many more types of statistical tests, there are an amazing number of experiments, both within psychological and biological sciences that you can answer with the designs/analyses listed below. Of course, there are many variations, since the examples in the modules are limited to two levels of the independent variables and two independent variables, but adding levels and independent variables is just a slight extension of what is covered. There are also cases in which there are no continuous variables, in which case you would often use a "non-parametric" technique, and complex modeling of many continuous variables which would require "multivariate" analyses. However, in cases where an experimenter uses a traditional method, in which groups are formed and variables are manipulated the designs and analyses covered in these modules will often work fine. Further, these more complex types of data analyses such as multivariate techniques are extensions of the basic "univariate" techniques coverd in the modules, so that this knowledge can serve as an important and necessary foundation for the understanding of these techniques. (Figure 7 is a map/flow chart to aid you in selecting the appropriate analysis for a given design among those covered in the PsychConnections.com modules. If you want to go to the module to review a given analysis click on the appropriate white square.) Figure 7. Flow Chart Representing Choice of Analysis Depending on Design Psychology World was created by Richard Hall in 1998 and is covered by a creative commons (by-nc) copyright

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Calculus/Differentiation/Differentiation Defined < Calculus‎ | Differentiation ← Differentiation/Contents Product and Quotient Rules → Differentiation/Differentiation Defined 1 What is Differentiation? 2 The Definition of Slope 2.1 Of a line 2.2 Of a graph of a function 2.3 Exercises 3 The Rate of Change of a Function at a Point 4 The Definition of the Derivative 4.2 Understanding the derivative notation 5 Differentiation Rules 5.1 Derivative of a constant function 5.1.1 Intuition 5.1.2 Proof 5.2 Derivative of a linear function 5.3 Constant multiple and addition rules 5.3.1 The Constant Rule 5.3.2 The Addition and Subtraction Rules 5.4 The Power Rule 5.5 Derivatives of polynomials What is Differentiation?[edit] Differentiation is an operation that allows us to find a function that outputs the rate of change of one variable with respect to another variable. Informally, we may suppose that we're tracking the position of a car on a two-lane road with no passing lanes. Assuming the car never pulls off the road, we can abstractly study the car's position by assigning it a variable, . Since the car's position changes as the time changes, we say that is dependent on time, or . This tells where the car is at each specific time. Differentiation gives us a function which represents the car's speed, that is the rate of change of its position with respect to time. Equivalently, differentiation gives us the slope at any point of the graph of a non-linear function. For a linear function, of form , is the slope. For non-linear functions, such as , the slope can depend on ; differentiation gives us a function which represents this slope. The Definition of Slope[edit] Historically, the primary motivation for the study of differentiation was the tangent line problem: for a given curve, find the slope of the straight line that is tangent to the curve at a given point. The word tangent comes from the Latin word tangens, which means touching. Thus, to solve the tangent line problem, we need to find the slope of a line that is "touching" a given curve at a given point, or, in modern language, that has the same slope. But what exactly do we mean by "slope" for a curve? The solution is obvious in some cases: for example, a line is its own tangent; the slope at any point is . For the parabola , the slope at the point is ; the tangent line is horizontal. But how can you find the slope of, say, at ? This is in general a nontrivial question, but first we will deal carefully with the slope of lines. Of a line[edit] Three lines with different slopes The slope of a line, also called the gradient of the line, is a measure of its inclination. A line that is horizontal has slope 0, a line from the bottom left to the top right has a positive slope and a line from the top left to the bottom right has a negative slope. The slope can be defined in two (equivalent) ways. The first way is to express it as how much the line climbs for a given motion horizontally. We denote a change in a quantity using the symbol (pronounced "delta"). Thus, a change in is written as . We can therefore write this definition of slope as: An example may make this definition clearer. If we have two points on a line, and , the change in from to is given by: Likewise, the change in from to is given by: This leads to the very important result below. The slope of the line between the points and is Alternatively, we can define slope trigonometrically, using the tangent function: where is the angle from the rightward-pointing horizontal to the line, measured counter-clockwise. If you recall that the tangent of an angle is the ratio of the y-coordinate to the x-coordinate on the unit circle, you should be able to spot the equivalence here. Of a graph of a function[edit] The graphs of most functions we are interested in are not straight lines (although they can be), but rather curves. We cannot define the slope of a curve in the same way as we can for a line. In order for us to understand how to find the slope of a curve at a point, we will first have to cover the idea of tangency. Intuitively, a tangent is a line which just touches a curve at a point, such that the angle between them at that point is zero. Consider the following four curves and lines:

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Search this site: Plus Blog News from the world of maths: The mathematics of monopoly on More or less The mathematics of monopoly on More or less More or Less, BBC Radio 4's program that takes you on a journey through the often abused but ever ubiquitous world of numbers, has recently returned to the airways, and next Monday (17th December 4.30 pm), regular Plus contributors Rob Eastaway and John Haigh are featuring on the program discussing the maths of Monopoly. Eastaway and Haigh have written for Plus many times on a range of topics including: Coincindences; The national lottery; The maths of cricket and football; Remembrance of numbers past; Maths and Magic. Plus spoke to Eastaway about the science of Monopoly, and without giving too much away, Eastaway commented that because the "Go to jail" square is the most frequently visited sqaure on the board, the orange properties are the best investments, as players leaving jail are most likely to then land on these properties. This means you should invest in Bow Street, Marlborough and Vine Street — or in the US version, St James Place, New York Avenue or Tennessee Avenue. News from the world of maths: Christopher Zeeman Medal for Maths Communication Christopher Zeeman Medal for Maths Communication The Christopher Zeeman Medal, the first award dedicated to recognising excellence in the communication of mathematics has been launched by the London Mathematical Society (LMS) and the Institute of Mathematics and its Applications (IMA). The LMS and IMA want to honour mathematicians who have excelled in promoting mathematics and engaging with the general public. They may be academic mathematicians based in universities, mathematics school teachers, industrial mathematicians, those working in the financial sector or indeed mathematicians from any number of other fields. Most importantly, these mathematicians will have worked exceptionally to bring mathematics to a non-specialist audience. Whether it is through giving public lectures, writing books, appearing on radio or television, organising events or through an entirely separate medium, the LMS and IMA want to celebrate the achievements of mathematicians who work to inspire others. In a joint statement, the presidents of the LMS and IMA said, "We are delighted to be able to show how much we need and value mathematicians who can promote their subject successfully. This role is vital to inspiring the next generation of mathematicians as well as helping the wider public to enjoy mathematics." The award is named after Professor Sir Christopher Zeeman, FRS, whose notable career was pioneering not only in his fields of topology and catastrophe theory, but who was also ground-breaking in bringing his beloved mathematics to the wider public. Sir Christopher was the first mathematician to be asked to deliver the Royal Institution Christmas Lectures in 1978, a full 160 years since they began. His "Mathematics into pictures" lectures, have been cited by many young UK mathematicians as their inspiration. They also led to the creation of the Ri's Mathematics Masterclasses, weekly lectures delivered to schoolchildren across the UK via a network of 50 centres. Sir Christopher's skill as a communicator has been recognised in the wider community. In 1988, he was the third recipient of the Royal Society's Faraday Prize, awarded annually to a scientist or engineer who has excelled in communicating science to public audiences. His award was made "for the contributions he has made to the popularization of mathematics". On the announcement of the medal, Sir Christopher said, "I am extremely honoured to have such an important award bear my name. I hope this medal will encourage more mathematicians to see communicating their work to the wider public as a key part of their role." Nominations for the medal are now invited. To receive a nomination form, please contact: The Secretary to the Christopher Zeeman Medal London Mathematical Society De Morgan House 57-58 Russell Square London WC1B 4HS Or email [email protected]. Forms should be returned by February 2008. posted by Plus @ 2:39 PM

数学

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Abstract linear spaces Algebra index History Topics Index Cartesian geometry, introduced by Fermat and Descartes around 1636, had a very large influence on mathematics bringing algebraic methods into geometry. By the middle of the 19th Century however there was some dissatisfaction with these coordinate methods and people began to search for direct methods, i.e. methods of synthetic geometry which were coordinate free. It is possible however to trace the beginning of the vector concept back to the beginning of the 19th Century with the work of Bolzano. In 1804 he published a work on the foundations of elementary geometry Betrachtungen über einige Gegenstände der Elementargoemetrie. Bolzano, in this book, considers points, lines and planes as undefined elements and defines operations on them. This is an important step in the axiomatisation of geometry and an early move towards the necessary abstraction for the concept of a linear space to arise. The move away from coordinate geometry was mainly due to the work of Poncelet and Chasles who were the founders of synthetic geometry. The parallel development in analysis was to move from spaces of concrete objects such as sequence spaces towards abstract linear spaces. Instead of substitutions defined by matrices, abstract linear operators must be defined on abstract linear spaces. In 1827 Möbius published Der barycentrische Calcul a geometrical book which studies transformations of lines and conics. The novel feature of this work is the introduction of barycentric coordinates. Given any triangle ABC then if weights a, b and c are placed at A, B and C respectively then a point P, the centre of gravity, is determined. Möbius showed that every point P in the plane is determined by the homogeneous coordinates [a,b,c], the weights required to be placed at A, B and C to give the centre of gravity at P. The importance here is that Möbius was considering directed quantities, an early appearence of vectors. In 1837 Möbius published a book on statics in which he clearly states the idea of resolving a vector quantity along two specified axes. Between these two works of Möbius, a geometrical work by Bellavitis was published in 1832 which also contains vector type quantities. His basic objects are line segments AB and he considers AB and BA as two distinct objects. He defines two line segments as 'equipollent' if they are equal and parallel, so, in modern notation, two line segments are equipollent if they represent the same vector. Bellavitis then defines the 'equipollent sum of line segments' and obtains an 'equipollent calculus' which is essentially a vector space. In 1814 Argand had represented the complex numbers as points on the plane, that is as ordered pairs of real numbers. Hamilton represented the complex numbers as a two dimensional vector space over the reals although of course he did not use these general abstract terms. He presented these results in a paper to the Irish Academy in 1833. He spent the next 10 years of his life trying to define a multiplication on the 3-dimensional vector space over the reals. Hamilton's quaternions, published in 1843, was an important example of a 4-dimensional vector space but, particularly with Tait's work on quaternions published in 1873, there was to be some competition between vector and quaternion methods. You can see a picture of the plaque commemorating where Hamilton discovered the Quaternions and a (fanciful) engraving of when he carved the rules. In 1857 Cayley introduced matrix algebras, helping the move towards more general abstract systems by adding to the different types of structural laws being studied. In 1858 Cayley noticed that the quaternions could be represented by matrices. In 1867 Laguerre wrote a letter to Hermite Sur le calcul des systèmes linéaires. His systèmes linéaires is a table of coefficients of a system of linear equations denoted by a single upper-case letter and Laguerre defines addition, subtraction and multiplication of of these linear sysyems. In this work Laguerre aims to unify algebraic systems such as complex numbers, Hamilton's quaternions and notions introduced by Galois and by Cauchy. Laguerre's work on linear systems was followed up by a paper by Carvallo in 1891. In this paper he defines operators on vector functions and draws a clear distinction between operators and matrices. To understand the difference between the notions of an operator and a matrix, it suffices to say that, if one changes the coordinate system, one obtains a different matrix to represent the same vector function, but the same operator. Another mathematician who was moving towards geometry without coordinates was Grassmann. His work is highly original but the notion of barycentric coordinates introduced by Möbius was his main motivation. Grassmann's contribution Die Ausdehnungslehre appeared in several different versions. The first was in 1844 but it was a very difficult work to read, and clearly did not find favour with mathematicians, so Grassmann tried to produce a more readable version which appeared in 1862. Clebsch inspired Grassmann to work on this new version. Grassmann studied an algebra whose elements are not specified, so are abstract quantities. He considers systems of elements on which he defines a formal operation of addition, scalar multiplication and multiplication. He starts with undefined elements which he calls 'simple quantities' and generates more complex quantities using specified rules. But ... I go further, since I call these not just quantities but simple quantities. There are other quantities which are themselves compounded quantities and whose characteristics are as distinct relative to each other as the characteristics of the different simple quantities are to each other. These quantities come about through addition of higher forms ... His work contains the familiar laws of vector spaces but, since he also has a multiplication defined, his structures satisfy the properties of what are today called algebras. The precise structures are now known as Grassmann algebras. The ideas of linearly independent and linearly dependent sets of elements are clearly contained in Grassmann's work as is the idea of dimension (although he does not use the term). The scalar product also appears in Grassmann's 1844 work. Grassmann's 1862 version of Die Ausdehnungslehre has a long introduction in which Grassmann gives a summary of his theory. In this introduction he also defends his formal methods which had clearly been objected to by a number of mathematicians. Grassmann's justification comes very close to saying that he is setting up an axiomatic theory and this shows that he is well ahead of his time. Cauchy and Saint-Venant have some claims to have invented similar systems to Grassmann. Saint-Venant's claim is a fair one since he published a work in 1845 in which he multiples line segments in an analogous way to Grassmann. In fact when Grassmann read Saint-Venant's paper he realised that Saint-Venant had not read his 1844 work and sent two copies of the relevant parts to Cauchy, asking him to pass one copy to Saint-Venant. However, rather typically of Cauchy, in 1853 he published Sur les clefs algébrique in Comptes Rendus which describes a formal symbolic method which coincides with that of Grassmann's method (but makes no reference to Grassmann). Grassmann complained to the Académie des Sciences that his work had priority over Cauchy's and, in 1854, a committee was set up to investigate who had priority. We still await the committee's report! The first to see the importance of Grassmann's work was Hankel. In 1867 he wrote a paper Theorie der complexen Zahlensysteme concerning formal systems where combination of the symbols are abstractly defined. He credits Grassmann's Die Ausdehnungslehre as giving the foundation for his work. The first to give an axiomatic definition of a real linear space was Peano in a book published in Torino in 1888. He credits Leibniz, Möbius's 1827 work, Grassmann's 1844 work and Hamilton's work on quaternions as providing ideas which led him to his formal calculus. Peano's 1888 book Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle operazioni della logica deduttiva is remarkable. It gives the basic calculus of set operation introducing the modern notation ∩, ∪, for intersection, union and an element of. It was many years before this notation was to become accepted, in fact Peano's book seems to have had very little influence for many years. It is equally remarkable for containing an almost modern introduction to linear spaces and linear algebra. In Chapter IX of the book Peano gives axioms for a linear space. It is hard to believe that Peano writes the following in 1888. It could almost come from a 1988 book! The first is for equality of elements (a = b) if and only if (b = a), if (a = b) and (b = c) then (a = c). The sum of two objects a and b is defined, i.e. an object is defined denoted by a + b, also belonging to the system, which satisfies: If (a = b) then (a + c = b + c), a + b = b + a, a + (b + c) = (a + b) + c, and the common value of the last equality is denoted by a + b + c. If a is an object of the system and m a positive integer, then we understand by ma the sum of m objects equal to a. It is easy to see that for objects a, b, ... of the system and positive integers m, n, ... one has If (a = b) then (ma = mb), m(a+b) = ma+mb, (m+n)a = ma+na, m(na) = mna, 1a = a. We suppose that for any real number m the notation ma has a meaning such that the preceeding equations are valid. Peano goes on to state the existence of a zero object 0 and says that 0a = 0, that a - b means a + (-b) and states it is easy to show that a - a = 0 and 0 + a = a. Peano defines a linear system to be any system of objects satisfying his four conditions. He goes on to define dependent objects and independent objects. He then defines dimension. Definition: The number of the dimensions of a linear system is the maximal number of linearly independent objects in the system. He proves that finite dimensional spaces have a basis and gives examples of infinite dimensional linear spaces. Peano considers entire functions f(x) of a variable x, defines the sum of f1(x) and f2(x) and the product of f(x) by a real number m. He says:- If one considers only functions of degree n, then these functions form a linear system with n + 1 dimensions, the entire functions of arbitrary degree form a linear system with infinitely many dimensions. Peano defines linear operators on a linear space, shows that by using coordinates one obtains a matrix. He defines the sum and product of linear operators. In the 1890's Pincherle worked on a formal theory of linear operators on an infinite dimensional vector space. However Pincherle did not base his work on that of Peano, rather on the abstract operator theory of Leibniz and d'Alembert. Like so much work in this area it had very little immediate impact and axiomatic infinite dimensional vector spaces were not studied again until Banach and his associates took up the topic in the 1920's. Although never attaining the level of abstraction which Peano had achieved, Hilbert and his student Schmidt looked at infinite dimensional spaces of functions in 1904. Schmidt introduced a move towards abstraction in 1908 introducing geometrical language into Hilbert space theory. The fully axiomatic approach appeared in Banach's 1920 doctoral dissertation. References (13 books/articles) Main index Biographies Index Famous curves index Chronology Time lines Mathematicians of the day Anniversaries for the year Search Form Societies, honours, etc JOC/EFR May 1996 http://www-history.mcs.st-andrews.ac.uk/HistTopics/Abstract_linear_spaces.html

数学

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Amereon Limited 1 The Time Traveller (for so it will be convenient to speak of him) was expounding a recondite matter to us. His grey eyes shone and twinkled, and his usually pale face was flushed and animated. The fire burned brightly, and the soft radiance of the incandescent lights in the lilies of silver caught the bubbles that flashed and passed in our glasses. Our chairs, being his patents, embraced and caressed us rather than submitted to be sat upon, and there was that luxurious after-dinner atmosphere when thought runs gracefully free of the trammels of precision. And he put it to us in this way—marking the points with a lean forefinger—as we sat and lazily admired his earnestness over this new paradox (as we thought it:) and his fecundity.“You must follow me carefully. I shall have to controvert one or two ideas that are almost universally accepted. The geometry, for instance, they taught you at school is founded on a misconception.”“Is not that rather a large thing to expect us to begin upon?” said Filby, an argumentative person with red hair.“I do not mean to ask you to accept anything without reasonable ground for it. You will soon admit as much as I need from you. You know of course that a mathematical line, a line of thickness nil, has no real existence. They taught you that? Neither has a mathematical plane. These things are mere abstractions.”“That is all right,” said the Psychologist.“Nor, having only length, breadth, and thickness, can a cube have a real existence.”“There I object,” said Filby. “Of course a solid body may exist. All real things—”“So most people think. But wait a moment. Can an instantaneous cube exist?”“Don’t follow you,” said Filby.“Can a cube that does not last for any time at all, have a real existence?”Filby became pensive. “Clearly,” the Time Traveller proceeded, “any real body must have extension in four directions: it must have Length, Breadth, Thickness, and—Duration. But through a natural infirmity of the flesh, which I will explain to you in a moment, we incline to overlook this fact. There are really four dimensions, three which we call the three planes of Space, and a fourth, Time. There is, however, a tendency to draw an unreal distinction between the former three dimensions and the latter, because it happens that our consciousness moves intermittently in one direction along the latter from the beginning to the end of our lives.”“That,” said a very young man, making spasmodic efforts to relight his cigar over the lamp; “that…very clear indeed.”“Now, it is very remarkable that this is so extensively overlooked,” continued the Time Traveller, with a slight accession of cheerfulness. “Really this is what is meant by the Fourth Dimension, though some people who talk about the Fourth Dimension do not know they mean it. It is only another way of looking at Time. There is no difference between Time and any of the three dimensions of Space except that our consciousness moves along it. But some foolish people have got hold of the wrong side of that idea. You have all heard what they have to say about this Fourth Dimension?”“I have not,” said the Provincial Mayor.“It is simply this. That Space, as our mathematicians have it, is spoken of as having three dimensions, which one may call Length, Breadth, and Thickness, and is always definable by reference to three planes, each at right angles to the others. But some philosophical people have been asking why three dimensions particularly—why not another direction at right angles to the other three?—and have even tried to construct a Four-Dimension geometry. Professor Simon Newcomb was expounding this to the New York Mathematical Society only a month or so ago. You know how on a flat surface, which has only two dimensions, we can represent a figure of a three-dimensional solid, and similarly they think that by models of three dimensions they could represent one of four—if they could master the perspective of the thing. See?”“I think so,” murmured the Provincial Mayor; and, knitting his brows, he lapsed into an introspective state, his lips moving as one who repeats mystic words. “Yes, I think I see it now,” he said after some time, brightening in a quite transitory manner.“Well, I do not mind telling you I have been at work upon this geometry of Four Dimensions for some time. Some of my results are curious. For instance, here is a portrait of a man at eight years old, another at fifteen, another at seventeen, another at twenty-three, and so on. All these are evidently sections, as it were, Three-Dimensional representations of his Four-Dimensioned being, which is a fixed and unalterable thing.“Scientific people,” proceeded the Time Traveller, after the pause required for the proper assimilation of this, “know very well that Time is only a kind of Space. Here is a popular scientific diagram, a weather record. This line I trace with my finger shows the movement of the barometer. Yesterday it was so high, yesterday night it fell, then this morning it rose again, and so gently upward to here. Surely the mercury did not trace this line in any of the dimensions of Space generally recognized? But certainly it traced such a line, and that line, therefore, we must conclude was along the Time-Dimension.”“But,” said the Medical Man, staring hard at a coal in the fire, “if Time is really only a fourth dimension of Space, why is it, and why has it always been, regarded as something different? And why cannot we move in Time as we move about in the other dimensions of Space?”The Time Traveller smiled. “Are you sure we can move freely in Space? Right and left we can go, backward and forward freely enough, and men always have done so. I admit we move freely in two dimensions. But how about up and down? Gravitation limits us there.”“Not exactly,” said the Medical Man. “There are balloons.”“But before the balloons, save for spasmodic jumping and the inequalities of the surface, man had no freedom of vertical movement.”“Still they could move a little up and down,” said the Medical Man.“Easier, far easier down than up.”“And you cannot move at all in Time, you cannot get away from the present moment.”“My dear sir, that is just where you are wrong. That is just where the whole world has gone wrong. We are always getting away from the present movement. Our mental existences, which are immaterial and have no dimensions, are passing along the Time-Dimension with a uniform velocity from the cradle to the grave. Just as we should travel down if we began our existence fifty miles above the earth’s surface.”“But the great difficulty is this,” interrupted the Psychologist. “You can move about in all directions of Space, but you cannot move about in Time.”“That is the germ of my great discovery. But you are wrong to say that we cannot move about in Time. For instance, if I am recalling an incident very vividly I go back to the instant of its occurrence: I become absent-minded, as you say. I jump back for a moment. Of course we have no means of staying back for any length of Time, any more than a savage or an animal has of staying six feet above the ground. But a civilized man is better off than the savage in this respect. He can go up against gravitation in a balloon, and why should he not hope that ultimately he may be able to stop or accelerate his drift along the Time-Dimension, or even turn about and travel the other way?”“Oh, this,” began Filby, “is all—-”“Why not?” said the Time Traveller.“It’s against reason,” said Filby.“What reason?” said the Time Traveller.“You can show black is white by argument,” said Filby, “but you will never convince me.”“Possibly not,” said the Time Traveller. “But now you begin to see the object of my investigations into the geometry of Four Dimensions. Long ago I had a vague inkling of a machine—-”“To travel through Time!” exclaimed the Very Young Man.“That shall travel indifferently in any direction of Space and Time as the driver determines.”Filby contented himself with laughter.“But I have experimental verification,” said the Time Traveller.“It would be remarkably convenient for the historian,” the Psychologist suggested. “One might travel back and verify the accepted account of the Battle of Hastings, for instance!”“Don’t you think you would attract attention?” said the Medical Man. “Our ancestors had no great tolerance for anachronisms.”“One might get one’s Greek from the very lips of Homer and Plato,” the Very Young Man thought.“In which case they would certainly plough you for the Little-go. The German Scholars have improved Greek so much.”“Then there is the future,” said the Very Young Man. “Just think! One might invest all one’s money, leave it to accumulate at interest, and hurry on ahead!”“To discover a society,” said I, “erected on a strictly communistic basis.”“Of all the wild extravagant theories!” began the Psychologist.“Yes, so it seemed to me, and so I never talked of it until—-”“Experimental verification!” cried I. “You are going to verify that?”“The experiment!” cried Filby, who was getting brain-weary.“Let’s see your experiment anyhow,” said the Psychologist, “though it’s all humbug, you know.”The Time Traveller smiled round at us. Then, still smiling faintly, and with his hands deep in his trousers pockets, he walked slowly out of the room, and we heard his slippers shuffling down the long passage to his laboratory.The Psychologist looked at us. “I wonder what he’s got?”“Some sleight-of-hand trick or other,” said the Medical Man, and Filby tried to tell us about a conjurer he had seen at Burslem; but before he had finished his preface the Time Traveller came back, and Filby’s anecdote collapsed.The thing the Time Traveller held in his hand was a glittering metallic framework, scarcely larger than a small clock, and very delicately made. There was ivory in it, and some transparent crystalline substance. And now I must be explicit, for this that follows—unless his explanation is to be accepted-is an absolutely unaccountable thing. He took one of the small octagonal tables that were scattered about the room, and set it in front of the fire, with two legs on the hearthrug. On this table he placed the mechanism. Then he drew up a chair, and sat down. The only other object on the table, was a small shaded lamp, the bright light of which fell upon the model. There were also perhaps a dozen candles about, two in brass candlesticks upon the mantel and several in sconces, so that the room was brilliantly illuminated. I sat in a low arm-chair nearest the fire, and I drew this forward so as to be almost between the Time Traveller and the fire-place. Filby sat behind him, looking over his shoulder. The Medical Man and the Provincial Mayor watched him in profile from the right, the Psychologist from the left. The Very Young Man stood behind the Psychologist. We were all on the alert. It appears incredible to me that any kind of trick, however subtly conceived and however adroitly done, could have been played upon us under these conditions.The Time Traveller looked at us, and then at the mechanism. “Well?” said the psychologist.“This little affair,” said the Time Traveller, resting his elbows upon the table and pressing his hands together above the apparatus, “is only a model. It is my plan for a machine to travel through time. You will notice that it looks singularly askew, and that there is an odd twinkling appearance about this bar, as though it was in some way unreal.” He pointed to the part with his finger. “Also, here is one little white lever, and here is another.”The Medical Man got up out of his chair and peered into the thing. “It’s beautifully made,” he said.“It took two years to make,” retorted the Time Traveller. Then, when we had all imitated the action of the Medical Man, he said: “Now I want you clearly to understand that this lever, being pressed over, sends the machine gliding into the future, and this other reverses the motion. This saddle represents the seat of a time traveller. Presently I am going to press the lever, and off the machine will go. It will vanish, pass into future Time, and disappear. Have a good look at the thing. Look at the table too, and satisfy yourselves there is no trickery. I don’t want to waste this model, and then be told I’m a quack.”There was a minute’s pause perhaps. The Psychologist seemed about to speak to me, but changed his mind. Then the Time Traveller put forth his finger toward the lever. “No,” he said suddenly. “Lend me your hand.” And turning to the Psychologist, he took that individual’s hand in his own and told him to put out his forefinger. So that it was the Psychologist himself who sent forth the model Time Machine on its interminable voyage. We all saw the lever turn. I am absolutely certain there was no trickery. There was a breath of wind, and the lamp flame jumped. One of the candles on the mantel was blown out, and the little machine suddenly swung round, became indistinct, was seen as a ghost for a second perhaps, as an eddy of faintly glittering brass and ivory; and it was gone—vanished! Save for the lamp the table was bare.Everyone was silent for a minute. Then Filby said he was damned.The Psychologist recovered from his stupor, and suddenly looked under the table. At that the Time Traveller laughed cheerfully. “Well?” he said, with a reminiscence of the Psychologist. Then, getting up, he went to the tobacco jar on the mantel, and with his back to us began to fill his pipe.We stared at each other. “Look here,” said the Medical Man, “are you in earnest about this? Do you seriously believe that that machine has travelled into time?”“Certainly,” said the Time Traveller, stooping to light a spill at the fire. Then he turned, lighting his pipe, to look at the Psychologist’s face. (The Psychologist, to show that he was not unhinged, helped himself to a cigar and tried to light it uncut.) “What is more, I have a big machine nearly finished in there”—he indicated the laboratory—“and when that is put together I mean to have a journey on my own account.”“You mean to say that that machine has travelled into the future?” said Filby.“Into the future or the past—I don’t, for certain, know which.”After an interval the Psychologist had an inspiration. “It must have gone into the past if it has gone anywhere,” he said.“Why?” said the Time Traveller.“Because I presume that it has not moved in space, and if it travelled into the future it would still be here all this time, since it must have travelled through this time.”“But,” I said, “if it travelled into the past it would have been visible when we came first into this room; and last Thursday when we were here; and the Thursday before that; and so forth!”“Serious objections,” remarked the Provincial Mayor, with an air of impartiality, turning towards the Time Traveller.“Not a bit,” said the Time Traveller, and, to the Psychologist: “You think. You can explain that. It’s presentation below the threshold, you know, diluted presentation.”“Of course,” said the Psychologist, and reassured us. “That’s a simple point of psychology. I should have thought of it. It’s plain enough, and helps the paradox delightfully. We cannot see it, nor can we appreciate this machine, any more than we can the spoke of a wheel spinning, or a bullet flying through the air. If it is travelling through time fifty times or a hundred times faster than we are, if it gets through a minute while we get through a second, the impression it creates will of course be only one-fiftieth or one-hundredth of what it would make if it were not travelling in time. That’s plain enough.” He passed his hand through the space in which the machine had been. “You see?” he said, laughing.We sat and stared at the vacant table for a minute or so. Then the Time Traveller asked us what we thought of it all.“It sounds plausible enough to-night,” said the Medical Man; “but wait until to-morrow. Wait for the common sense of the morning.”“Would you like to see the Time Machine itself?” asked the Time Traveller. And therewith, taking the lamp in his hand, he led the way down the long, draughty corridor to his laboratory. I remember vividly the flickering light, his queer, broad head in silhouette, the dance of the shadows, how we all followed him, puzzled but incredulous, and how there in the laboratory we beheld a larger edition of the little mechanism which we had seen vanish from before our eyes. Parts were of nickel, parts of ivory, parts had certainly been filed or sawn out of rock crystal. The thing was generally complete, but the twisted crystalline bars lay unfinished upon the bench beside some sheets of drawings, and 1 took one up for a better look at it. Quartz it seemed to be.“Look here,” said the Medical Man, “are you perfectly serious? Or is this a trick—like that ghost you showed us last Christmas?”“Upon that machine,” said the Time Traveller, holding the lamp aloft, “I intend to explore time. Is that plain? I was never more serious in my life.”None of us quite knew how to take it.I caught Filby’s eye over the shoulder of the Medical Man, and he winked at me solemnly. All new material in this edition is Copyright © 1986 by Tom Doherty Associates, LLC. Excerpted from The Time Machine by H. G. Wells Copyright © 0001 by H. G. Wells. Excerpted by permission. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

数学

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Search this site: Plus Blog Maths in a minute: Countable infinities An infinite set is called countable if you can count it. In other words, it's called countable if you can put its members into one-to-one correspondence with the natural numbers 1, 2, 3, ... . For example, a bag with infinitely many apples would be a countable infinity because (given an infinite amount of time) you can label the apples 1, 2, 3, etc. Two countably infinite sets A and B are considered to have the same "size" (or cardinality) because you can pair each element in A with one and only one element in B so that no elements in either set are left over. This idea seems to make sense, but it has some funny consequences. For example, the even numbers are a countable infinity because you can link the number 2 to the number 1, the number 4 to 2, the number 6 to 3 and so on. So if you consider the totality of even numbers (not just a finite collection) then there are just as many of them as natural numbers, even though intuitively you'd think there should only be half as many. Something similar goes for the rational numbers (all the numbers you can write as fractions). You can list them as follows: first write down all the fractions whose denominator and numerator add up to 2, then list all the ones where the sum comes to 3, then 4, etc. This is an unfailing recipe to list all the rationals, and once they are listed you can label them by the natural numbers 1, 2, 3, ... . So there are just as many rationals as natural numbers, which again seems a bit odd because you'd think that there should be a lot more of them. It was Galileo who first noticed these funny results and they put him off thinking about infinity. Later on the mathematician Georg Cantor revisited the idea. In fact, Cantor came up with a whole hierarchy of infinities, one "bigger" than the other, of which the countable infinity is the smallest. His ideas were controversial at first, but have now become an accepted part of pure mathematics. You can find out more about all this in our collection of articles on Postcard from New York A buckyball in Madison Square. Yesterday we opened the Plus New York office, amidst snow covered streets at the foot of the Empire State Building! The day started with a trip to MoMath, the recently opened maths museum in central New York. It was filled with a fascinating array of interactive exhibits demonstrating the beauty and playfulness of mathematics. And as one of the volunteers told us, playfulness is what it's all about. There were musical spheres demonstrating the maths of music, a fractal machine with cameras creating fractals from their surroundings, and a chance to discover the paths of mathematical rolling stones. It was full of children and the young at heart discovering the joy of maths for themselves. We also discovered an illuminated buckyball in the park just across from our hotel and the arithmetic of relationships in the High Line park. Maths is everywhere in NYC! The Institute for Advanced Study in Princeton. Today we had a very early start, taking the train from New York Penn Station to Princeton to visit the Institute for Advanced Studies. We were very lucky to speak with Freeman Dyson and Edward Witten about quantum field theory (QFT), the mathematical framework that has made much of the advancements in physics possible in the last century. This is the main reason for our trip to the States and we are looking forward to more interviews this week with other luminaries of theoretical physics to continue our series telling the story of QFT. You can read our first articles here. We'd like to thank Jeremy Butterfield and Nazim Boutta, our gurus in QFT for all their help in preparing for the trip! After a Manhattan or two tonight we're heading to Boston tomorrow to continue our quantum adventure! The arithmetic of relationships on the High Line.

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A Neighborhood of Infinity The Curious Rotational Memory of the Electron, Part 1 There's a curious and bizarre fact about the universe that is introduced in physics courses without anyone stopping to point out just how curious and bizarre it is. I think it deserves to be more widely known to non-physicists. It's a little esoteric, but not so esoteric it can't be grounded in ordinary everyday concepts. And it isn't something hypothetical but something that is fundamental to modern physics and testable in the lab. It is also the reason why I chose to use theta/2 instead of theta in my quantum computing code.This post will be in two parts. This part will mainly be about geometry and topology, and in the second part I'll talk about the physics.In order to explain the idea I want to introduce the concept of a universal cover. If you look at the definition of this notion anywhere it can seem pretty daunting to non-mathematicians, but in actual fact the key idea is so simple that even a child could understand it. I can say that without fear of exaggeration because universal covers used to play a role in a childhood fantasy of mine. (OK, I was a weird kid, but not that weird.)My crazy idea as a kid was this: if there were two ways to get from A to B, maybe the B you reached by taking either of the two routes were actually slightly different, even though they looked the same. I was actually more concerned with journeys from A to A. Suppose the kitchen has a door at each end and that you start in the bedroom, go down to the kitchen at one end, come out the other end, and return by a different path to the bedroom. How do you know this is the same bedroom? (I also admit I read a lot of science fiction as a kid.)Suppose instead you took a simpler path: you left the bedroom and went to another room and then returned by the path you went. If you had taken a piece of string and tied one end to the bedroom and walked this path playing out the string behind you as you went then when you returned to your bedroom you would have both ends of the string visible. You could then reel in the string from your journey, without letting go of the ends, and see that indeed, both ends were in fact the ends of the same string and so you had retuned to the start of your journey. But if you'd taken the round trip through the double-doored kitchen you wouldn't be able to pull the string tight and so you could no longer be sure.In what follows we consider two paths from A to B equivalent in this way: mark the two paths with strings going from A to B. If you can pull one string to line up exactly with the other (you're allowed to stretch) without breaking the string or moving the endpoints then they are considered equivalent. (The correct technical term is homotopic.) So if a path that appears to go from A to A is homotopic to the trivial zero length path, then you know that the path really is from A back to A. But if you can't shrink the path down to zero length you're faced with the possibility that the string is extended from A to a similar looking place, A'. (Of course this implies a considerable conspiracy with someone tying a distinct identical string to an identical bedroom, but if you can't see both ends, alien beings from a higher dimension could do anything...)The best case scenario is that all of the bedrooms are the same - a conventional house. Call this house X. But what's the worst case scenario? How many different copies of the bedroom might there be? Well let's call the starting bedroom 0. If we walk anti-clockwise round the house, going through the kitchen, we end up back at a bedroom. In the worst case scenario this will be another bedroom. Let's call it 1. Walk round again and we get to another copy of the bedroom, call it 2 and so on. So we have an infinite sequence of bedrooms, (0,1,2,…). But we could have walked clockwise. In that case we'd get a sequence (-1,-2,…). So now think about the set of all points in this infinite house. Each point can be described as a pair (x,n) where x is a point in the house X and n labels which of the infinite number of paths you could have taken to get there. (Remember, if the two paths are homotopic we consider them to be the same path.)If using a house as an example seems too far-fetched think of a mult-storey car park. It's easy to walk what you think is a closed loop but find that you've in fact walked to almost identical looking, but different, floor.(Incidentally, this kind of infinite house is the kind of thing that can be created in a Portal engine such as in the game Portal.)Let's think about this more generally. Given a topological space X we can form another space X' from it by (1) picking a point p in it and (2) forming the set of all pairs (x,n) where x is a point in X and n is a path (up to homotopy) from p to x. X' is known as the universal cover of X.Enough with houses, let's try a more purely geometric example. Consider the space of rotations in 2D, otherwise known as SO(2). A rotation can simply be described by an angle with the proviso that a rotation through an angle θ is the same as a rotation through θ+2nπ where n is an integer. The space of rotations is essentially the same thing as the circle (by circle I mean the 1-dimensional boundary of a 2-dimensional disk). Looping once round the circle brings you back to where you started. So how can we visualise the universal cover of SO(2)? There's a nice trick for this. Imagine a disk that's free to rotate around one axis. SO(2) describes the rotations of this disk. But now imagine a belt, fixed at one end, connected at the other end to the centre of this disk like this:The rule is that we're allowed to twist the belt as much as we like as long as the centre line of the belt is along the axis. Notice how if we rotate the disk through an angle of 2nπ the disk returns to its starting point, but the belt 'remembers' how many twists we took to get there. Notice another thing: if we travel along the belt from one end to the other, the orientation of the belt at each point gives a 2D rotation. In other words, the belt itself represents a path in SO(2) with one one representing a fixed point in SO(2) and the other end matching the current orientation of the disk. So if the orientations of the disk represent elements of SO(2), the possible twistings of the belt represent elements in the universal cover of SO(2). And what the twists do is allow us to differentiate between rotations through θ and rotations through θ+2nπ. In other words, the universal cover is just the real line, but not folded down so that θ is the same as θ+2nπ.The universal cover is the worst case scenario where all non-homotopic paths are considered to represent distinct points. But it's also possible to construct an in-between scenario where some paths that aren't homotopic to each other are still considered to lead to the same place. For example, take the real line again. If we identify 0 and 2π we get SO(2). If we consider 0 and 2nπ to be distinct for n≠0 we get the universal cover. But we pick an N≠1 and consider 2Nπ to be the same as 0, but that 2nπ is distinct from 0 for 0<n<N. What we now end up with is an N-fold cover of SO(2). Unfortunately I don't know a way construct a physical example of a general N-fold cover using belts and discs. (At least for N>2. For N=2 you should easily be able to come up with a mechanism based on what's below.)With the examples so far, the universal cover of a space has always had an infinite number of copies of the original space. That isn't always the case and now we'll meet an example where the universal cover is a double cover: the real projective plane, RP2. The easiest way to visualise this is to imagine the game of Asteroids played on a circular TV screen. In the usual Asteroids, when you go off one side of the screen you reappear on the opposite side. In this version it's much the same, points on the edge that are diametrically opposite are considered to be the same points. Such a universe is a real projective plane. Let P be the point at the centre of the screen. Suppose we attach one end of our string to P and fly off towards the edge, reappearing on the other side. Eventually we get back to P again. Can we shrink this path down to a point? Unfortunately, any path that crosses the edge of our screen will always cross the edge no matter how much we try to deform it. If we move the point at which it crosses one edge of the screen, the point at the other side always remains diametrically opposite.This means you can never bring these points together in order to collapse down a loop. On the other hand, suppose we loop twice round the edge. The following diagram shows how this path can be shrunk down to a loop:So given two distinct points in RP2 there are two inequivalent paths between them, and so the universal cover of RP2 is made from two copies of the space and is in fact a double cover.Now we'll try a 3D space - the space of 3D rotations, otherwise known as SO(3). This space is 3-dimensional because it takes 3 numbers to define a rotation, for example we can specify a rotation using a vector whose direction is the axis of rotation and whose length is the angle of rotation. Now all possible rotations are rotations through an angle 0≤θ≤π for some exis. So every rotation can be represented as a point in the ball of radius π. But a rotation around the axis a through an angle of π is also the same as a rotation around the axis -a through π. In other words, SO(3) is the closed 3-dimensional ball with diametrically opposite points identified. This is very similar to the situation with RP2. In fact, almost exactly the same argument shows that the universal cover of SO(3) is the double cover. (For an alternative wording, see this posting.) There is also another way of seeing this that is analogous to the disk with belt example above. This time we have a sphere instead of a disk. The belt is fixed at one end and the other end is attached to the sphere. We are free to rotate the sphere however we want, and we're no longer confined to having the belt along one axis. It is an amazing but easily demonstrable fact that if you rotate the sphere through 2π, you cannot untwist the belt, but if you rotate it through 4π you can! This is known as the Dirac Belt Trick, and if you don't believe it, watch the Java applet written by science fiction author Greg Egan.If you've done much computer graphics you'll have already met the double cover of SO(3). Computer graphics people sometimes like to represent rotations using unit quaternions. The unit quaternions form a 3-dimensional sphere but there are two such quaternions for each rotation. So the unit quaternions actually represent this double cover, and SO(3) is a 3-dimensional sphere with antipodean points identified.And that's the geometry dealt with. In the next installment I'll say what this has to do with electrons and my quantum computing code.FootnoteBalls and Spheres. The closed n-dimensional ball is the set of points (x1,...,xn) such that x12+...+xn2≤1. The (n-1)-dimensional sphere is the set of points (x1,...,xn) such that x12+...+xn2=1. Derek Elkins Another way to demonstrate the double cover that you can do in the comfort of your own home is to rotate your hand at the wrist keeping your palm facing upward at all times through 360 and then 720 degree rotation. At 360 degrees your arm certainly will not be in it's original state, but it will at 720. Saturday, 31 March, 2007 sigfpe Derek,That's the Feyman plate trick which I was lucky enough to see Feynman perform many years ago. How about, for an N-fold cover, take a belt that's N times the circumference of a wheel, and span it out with another wheel, mounted just to help keeping things tight - resulting in something vaguely like a bicycle chain mechanism.Now, for each revolution of the wheel, viz every 2π rotation, you end up at a different point along the belt, but after N of them, you get back to the point you started.Would this capture the structure needed? My quantum mechanics prof demonstrated SU(2) using a belt as well. I thought he was just a little eccentric, I didn't realise it was a trick due to Dirac.There was a talk at Perimeter Institute a while ago by Sundance Bilson-Thompson, who is trying to build a preon model out of braids and ribbons. If his model is true, the electron is described by SU(2) because it is really made of tiny belts! Nice exposition!Possibly worth pointing out that the universal cover is actually "universal" in the technical sense that it's initial in some appropriate category, here the category of covers of X and homotopy classes of (surjective?) maps. The Curious Rotational Memory of the Electron, Par... The Angel Problem has been Solved! (Maybe) The Shor Quantum Error Correcting Code (and a Mona... Independence, entanglement and decoherence with th... Monads, Vector Spaces and Quantum Mechanics pt. II... The allure of LaTeX Monads in C pt. III Rubrication Richard Borcherds: Mathematics and physics The n-Category Cafe Blog: A Neighborhood of InfinityCode: GithubTwitter: sigfpeHome page: www.sigfpe.com

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Search this site: Plus Blog News from the world of maths: You can come out now, it's safe... You can come out now, it's safe... Well it's official, the first beam in the Large Hadron Collider at CERN has safely made its way around the 27km tunnel at around 1030 this morning, local time. It was a historic moment, the culmination of over 20 years' work building the biggest experiment the world has seen, and one that many hope will give us a glimpse into the beginnings of the universe and give experimental evidence to long-held theories fundamental to physics. "It’s a fantastic moment,” said LHC project leader Lyn Evans, “we can now look forward to a new era of understanding about the origins and evolution of the universe.” Starting up a major new particle accelarator takes much more than just flipping a switch. Thousands of individual elements have to work in harmony and timings have to be synchronized to under a billionth of a second. The second beam was fired at around 2pm local time, and is now making its way around in the opposite direction. Over the next few weeks, as the people at the LHC learn how to drive their new toy, they will steer the two beams, finer than a human hair, into a head-on collision. It will be these collisions that will allow the research programme to begin properly. Once colliding beams have been established, there will be a period of measurement and calibration for the LHC’s four major experiments, and new results could start to appear in about a year's time. Experiments at the LHC will allow physicists to complete a journey that started with Newton's description of gravity. Gravity acts on mass, but so far science is unable to explain the mechanism that generates mass. Experiments at the LHC will provide the answer. LHC experiments will also try to probe the mysterious dark matter of the universe – visible matter seems to account for just 4% of what must exist, while about a quarter is believed to be dark matter. They will investigate the reason for nature's preference for matter over antimatter, and they will probe matter as it existed at the very beginning of time. “The LHC is a discovery machine,” said CERN Director General Robert Aymar, “its research programme has the potential to change our view of the Universe profoundly, continuing a tradition of human curiosity that’s as old as mankind itself.” You can read more about the LHC and the science it is exploring on Plus The LHC for dummies The physics of elementary particles The search for Higgs Secrets of the Universe Stringent tests Time to get packing? String theory: from Newton to Einstein and beyond Tying it all up PS. Oh and for the science-scaredy-cats, you can come out from under the bed for now, no black holes have been created as of yet! News from the world of maths: Prime record broken? Prime record broken? Volunteers have claimed to have found the largest prime number yet — twice within a fortnight! The two new record breakers are both Mersenne primes: numbers which can

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Search this site: Plus Blog Bridges of Königsberg: The movie We've read the book. We've bought the T-shirt. And now, finally, here it is: the movie of one of our favourite maths problems, the bridges of Königsberg. Though admittedly, we made it ourselves. We learnt several interesting lessons in the process. For example that a bin doesn't make a good supporting character and that people who shouldn't be in the frame should get out of it. But other than that, we're well on course for an Oscar this weekend! You can read more about the bridges of Königsberg here. Congratulations to the new Stephen W Hawking Professor of Cosmology! There was a brief pause in research at the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge (DAMTP, also the home of Plus) this afternoon to celebrate the newly established Chair in Cosmology. The Chair, funded by a $US 6 million donation from Avery-Tsui Foundation, is named after Stephen Hawking and he will be the first to hold the Professorship. Paul Shellard, Director of the Centre for Theoretical Cosmology, said that the honour recognised Hawking's contributions to changing our understanding of the Universe. Stephen Hawking experiencing zero gravity (Image: NASA) "When I arrived at DAMTP in 1962 cosmology was a speculative science and we didn't know if the Universe had a beginning or had existed forever in a steady state," Hawking said. He went on to say that the new Professorship recognised the role of the department in taking cosmology from this speculative start to the remarkably successful field it is today. We'd like to congratulate Hawking on his new post (and thank him for the cake and champagne!) and look forward to the next exciting discovery from our cosmologist neighbours. You can read more about Hawking's life and work in his articles 60 years in a nutshell and A brief history of mine, and in our coverage of his 60th and 70th birthday symposia. And of course, there's much more about cosmology on Plus. Maths in a minute: Counting numbers Are there more irrational numbers than rational numbers, or more rational numbers than irrational numbers? Well, there are infinitely many of both, so the question doesn't make sense. It turns out, however, that the set of rational numbers is infinite in a very different way from the set of irrational numbers. As we saw here, the rational numbers (those that can be written as fractions) can be lined up one by one and labelled 1, 2, 3, 4, etc. They form what mathematicians call a countable infinity. The same isn't true of the irrational numbers (those that cannot be written as fractions): they form an uncountably infinite set. In 1873 the mathematician came up with a beautiful and elegant proof of this fact. First notice that when we put the rational numbers and the irrational numbers together we get all the real numbers: each number on the line is either rational or irrational. If the irrational numbers were countable, just as the rationals are, then the real numbers would be countable too — it's not too hard to convince yourself of that. So let’s suppose the real numbers are countable, so that we can make a list of them, for example 1.

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Search this site: Plus Blog Mathematical theatre at the Science Museum: X&Y What is the shape of the Universe? Is it finite or infinite? Does it have an edge? In their new show X&Y Marcus du Sautoy and Victoria Gould use mathematics and the theatre to navigate the known and unknown reaches of our world. Through a series of surreal episodes, X and Y, trapped in a Universe they don't understand and confronted for the first time with another human being, tackle some of the biggest philosophical and scientific questions on the books: where did the Universe come from, does time have an end, is there something on the other side, do we have free will, can we ever prove anything about our Universe for sure or is there always room for another surprise? Marcus and Victoria met while working on A disappearing number, Complicite's multi award-winning play about mathematics. X&Y has developed from that collaboration and pursues many of the questions at the heart of A disappearing number. X&Y is on at the Science Museum in London 10 - 16 October 2013. Click here to book tickets. You can read about A disappearing number, an interview with Victoria Gould and several articles by Marcus du Sautoy on Plus. The Magic Cube: Get puzzling! If you like the Rubik's cube then you might love the Magic Cube. Rather than having colours on the little square faces it has number on it. So your task is not only to put the large faces together in the right way, but also to figure out what this right way is. Which numbers should occur together on the same face and in what order? Jonathan Kinlay, the inventor of the Magic Cube, has estimated that there are 140 x 1021 different configurations of the Magic Cube. That's 140 followed by 21 zeroes and 3000 more configurations than on an ordinary Rubik's cube. To celebrate the launch of the Magic Cube, Kinlay's company Innovation Factory is running a competition to see who can solve the cube first. To start it off they will be shipping a version the puzzle directly to 100 of the world's leading quantitative experts, a list that includes people at MIT, Microsoft and Goldman Sachs. You can join too by nominating yourself (or someone else). Innovation Factory will accept up to 20 nominees (in addition to those that have already been picked). The competition will launch in September and run for 60 days. To nominate someone please send an email to [email protected], giving the name and email, mailing address of the nominee and a brief explanation of why you think they should be included in the competition. If you don't get accepted, don't worry — the Magic Cube will go on sale after the competition has ended. The winner will receive lots of glory and a metal version of the Magic Cube precision-machined from solid aluminium, and they will be featured on the Innovation Factory website. As a warm-up you can read about the ordinary Rubik's cube on Plus. The paper galaxy August 16,

数学

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Search this site: Plus Blog Plus Advent Calendar Door #19: Beautiful buildings The dramatic curved surfaces of some of the iconic buildings created in the last decade, such as 30 St Mary's Axe (AKA the Gherkin) in London, are only logistically and economically possible thanks to mathematics. Curved panels of glass or other material are expensive to manufacture and to fit. Surprisingly, the curved surface of the Gherkin has been created almost entirely out of flat panels of glass — the only curved piece is the cap on the very top of the building. And simple geometry is all that is required to understand how. A geodesic sphere. One way of approximating a curved surface using flat panels is using the concept of geodesic domes and surfaces. A geodesic is just a line between two points that follows the shortest possible distance — on the Earth the geodesic lines are great circles, such as the lines of longitude or the routes aircraft use for long distances. A geodesic dome is created from a lattice of geodesics that intersect to cover the curved surface with triangles. The mathematician Buckminster Fuller perfected the mathematical ideas behind geodesic domes and hoped that their properties — greater strength and space for minimum weight — might be the future of housing. To try to build a sphere out of flat panels, such as a geodesic sphere, you first need to imagine an icosahedron (a polyhedron made up of 20 faces that are equilateral triangles) sitting just inside your sphere, so that the points of the icosahedron just touch the sphere's surface. An icosahedron, with its relatively large flat sides, isn't going to fool anyone into thinking it's curved. You need to use smaller flat panels and a lot more of them. Divide each edge of the icosahedron in half, and join the points, dividing each of the icosahedron's faces into four smaller triangles. Projecting the vertices of these triangles onto the sphere (pushing them out a little til they two just touch the sphere's surface) now gives you a polyhedron with 80 triangular faces (which are no longer equilateral triangles) that gives a much more convincing approximation of the curved surface of the sphere. You can carry on in this way, dividing the edges in half and creating more triangular faces, until the surface made up of flat triangles is as close to a curved surface as you would like. Find out more about the Gherkin and other iconic buildings on Plus. Plus Advent Calendar Door #18: Clever sums How would you go about adding up all the integers from 1 to 100? Tap them into a calculator? Write a little computer code? Or look up the general formula for summing integers? Carl Friedrich Gauss as depicted on the (now defunct) German 10 Marks note. Legend has it that the task of summing those numbers was given to the young Carl Friedrich Gauss by his teacher at primary school, as a punishment for misbehaving. Gauss didn't have a calculator or computer, no one did at that time, but he came up with the correct answer within seconds. Here's how he did it. Notice that you can sum the numbers in pairs, starting at either end. First you add 1 and 100 to get 101. Next it's 2 and 99, giving 101 again. The same for 3 and 98. Continuing like this, the last pair you get is 50 and 51 and they give 101 again. Altogether there are 50 pairs all adding to 101, so the answer is 50 x 101 = 5050. Easy — if you're Gauss.

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Search this site: Plus Blog News from the world of maths: Sad news: Paul Cohen dies Sad news: Paul Cohen dies The distinguished mathematician Paul Cohen sadly died on Friday the 23rd of March 2007, just a few days before his 73rd birthday. Cohen worked on a range of topics, but is best-known for his work on set theory. His work in this area cuts right down to the foundations of mathematics. The mathematician David Hilbert famously believed that it should be possible to phrase all of mathematics in a single and completely formal theory. Based on a collection of axioms — pre-determined facts that are so self-evident they do not themselves need to be proved — and the rules of logic, it should be possible to formally prove every mathematical truth and to arrive at a complete theory that is free from contradiction. Set theory provides a language in which such a formal system might be phrased. In maths a set is simply a collection of objects. The objects themselves are allowed to remain abstract and so you can talk about all mathematical objects — whether they are functions, numbers or anything else — in terms of sets. In the beginning of the twentieth century, mathematicians laid down a list of axioms and rules of logic that resulted in a formal and rigorous theory of sets. The system is known as ZF theory, after Ernst Zermelo and Abraham Fraenkel. On the face of it, sets are simple objects. But once you allow them to have an infinite number of elements, things become complicated. If you take the set of whole numbers, for example, and compare it to the set of all numbers, you'll notice that although both sets are infinite, they are fundamentally different from each other. The set of whole numbers consists of isolated objects, whereas you can think of all the numbers as merging together to give a continuum. In some sense, the set of all numbers is "bigger" than the set of whole numbers, so we need a notion of size for infinite sets also. The mathematician Georg Cantor (1845 - 1918) formalised such a notion, called cardinality, and in doing so came up with a conjecture that became known as the continuum hypothesis: that there is no set that is "larger" than the set of whole numbers and "smaller" than the set of all numbers. However, a proof of this fact illuded Cantor and the continuum hypothesis became the first on Hilbert's list of mathematical challenges for the 20th century. When it comes to the axioms of set theory, things aren't all that clear-cut either. Some axioms are clear as day, for example the one which states that two sets are the same if all their elements are the same. Others are more controversial though, and one of them is the axiom of choice. It states that if you have a collection of sets, then you can form a new set by picking one element from each. Again, this is clearly possible when you've got a finite collection of sets, but if there are infinitely many, it's not clear that a mechanism for picking an element from each always exists. Although the majority of mathematicians accept the axiom of choice, there is a school of thought which doubts it. Paul Cohen proved two significant — and to the Hilbert school of thought disappointing — results in this area. He showed that neither the continuum hypothesis nor the axiom of choice can be proved from the axioms of ZF theory. Together with results previously proved by Kurt Gödel, this means that neither can be proved to be either true or false within ZF theory. Within ZF theory, the continuum hypothesis will forever remain a mystery. As far as the axiom of choice is concerned, the result means that there's no clear indication as to whether you should include it or its negation as one of your initial axioms of set theory. In both cases you come up with a sound system, so the decision whether to include it or not has to be made on different grounds. In proving these results, Cohen not only contributed to the philosophical debate about the foundations of maths, but also developed a whole new set of tools to deal with questions on what can and cannot be proved within formal mathematical systems. His work was honoured with a Fields medal in 1966. The world of maths has lost one of its most distinguished members. posted by Plus @ 10:27 AM

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Ivo Milan Babuska Born: 22 March 1926 in Prague, Czechoslovakia (now Czech Republic) Ivo Babuska lived through difficult times in Czechoslovakia as he was growing up. The German invasion of the west of the country in September 1938 was followed six months later by the whole country being taken over by Germany at the start of World War II. Babuska was thirteen years old at this time and most of his secondary education, therefore, took place under German occupation. Only after the war ended in 1945 was he able to begin his university education. Babuska studied civil engineering at the Czech Technical University in Prague and was awarded a Dipl. Ing in 1949. He undertook research in engineering, advised by F Faltus, and was awarded a Dr. Tech. by the Faculty of Civil Engineering of the Czech Technical University in 1951. However, simultaneously with this research in engineering, Babuska was a mathematics student at the Central Mathematical Institute in Prague studying under Vladimir Knichal. While Babuska was studying at the Institute, its name changed in 1953 to the Mathematical Institute of the Czechoslovak Academy of Sciences. At the Mathematical Institute, in addition to Knichal, he was strongly influenced by Eduard Čech who was appointed Director of the Central Mathematical Institute in 1950, and Director of the Czechoslovak Academy of Sciences in 1952. In 1955 Babuska was awarded his Candidate's degree (equivalent to a Ph.D.). Given Babuska's training, coming first to engineering and, slightly later to mathematics, it is no surprise to see his publications being in the engineering area but become more slanted towards advanced mathematical techniques to solve engineering problems. His first papers, all written in Czech, were Welding stresses and deformations (1952), Plane elasticity problem (1952), A contribution to the theoretical solution of welding stresses and some experimental results (1953), A contribution to one method of solution of the biharmonic problem (1954), Solution of the elastic problem of a half-plane loaded by a sequence of singular forces (1954), (with L Mejzlik) The stresses in a gravity dam on a soft bottom (1954), On plane biharmonic problems in regions with corners (1955), (with L Mejzlik) The method of finite differences for solving of problems of partial differential equations (1955), and Numerical solution of complete regular systems of linear algebraic equations and some applications in the theory of frameworks (1955). His 1954 paper investigating stresses in a gravity dam was a direct consequence of a project that he led between 1953 and 1956 on using computational techniques to examine the technology involved in building the Orlik Dam on the Vltava River about 80 km from Prague. This river, the longest in the Czech Republic, is a major source of hydroelectric power with several important dams creating artificial lakes. The Orlik Dam [6]:- ... is a gravitational concrete dam 91 m high. The group, consisting of civil engineers, material scientists, mathematicians, and desk calculator operators, concentrated on the technology without artificial cooling, which is usually used to remove the heat created during the hardening of concrete. All the computations were carried out on mechanical desk calculators. Basically, the mathematical problem Babuska's group had to solve was to find a numerical solution to a nonlinear partial differential equation. In 1955, after the award of his doctorate, Babuska was appointed as head of the Department of Constructive Methods of Mathematical Analysis of the Mathematical Institute of the Czechoslovak Academy of Sciences. In the same year, in collaboration with Karel Rektorys and Frantisek Vycichlo, he published his first book The mathematical theory of plane elasticity (Czech). A German translation was published in 1960. Frantisek Kroupa writes in a review of the original Czech edition:- The book is devoted to the application of the theory of functions of a complex variable to solving plane problems of the classical mathematical theory of elasticity (for static problems without the effect of body forces). From the mathematical point of view it deals with the special method of solving a biharmonic equation for given boundary conditions. The book gives and further develops some of the results of N I Muskelishvili and collaborators. An original contribution is the axiomatic construction of the fundamentals of plane elasticity, the accuracy and generality of the mathematical procedures and some new numerical methods of solution. In the following year, 1956, Babuska founded the journal Applications of Mathematics (Applikace Matematiky). In 1960 Babuska was awarded a Dr Sc. (the highest possible degree in Czechoslovakia, equivalent to a D.Sc.) by the Czechoslovak Academy of Sciences. His next important book, published in collaboration with Milan Práger and Emil Vitásek in 1964, was Numerical Solution of Differential Equations (Czech). It was translated into English and published under the title Numerical processes in differential equations two years later. Richard Hamming reviewed the English translation and writes:- This book shows both how much mathematics has to contribute to computing when competent mathematicians actually look at what computing is (rather than treating it as if it were a branch of mathematics), and how much they can miss the current temper of computing. ... They make frequent experimental verifications of their theories, thus showing that they regard computing as a science whose results are to be accepted or rejected by the final authority of experience. ... The book is a significant contribution. The political situation in Czechoslovakia has not been mentioned up to now in Babuska's biography since it has played a relatively small part. The Communists had seized control of the country in 1948 and it was under strong Soviet influence over the following years. Mathematics was allowed to develop without interference, however, and the applied and computational methods developed by Babuska found favour. Beginning in 1964 reformers had won many concessions which became more clear-cut in early 1968 when the country began to implement "socialism with a human face". The reforms came to a sudden end, however, in August 1968 when Soviet tanks rolled into Prague. Babuska had just been appointed as a professor at the Charles University of Prague but, given the political situation, he travelled with his family to the United States where he spent a year as a visiting professor at the Institute for Fluid Dynamics and Applied Mathematics at the University of Maryland at College Park. He was given a permanent appointment as a professor at the University of Maryland in the following year and he held this position until 1995. He was then appointed Professor of Aerospace Engineering and Engineering Mechanics, Professor of Mathematics, and appointed to the Robert Trull Chair in Engineering at the University of Texas at Austin. Although now over 85 years of age, he continues to hold these positions. After coming to the United States, Babuska became the world-leading expert in finite element analysis. The authors of [2] summarise his important contributions to this area:- During his 27 year career at the University of Maryland, Professor Babuska established himself as the unquestionable leader of the international finite element community. In his landmark paper in 1971, Ivo introduced the discrete inf-sup condition, generalizing the results of J Cea and R Varga, and setting the theoretical framework for stability and convergence analysis of arbitrary linear problems. Three years later, F Brezzi reinforced the formalism for problems with constraints, and the name of the famous discrete BB condition was coined. ... Ivo has had a unique ability to foresee the development of the field of finite elements. He has been the force behind many developments in this field. His work with W Rheinboldt on a-posteriori error estimation has essentially started the field on adaptive finite element methods. In a landmark paper in 1979 with B Kellogg and J Pitkäranta, the effect of h-adaptivity on the convergence rates for problems with singularities was explained. In the late seventies, Barna Szabo convinced Ivo to reexamine the then established concept of higher order methods, and the p-version of the Finite Element Method was born. The p-method turned out to be much less sensitive to incompressibility constraints (work with M Vogelius) and locking effects in the analysis of thin-walled structures. The monograph on the p-method with B Szabo (1991) reaches far outside of the constraints of the mathematical community and has become a standard reference for engineers practicing higher order methods. The monograph referred to in this quote, written in collaboration with Barna Szabo, is Finite Element Analysis (1991). The authors write in the Preface:- Our purpose in writing this book is to introduce the finite element method to engineers and engineering students in the context of the engineering decision-making process. Basic engineering and mathematical concepts are the starting points. Key theoretical results are summarized and illustrated by examples. Focus is on the developments in finite element analysis technology during the 1980s and their impact on reliability, quality assurance procedures in finite element computations, and performance. The principles that guide the construction of mathematical models are described and illustrated by examples. The reviewer [5] writes:- Numerous books on the finite element method with a variety of objectives have appeared recently, so many that it would be quite lengthy to compare even a representative number of them. The present book has extensive detail with regard to examples, and its coverage of topics in linear elasticity is exhaustive. Both of these are essential for the book to be successful with an engineering audience. Richard Scott writes in a review:- I like this book. It is a very nice, and somewhat unique, blend of theory and engineering practice. ... I think the authors have succeeded admirably in their goals and I recommend the text for serious consideration for a first graduate course on finite elements. In 2001, in collaboration with Theofanis Strouboulis, Babuska published The finite element method and its reliability. Carsten Carstensen writes:- The reliability of a given numerical approximation is one essential task in applied science and engineering. Here, two leading scientists devote six chapters on eight hundred pages to it and fix the state of the art of rigorous 'a posteriori' finite element error analysis. Babuska, with Ivan Hlavácek and Jan Chleboun, published Uncertain input data problems and the worst scenario method in 2004. Among the many other services to mathematics which Babuska has given, we mention the many journals which have benefited by his accepting a position on their editorial board: Communications in Applied Analysis; Communications in Numerical Methods in Engineering; Computer & Mathematics; Computer Methods in Applied Mechanics and Engineering; Computers and Structures; Communications in Applied Analysis; International Journal for Numerical Methods in Engineering; Modelling and Scientific Computing; Numerical Mathematics - A Journal of Chinese Universities; Numerical Methods for Partial Differential Equations; and Siberian Journal of Computer Mathematics. Babuska has received many awards for his contributions: the Czechoslovak State Prize for Mathematics (1968); the Humbolt Senior US Scientist Award of Federal Republic of Germany (1976); the Medal for Merit in the Development of Mechanics of the Czech Society for Mechanics (1993); and the George David Birkhoff Prize in Applied Mathematics awarded jointly by the American Mathematical Society and the Society for Industrial and Applied Mathematics (1994):- ... for important contributions to the reliability of finite element methods, the development of a general framework for finite element error estimation, and the development of p and h-p finite element methods... In 1995 Babuska was awarded the John von Neumann Award by the Association for Computational Mechanics for:- ... extraordinary contributions and the breadth and depth of his work, and their importance to the broad fields of computational mechanics Babuska ended his Acceptance Speech, which examined the legacy of von Neumann, with these words:- Mr President, I would like to thank you again for the great honour that has been bestowed upon me and to express my opinion that nearly 40 years after the death of von Neumann, a towering scientific figure of the 20th Century, we, who work in computational mechanics, can still learn tremendously from the legacy, ideas and philosophy of John von Neumann. The Union of Czech Mathematicians and Physicists made him an honorary member in 1996 and, in the same year, presented him with their Commemorative Medal. He received the Bolzano Medal from the Czech Academy of Sciences in 1997 and the Honorary Medal "De Scientia et Humanitate Optime Meritis" from the same Academy in 2006. He has received honorary degrees from the University of Westminster, London (1996), Brunel University, London (1996), Charles University, Prague (1997), and the Helsinki University of Technology (2000). He was elected a fellow of the International Association of Computational Mechanics in 2002 and of the World Innovation Foundation in 2004. He was elected a member of the European Academy of Sciences in 2003 and of the National Academy of Engineering in 2005. The International Astronomical Union named asteroid No. 36030 "Babuska" in his honour in 2002. Ivo Babuska is married to Renots. Article by: J J O'Connor and E F Robertson [http://www-history.mcs.st-andrews.ac.uk/Biographies/Babuska.html]

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Search this site: Plus Blog News from the world of maths: Maths on telly and radio Maths on telly and radio BBC Four will launch a new three-part TV series on maths at 9pm this coming Monday the 6th of October. In The story of maths Marcus du Sautoy, Oxford mathematician and one of the UK's finest maths popularisers, describes the often surprising lives of the great mathematicians, explains the development of the key mathematical ideas and shows how — in a multitude of unusual ways — those ideas underpin the science, technology and culture that shape our world. The first episode will look at the contributions from the ancient Greeks. To get you into the mood, have a look at du Sautoy's Plus articles The prime number lottery, The music of the primes, and Beckham in his prime number. If you prefer the radio, then tune into BBC Radio 4's In our time on Thursday the 9th of October at 9am and again at 9.30pm. Melvyn Bragg together with experts including John D Barrow will discuss Gödel's infamous incompleteness theorem. posted by Plus @ 11:46 AM 6 Comments: At 1:46 PM, Anonymous Do you think "The story of maths" on BBC 4, will be viewable in some format for those outside the UK? I believe Simon Singh's "Fermat's Last Theorem" documentary originally aired on BBC, but was distributed to PBS US. I enjoyed it so much, I went out and bought the book. It would be nice to listen/watch these intelligent programs in a more timely fashion, rather than several years later :-) I've been able to download a few BBC mp3 files and listen to them on the go. Sounds very interesting! Is it possible to view BBC4 without living in the UK? At 11:29 AM, Plus said... As far as we can tell there's no easy way to watch BBC Four from abroad, but we're trying to find out how to get hold of the programme. We'll let you know when we know.... At 12:38 PM, &

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Search this site: Plus Blog Plus Advent Calendar Door #1: I'm free... aren't I? Are you reading this page because you decided to or because you were destined to from the start of time? Plus shares Mick's sentiment that we are free to do what we want, any old time. But what does physics and mathematics have to say about free will? In one of our most mind-bending yet enjoyable investigations (published in January this year), we spoke to philosopher of physics Jeremy Butterfield, quantum physicist Anton Zeilinger, cosmologist and mathematician George Ellis and mathematician John Conway to find out more. Freedom and physics — Most of us think that we have the capacity to act freely. Our sense of morality, our legal system, our whole culture is based on the idea that there is such a thing as free will. It's embarrassing then that classical physics seems to tell a different story. And what does quantum theory have to say about free will? Free, from top to bottom? — A traditional view of science holds that every system — including ourselves — is no more than the sum of its parts. To understand it, all you have to do is take it apart and see what's happening to the smallest constituents. But the mathematician and cosmologist George Ellis disagrees. He believes that complexity can arise from simple components and physical effects can have non-physical causes, opening a door for our free will to make a difference in a physical world. John Conway: discovering free will (part I) — On August 19, 2004, John Conway was standing with his friend Simon Kochen at the blackboard in Kochen’s office in Princeton. They had been trying to understand a thought experiment involving quantum physics and relativity. What they discovered, and how they described it, created one of the most controversial theorems of their careers: The Free Will Theorem. John Conway: discovering free will (part II) — In this, the second part of our interview, John Conway explains how the Kochen-Specker Theorem from 1965 not only seemed to explain the EPR Paradox, it also provided the first hint of Conway and Kochen's Free Will Theorem. John Conway: discovering free will (part III) — In the third part of our interview John Conway continues to explain the Free Will Theorem and how it has changed his perception of the Universe. Back to the 2012 Plus Advent Calendar Mathematical fire fighters.... Siberia needs you! Make your childhood dreams come true and join the Paris Fire Brigade as part of the 2012-13 Mathematical Competitive Game. This year's game is brought to you by the French Federation of Mathematical Games, Société de Calcul Mathématique SA as well as The Paris Firemen Brigade and is open to anyone to enter either individually or in groups. The previous games have also all been based on real life problems – designing a bus or electricity network or searching for the best car itinerary in partnership with transport or electricity companies – and solving them usually takes several months of work. This year's game is to decide how best to use your limited resources of firefighters and equipment to fight fires and minimise their damage across the whole of Siberia. This huge region is broken up into cells defined by latitude and longitude, so these vary in size as well as the their level of population and landscape. There are mathematical descriptions of the ways the fires start, spread and the damage they cause as well as for modelling your firefighting capacities and the way they can combat the fires. Your job is to decide how to distribute your firefighters and specially equipped planes and then estimate the cost from damage from the fires in one summer, the cost of your firefighting strategy and the probability that your strategy to keep costs to this level will work. And there's not only mathematical glory and the gratitude of Siberia on offer, there's also up to 500 Euros in prizes for the winners. You can find the details of this year's competition

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Search this site: Plus Blog Plus Advent Calendar Door #19: Beautiful buildings The dramatic curved surfaces of some of the iconic buildings created in the last decade, such as 30 St Mary's Axe (AKA the Gherkin) in London, are only logistically and economically possible thanks to mathematics. Curved panels of glass or other material are expensive to manufacture and to fit. Surprisingly, the curved surface of the Gherkin has been created almost entirely out of flat panels of glass — the only curved piece is the cap on the very top of the building. And simple geometry is all that is required to understand how. A geodesic sphere. One way of approximating a curved surface using flat panels is using the concept of geodesic domes and surfaces. A geodesic is just a line between two points that follows the shortest possible distance — on the Earth the geodesic lines are great circles, such as the lines of longitude or the routes aircraft use for long distances. A geodesic dome is created from a lattice of geodesics that intersect to cover the curved surface with triangles. The mathematician Buckminster Fuller perfected the mathematical ideas behind geodesic domes and hoped that their properties — greater strength and space for minimum weight — might be the future of housing. To try to build a sphere out of flat panels, such as a geodesic sphere, you first need to imagine an icosahedron (a polyhedron made up of 20 faces that are equilateral triangles) sitting just inside your sphere, so that the points of the icosahedron just touch the sphere's surface. An icosahedron, with its relatively large flat sides, isn't going to fool anyone into thinking it's curved. You need to use smaller flat panels and a lot more of them. Divide each edge of the icosahedron in half, and join the points, dividing each of the icosahedron's faces into four smaller triangles. Projecting the vertices of these triangles onto the sphere (pushing them out a little til they two just touch the sphere's surface) now gives you a polyhedron with 80 triangular faces (which are no longer equilateral triangles) that gives a much more convincing approximation of the curved surface of the sphere. You can carry on in this way, dividing the edges in half and creating more triangular faces, until the surface made up of flat triangles is as close to a curved surface as you would like. Find out more about the Gherkin and other iconic buildings on Plus. Plus Advent Calendar Door #18: Clever sums How would you go about adding up all the integers from 1 to 100? Tap them into a calculator? Write a little computer code? Or look up the general formula for summing integers? Carl Friedrich Gauss as depicted on the (now defunct) German 10 Marks note. Legend has it that the task of summing those numbers was given to the young Carl Friedrich Gauss by his teacher at primary school, as a punishment for misbehaving. Gauss didn't have a calculator or computer, no one did at that time, but he came up with the correct answer within seconds. Here's how he did it. Notice that you can sum the numbers in pairs, starting at either end. First you add 1 and 100 to get 101. Next it's 2 and 99, giving 101 again. The same for 3 and 98. Continuing like this, the last pair you get is 50 and 51 and they give 101 again. Altogether there are 50 pairs all adding to 101, so the answer is 50 x 101 = 5050. Easy — if you're Gauss.

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Search this site: Plus Blog Maths in a minute: Not always 180 Over 2000 years ago the Greek mathematician Euclid came up with a list of five postulates on which he thought geometry should be built. One of them, the fifth, was equivalent to a statement we are all familiar with: that the angles in a triangle add up to 180 degrees. However, this postulate did not seem as obvious as the other four on Euclid's list, so mathematicians attempted to deduce it from them: to show that a geometry obeying the first four postulates would necessarily obey the fifth. Their struggle continued for centuries, but in the end they failed. They found examples of geometries that do not obey the fifth postulate. Spherical geometry Image: Lars H. Rohwedder. In spherical geometry the Euclidean idea of a line becomes a great circle, that is, a circle of maximum radius spanning right around the fattest part of the sphere. It is no longer true that the sum of the angles of a triangle is always 180 degrees. Very small triangles will have angles summing to only a little more than 180 degrees (because, from the perspective of a very small triangle, the surface of a sphere is nearly flat). Bigger triangles will have angles summing to very much more than 180 degrees. One funny thing about the length of time it took to discover spherical geometry is that it is the geometry that holds on the surface of the Earth! But we never really notice, because we are so small compared to the size of the Earth that if we draw a triangle on the ground, and measure its angles, the amount by which the sum of the angles exceeds 180 degrees is so tiny that we can't detect it. But there is another geometry that takes things in the other direction: Hyperbolic geometry Hyperbolic geometry isn't as easy to visualise as spherical geometry because it can't be modelled in three-dimensional Euclidean space without distortion. One way of visualising it is called the Poincaré disc. Take a round disc, like the one bounded by the blue circle in the figure on the right, and imagine an ant living within it. In Euclidean geometry the shortest path between two points inside that disc is along a straight line. In hyperbolic geometry distances are measured differently so the shortest path is no longer along a Euclidean straight line but along the arc of a circle that meets the boundary of the disc at right angles, like the one shown in red in the figure. A hyperbolic ant would experience the straight-line path as a detour — it prefers to move along the arc of such a circle. A hyperbolic triangle, whose sides are arcs of these semicircles, has angles that add up to less than 180 degrees. All the black and white shapes in the figure on the left are hyperbolic triangles. One consequence of this new hyperbolic metric is that the boundary circle of the disc is infinitely far away from the point of view of the hyperbolic ant. This is because the metric distorts distances with respect to the ordinary Euclidean one. Paths that look the same length in the Euclidean metric are longer in the hyperbolic metric the closer they are to the boundary circle. The figure below shows a tiling of the hyperbolic plane by regular heptagons. Because of the distorted metric the heptagons are all of the same size in the hyperbolic metric. And as we can see the ant would need to traverse infinitely many of them to get to the boundary circle — it is infinitely far away! Image created by David Wright. Hyperbolic geometry may look like a fanciful mathematical construct but it has real-life uses. When Einstein developed his special theory of relativity in 1905 he found that the symmetries of hyperbolic geometry were exactly what he needed to formulate the theory. Today mathematicians believe that hyperbolic geometry may help to understand large networks like Facebook or the Internet. You can read more about hyperbolic geometry in non-Euclidean geometry and Indra's pearls. Happy Tau Day!!

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From ChanceWiki Revision as of 15:19, 1 August 2011 by Bill Peterson (Talk | contribs) 3 Discussion of Ariely 4 The perils of genetic testing 5 In measuring hunger, quality may be more important than quantity 6 More on US visa lottery "I suspect the amygdala did not evolve to store odds ratios and heterogeneity P scores, but when an adverse event has prompted me to review the literature, I come away with a clearer understanding. There’s nothing like a baby free-floating in the abdomen to drive home the lessons from a prospective study of risk factors for uterine rupture. And that clarity of understanding will serve the next at-risk patient I encounter." Alison M. Stuebe, quoted from her article posted on the New England Journal of Medicine Health Policy and Reform blog. Submitted by Steve Simon. "On the nature of common errors. ...[M]ost of these mixups involve simple switches or offsets. These mistakes are easy to make, particularly if working with Excel or if working with 0/1 labels instead of names... [A] mixup in annotation affecting roughly a third of the data which was driven by a simple one-cell deletion from an Excel file coupled with an inappropriate shifting up of all values in the affected column only." Keith Baggerly and Kevin Coombes, discussion section in Deriving Chemosensitivity from cell lines: Forensic bioinformatics and reproducible research in high-throughput biology . “As every student of statistics quickly learns, statistical significance is not the same as substantive significance. …. The latter is often addressed by going beyond the question of whether an association exists to ask how strong the relationship is. But there is an even more basic problem in this case – statistical significance tests are inappropriate for this data analysis. They can determine whether two values derived from a random sample are different enough that they cannot be attributed to chance (sampling error). But the FARS [Fatality Analysis Reporting System] data are not a sample of some larger data set; they represent the entire data set of fatal auto accidents. The authors [of this study] note this problem but choose to use this approach anyway, following ‘official recommendations and our previous practice.’” Robert Lichter, “Alcohol and Accidents: Can One Drink Kill?” STATS, July 14, 2011 Submitted by Margaret Cibes "Coefficients for physical traits are on the average +0.2--not so high as for personality traits (+0.4) or religion (+0.9), but still significantly higher than zero. For a few physical traits the correlation is even higher that 0.2--e.g., an astonishing 0.61 for length of middle finger. At least unconsciously, people care more about their spouse's middle-finger length than about his or her hair color and intelligence!" Jared Diamond, in The Third Chimpanzee: The Evolution and Future of the Human Animal (p. 102) (Note: See 2D:4D in Chance News 44 for some other musings about the mysteries of finger length distributions.) Freshman composition, for example, “does not demand that faculty ask existential questions.” Ditto for courses in “Security and Protective Services,” and “Business Statistics.” These are, she says, “fields of study with fairly definitive answers” and it would be hard to argue that they are “essential to civilization.” Those who teach these and similarly vocational subjects “don’t really need the freedom to ask controversial questions in discussing them.” Naomi Schaefer Riley, The Faculty Lounges: and Other Reasons Why You Won't Get the College Education You Paid For as quoted in Vocationalism, academic freedom and tenure, by Stanley Fish, New York Times, 11 July 2011 Discussion of Ariely A post in Chance News 74 described Dan Ariely's 2008 book Predictably Irrational: The Hidden Forces That Shape Our Decisions as a great summer read [1], while pointing out that it was not written as an academic work. Paul Alper wrote to say that he had occasion to review the book in the context of some related work, and had identified some statistical concerns. As Paul writes: Ariely enjoys concocting experiments to demonstrate the irrationality. For example, he finds that satisfaction with a product depends on the price paid for the product--for example Bayer aspirin vs. the identical generic. Or, the enticing but utterly misleading “Free gift” will alter a decision. Reviewers loved his book. Nonetheless, there are some serious shortcomings. He invariably gives the average value of one group (e.g., satisfaction of Bayer aspirin users) compared to the other group (e.g., satisfaction of generic aspirin users) but he almost never indicates the variability. Averages alone are meaningless. Almost never does he state how many subjects are involved in each arm of a study. Almost all of his samples are convenience ones, rather than random samples. Almost all of his samples are MIT students, but his implicit inference is to the world at large. His examples of predictable irrationality appear unfailingly successful leading me to suspect a “file-drawer” issue--experiments which showed nothing in particular or the negative of what he theorizes, are put aside and not counted. The earlier post also noted that Ariely has a new book, The Upside of Irrationality: The Unexpected Benefits of Defying Logic at Work and at Home. This was reviewed by the New York Times; you can find a link to the review and read Ariely's reaction on his blog here. He notes that that he had consciously adopted a more conversational style in the book, and that this had drawn some criticism from the Times. He invited readers to submit their own opinions on this. Readers come down on both sides, and it is interesting to read the comments. One statistically minded reader wrote: Of course your [sic] irrationally asking for personal thoughts in comments instead of a (slightly) more accurate poll or a (very) accurate scientific survey. Submitted by Bill Peterson, based on a message from Paul Alper The perils of genetic testing How Bright Promise in Cancer Testing Fell Apart by Gina Kolata, The New York Times, July 7, 2011. We have seen a lot of advances in genetics recently, and there has been the hope that these would translate into better clinical care. But making the bridge from the laboratory to clinical practice has been much more difficult than expected. A program at Duke, for example, was supposed to identify weak spots in a cancer genome so that drugs could be targeted to that weak spot rather than just trying a range of different drugs in sequence. But the research at Duke turned out to be wrong. Its gene-based tests proved worthless, and the research behind them was discredited. Ms. Jacobs died a few months after treatment, and her husband and other patients’ relatives are suing Duke. The problems at Duke are not an isolated problem. The Duke case came right after two other claims that gave medical researchers pause. Like the Duke case, they used complex analyses to detect patterns of genes or cell proteins. But these were tests that were supposed to find ovarian cancer in patients’ blood. One, OvaSure, was developed by a Yale scientist, Dr. Gil G. Mor, licensed by the university and sold to patients before it was found to be useless. The other, OvaCheck, was developed by a company, Correlogic, with contributions from scientists from the National Cancer Institute and the Food and Drug Administration. Major commercial labs licensed it and were about to start using it before two statisticians from M. D. Anderson discovered and publicized its faults. The two statisticians, Keith Baggerly and Kevin Coombes, have made a career of debunking medical claims. In 2004, they (along with another M.D. Anderson statistician, Jeffrey Morris, published a paper that demonstrated serious flaws in the use of proteomic mass spectra to identify early ovarian cancer from normal tissue. The complex method proposed by Petrocoin et al in 2002 was apparently an artefact of equipment drift that would have been prevented if the original researchers had taken simple steps like randomizing the order of analysis of cancer and normal tissues. Baggerly and Coombes had also found problems with the data supporting the Duke test. Dr. Baggerly and Dr. Coombes found errors almost immediately. Some seemed careless — moving a row or a column over by one in a giant spreadsheet — while others seemed inexplicable. The Duke team shrugged them off as "clerical errors." Even though Baggerly and Coombes published a critique in a statistics journal, the Duke team continued to promote their genetic test. It was something else entirely that caused the problems with the Duke test to be treated seriously by the broader research community. The situation finally grabbed the cancer world’s attention last July, not because of the efforts of Dr. Baggerly and Dr. Coombes, but because a trade publication, The Cancer Letter, reported that the lead researcher, Dr. Potti, had falsified parts of his résumé. He claimed, among other things, that he had been a Rhodes scholar. Researchers in this area have a new-found sense of humility. With such huge data sets and complicated analyses, researchers can no longer trust their hunches that a result does — or does not — make sense. In measuring hunger, quality may be more important than quantity A Taste Test for Hunger Robert Jensen and Nolan Miller, The New York Times, July 9, 2011. The traditional measure of global hunger is the number of calories consumed. If you consume less calories than you need, then you are classified as hungry. But this had led to some paradoxical results. There is an alternative way of measuring hunger. You need to start with a baseline, namely the share of calories people get from the cheapest foods available to them: typically staples like rice, wheat or cassava. We call this the “staple calorie share.” We measure how many calories people get from these low-cost foods and how much they get from more expensive foods like meat. The greater the share of calories they receive from the former, the hungrier they are. The rationale for this is simple. Imagine you are a poor consumer in a developing country. You have very little money in your pocket, not enough to afford all the calories you need. And suppose you have only two foods to choose from, rice and meat. Rice is cheap and has a lot of calories, but you prefer the taste of meat. If you spent all your money on meat, you would get too few calories. You might do this every so often, but usually you would spend almost all of your money on rice; when faced with true hunger, taste is a luxury you can’t afford. But suppose you had a bit more money. You would probably add some meat to your diet, because now you can afford to do so while still getting the calories you need. You might even like meat so much that you start to switch away from rice even if you haven’t quite met your complete calorie needs, as long as you aren’t too far below. The authors argue that this approach removes some of the variations associated with the traditional calorie count method: some people need more calories than others, for example. They illustrate how the new measure of hunger performs better than the traditional measure in explaining trends in hunger in China from 1991 to 2001. 1. What are some other ways that you might assess hunger on a global scale? 2. What are some possible pitfalls to the use of this new measure of hunger. More on US visa lottery "Plaintiffs Lose Fight Over Visa Lottery" by Miriam Jordan, The Wall Street Journal, July 15, 2010 Applicants who had been notified that they had won a US visa in the May 2011 lottery drawing have lost a federal court case in which they tried to stop a new drawing of the 2011 lottery. (See Chance News 74[2].) It turns out that the May sampling process had violated a legal requirement of randomness: "a computer error caused 90% of the 50,000 winners to be selected from the first two days of applications instead of from the entire 30-day registration period." The court hearing earlier this week focused on the meaning of "random." Lawyers for the plaintiffs argued that they had been randomly chosen and that they didn't know that by filling their applications in the first two days that they would gain any advantage. They also contended that the "outcome was indeed not uniform, but nevertheless still random as required by law." The State Department argued that the results didn't represent a fair, random selection. This example reminds me of the 1970 Vietnam War draft lottery issues related to sampling in an inadequately stirred-up jar of birthday capsules. For this first draft lottery since 1940, 366 balls were dumped from a box into a glass jar, from which balls were drawn. The result was that higher numbers (fall birthdays) were somewhat more likely to have been chosen first and, consequently, their holders somewhat more likely to have been drafted into military service. See a 10-minute video of the actual Fall 1969 drawing for the 1970 draft, “CBS news draft lottery nov 1969”, and/or Norton Starr's 1997 article, "Nonrandom Risk: The 1970 Draft Lottery", for the raw data and a statistical discussion of it. Submitted by Margaret Cibes 1. The article does not specify the daily distribution of applications over the 30-day application period. (a) If the daily distribution of applications had been uniform over the 30 days, what percent of the 50,000 winners would you have expected from a random selection process? (b) If the distribution of applications had been skewed so that an overwhelming majority of them had been received during the first two days, could the resulting 22,000 "winners" have been expected? 2. Would you consider the May result an "error"? Would you attribute the "error" to a "computer"? 3. Might you have agreed with the plaintiffs that the May outcome was "still random"? Note: See Diversity Visa Lottery 2011 Results for official figures and other information from the U.S. State Department. Retrieved from "http://test.causeweb.org/wiki/chance/index.php?title=Chance_News_75&oldid=13931" Personal tools

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Math of Planet Earth 2013 Set to Debut at JMM Polar melt ponds, April 2012. The launch of Mathematics of Planet Earth 2013 is slated for January at the Joint Math Meetings in San Diego, where a highlight of the scheduled events is the Porter Public Lecture by Ken Golden of the University of Utah. Golden has been to Antarctica seven times and the Arctic eight times in pursuit of career-long scientific interest in sea ice; among his other research interests are climate change, composite materials, phase transitions, and inverse problems. He will devote the Porter lecture in part to a discussion of "how mathematical models of composite materials and statistical physics are being used to study key sea ice processes, such as freezing and melting." A better understanding of such processes, he says, is needed "to improve projections of the fate of Earth's sea ice packs, and the response of polar ecosystems."Named a SIAM Fellow in 2009 "for extraordinary interdisciplinary work on sea ice," Golden sent SIAM a selection of photos from some of his recent expeditions, including the one shown here. Another, showing him at work in the field, appears below, along with a brief introduction to MPE2013.Math of Planet Earth 2013You have probably heard that 2013 is the year of Mathematics of Planet Earth, but do you know what that means?MPE2013 began as an initiative of North American math institutes, a way of highlighting contributions of the mathematical sciences to understanding our planet and biosphere. In the short time since MPE2013 was founded, more than 100 organizations in the mathematical sciences, including SIAM, have become partners in this worldwide effort to promote the contributions of mathematics to areas of the geosciences, biology, climate, and related fields.The U.S. kickoff for MPE2013 will occur in January at the Joint Math Meetings in San Diego. Among the many MPE2013-related sessions on the program is the Porter Public Lecture, a joint project of the AMS, MAA, and SIAM that was chosen to fit the MPE2013 theme. Ken Golden (who in 2009 gave the SIAM invited lecture at the JMM) will speak about his team's work in Antarctica, including the study of sea ice and its role in climate modeling. Many additional special sessions at the Joint Math Meetings will be related to MPE2013. Events scheduled for the year include sponsored public lectures, workshops, and summer schools on MPE2013-related themes. In addition, many conferences in the mathematical sciences have elected to make Math of Planet Earth a major theme; among them are the 2013 SIAM Annual Meeting (San Diego, July 8�12) and the SIAM Conference on Geosciences (June 17�20 in Padova, Italy). The organizers hope that MPE2013 will attract the attention of the media and, with plans for a daily blog, hope to enlist people from the community to serve as bloggers (go to http://mpe2013.org/ to volunteer).An example of an activity scheduled for MPE2013 is a series of public lectures to be given around the world. Held in traditional lecture format in a public hall, each of the lectures will be streamed live to viewers around the world. Topics include biodiversity ("Motility: Molecules, Mechanics, Mathematics and Machines," by L. Mahadevan), epidemiology ("Puzzles in the Pattern of Plagues," by David Earn), and climate ("Using Mathematics to Combat Climate Change," by Ron Dembo).Stay tuned. You may see MPE2103 featured on public television in the U.S. or highlighted in the media. We hope that readers will seek ways to participate and help show the contributions that our community makes to understanding our world and the living things around us.---JMCA recent request from Jim Crowley for details about the 2013 Porter lecture brought the following reply: "Ken Golden is on an Australian icebreaker in Antarctica until November 5. . . ." Shown here coring ice during an earlier polar expedition, Golden is clearly a hands-on applied mathematician. The Porter lecture promises to be both visually spectacular (with video from recent Antarctic expeditions to be shown) and completely up to date!

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Buffon's Needle, the Easy Way Buffon's needle is a popular probability problem. Rule lines on the floor a distance d apart and toss a needle of length l<d onto it. What is the probability that the needle crosses a line? A solution is described at wikipedia but it involves a double integral and some trigonometry. Nowhere does it mention that there is a less familiar but much simpler proof, though if you follow the links you'll find it. In addition, the usual solution involves π but gives little intuition as to why π appears. The simpler proof reveals that it appears naturally as a ratio of the circumference of a circle to its diameter. I've known this problem since I was a kid and yet I hadn't seen the simpler proof until a friend sold me his copy of Introduction to Geometric Probability for $5 a few days ago.So instead of solving Buffon's needle problem we'll solve what appears to be a harder problem: when thrown, what is the expectation of the number of times a rigid curved (in a plane) wire length l (no restriction on l) crosses one of our ruled lines d apart? Here's an example of one of these 'noodles'. It crosses the ruled lines three times:Expectation is linear in the sense that E(A+B) = E(A)+E(B). So if we imagine the wire divided up into N very short segments of length l/N the expectation for the whole wire must be the sum of the expectations for all of the little pieces. If the wire is well behaved, for N large enough the segments are close to identical straight line segments. Here's a zoomed up view of a piece of our noodle:For a small straight line segment the expectation must simply be a function of the length of the segment. The expectation for the whole wire is the expectation for one segment multiplied by the number of segments. In other words, the expectation is proportional to the length of the wire and we can write E(l)=kl for some constant k.Now we know it's proportional to the length, we need to find the constant of proportionality, k. We need to 'calibrate' by thinking of a noodle shape where we know in advance exactly how many times it will cross the ruled lines. The following picture gives the solution:A circle of diameter d will almost always cross the lines in two places. The length of this wire is πd so E(πd)=2 and k=2/πd.The expected number of crossings for a wire of length l is 2l/πd. A needle of length l<d can intersect only zero or one times. So the expected value is in fact the probability of intersecting a line. The solution is 2l/πd.No integrals needed.The expected number of crossings is an example of an invariant measure, something I've talked about before. There are only a certain number of functions of a noodle that are additive and invariant under rotations and just knowing these facts is almost enough to pin down the solution.PuzzleNow I can leave you with a puzzle to solve. In the UK, a 50p coin is a 7 sided curvilinear polygon of constant width. Being constant width means a vending machine can consistently measure its width no matter how the coin is oriented in its plane. Can you use a variation of the argument above to compute the circumference of a 50p coin as a function of its width? ketil Expectation is linear in the sense that E(A+B) = E(A)+E(B). So if we imagine the wire divided up into N very short segments of length l/N the expectation for the whole wire must be the sum of the expectations for all of the little pieces.I don't understand this - that's only when A and B are independent, isn't it? Clearly, this isn't the case here -- as you discuss later, a line forming a circle of diameter d has probability 1 of crossing a line, a straight line of length pi*d will have a non-zero probability of not crossing a line.Or? Fergal Cool proof (but I wonder much effort does it take to formalise the linear approximation bit).What do you mean by "constant width"? Do you mean, given a 50p coin and a ruler both in fixed orientatio but you can slide them around, that the greatest width measurable is the same, no matter the orientation? ketil,Remember expectation is a sum or integral so it's always linear regardless of independence. Fergal,Imagine putting the coin in a box and sliding the left and right sides of the box as close as possible. The distance between the sides is the width. For a 50p coin the minimum width is independent if the orientation. "For a 50p coin the minimum width is independent if the orientation."Surely maximum. When you can't squeeze the sides of the box together any further you have minimised the width of the box, not the coin.It wasn't until I'd gone and figured out the answer via basic geometry that I realised your clarification of "width" almost gives away the answer. That said, proving it via geometry is quite easy given one theorem about angles in a circle. @ketil: A straight line of length pi*d also has a non-zero chance of crossing 3 lines. This must happen often enough to offset the times it crosses 0 or 1 lines. Leon Smith I don't think this argument is complete. Let me ask the following (possibly stupid) question:Let's modify the problem so that we are dealing with one line, not an infinite number of parallel lines spaced a uniform distance apart. And let's restrict ourselves to a line segment, not some arbitrary well-behaved curve. Now, what's the expected number of intersections between the line and the segment?Clearly, the expected value must be between 0 and 1, because the only possible outcomes is that they don't intersect, or that they intersect in exactly one location. However, your reasoning would seem to suggest that by choosing a sufficiently long line segment, the expected number of intersections could be greater than one.Now, I haven't worked out the resolution in detail, but I suspect the resolution to this paradox is carefully considering the probability distribution for the location of the line segment: first of all you can't have a uniform distribution over an infinite plane, so let's say we choose the distance from an endpoint of the segment to the line according to a Gaussian distribution centered on the line, and choose the orientation uniformly. If we were to "tack on" another line segment onto the end of the first, it's probability distribution should be a flatter curve, again centered on the line. It seems to me that your argument would have to depend upon the translational symmetry of your parallel lines: in effect you can have a uniform distribution over the entire plane by choosing a uniform distribution inside a given box that can tile the plane. Leon,You're right that fully writing out a pair consisting of (1) a rigorous statement of the problem and (2) a rigorous proof of the solution, would require quite a bit more work.This assumes translational symmetry and clearly there is no translationally symmetric probability distribution. Nonetheless, if someone gave me this as a problem in the real world (for a big enough floor) I wouldn't hesitate to consider a uniform distribution for the position relative to the nearest line.Interestingly, Geometric Probability Theory seems to take the line that it's expectation that's fundamental. In that case there's no difficulty with assuming translation invariance. But I need to read more of the book to find out, so take that with a pinch of salt. randomdeterminism Reminds me of a story that I heard about a probability prof trying to make the same point in a class using a more emphatic approach:The prof brought a plate of spaghatti to class, threw the spaghatti on the floor, drew a square, counted the number of strands that cross the square and gave an estimate for pi! Yoo randomdeterminism, since a square is not a collection of equally spaced parallel lines, I'm confused about how the professor's spaghetti can reveal an approximation of pi. jqb "A circle of diameter d will almost always cross the lines in two places."Not "almost".As for "little intuition as to why π appears" -- when I first learned of Buffon's needle I thought it was obvious, because a vertical needle of length d is sure to cross a line and a horizontal needle is sure not to, and the probability is obviously related to the angle, most likely to sin(θ)."since a square is not a collection of equally spaced parallel lines"Uh, seriously? "What Category do Haskell Types and Functions Live... Vectors, Invariance, and Math APIs

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