Posts Tagged ‘statistics’
“We couldn’t build quantum computers unless the universe were quantum and computing… We’re hacking into the universe.”*…
… in the process of which, as Ben Brubaker explains, we learn some fascinating things…
If you want to tile a bathroom floor, square tiles are the simplest option — they fit together without any gaps in a grid pattern that can continue indefinitely. That square grid has a property shared by many other tilings: Shift the whole grid over by a fixed amount, and the resulting pattern is indistinguishable from the original. But to many mathematicians, such “periodic” tilings are boring. If you’ve seen one small patch, you’ve seen it all.
In the 1960s, mathematicians began to study “aperiodic” tile sets with far richer behavior. Perhaps the most famous is a pair of diamond-shaped tiles discovered in the 1970s by the polymathic physicist and future Nobel laureate Roger Penrose. Copies of these two tiles can form infinitely many different patterns that go on forever, called Penrose tilings. Yet no matter how you arrange the tiles, you’ll never get a periodic repeating pattern.
“These are tilings that shouldn’t really exist,” said Nikolas Breuckmann, a physicist at the University of Bristol.
For over half a century, aperiodic tilings have fascinated mathematicians, hobbyists and researchers in many other fields. Now, two physicists have discovered a connection between aperiodic tilings and a seemingly unrelated branch of computer science: the study of how future quantum computers can encode information to shield it from errors. In a paper posted to the preprint server arxiv.org in November, the researchers showed how to transform Penrose tilings into an entirely new type of quantum error-correcting code. They also constructed similar codes based on two other kinds of aperiodic tiling.
At the heart of the correspondence is a simple observation: In both aperiodic tilings and quantum error-correcting codes, learning about a small part of a large system reveals nothing about the system as a whole…
Fascinating: “Never-Repeating Tiles Can Safeguard Quantum Information,” from @benbenbrubaker in @QuantaMagazine.
Plus- bonus background on tiling.
* “We couldn’t build quantum computers unless the universe were quantum and computing. We can build such machines because the universe is storing and processing information in the quantum realm. When we build quantum computers, we’re hijacking that underlying computation in order to make it do things we want: little and/or/not calculations. We’re hacking into the universe.” –Seth Lloyd
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As we care for qubits, we might send carefully-calculated birthday greetings to Herman Hollerith; he was born on this date in 1860. A statistician and inventor, he was a seminal figure in the development of data processing: he invented (for the 1890 U.S. Census) an electromechanical tabulating machine for punched cards to assist in summarizing information (and, later, for use in accounting). His invention of the punched card tabulating machine, which he patented in 1884, marked the beginning of the era of mechanized binary code and semiautomatic data processing systems– and his approach dominated that landscape for nearly a century.
The company that Hollerith founded to exploit his invention was merged in 1911 with several other companies to form the Computing-Tabulating-Recording Company. In 1924, the company was renamed “International Business Machines” (or, as we know it, IBM).
“Don’t let us forget that the causes of human actions are usually immeasurably more complex and varied than our subsequent explanations of them”*…
Further, in a fashion, to yesterday’s post: Patricia Fara explains how the tension between religion and science as arbiters of knowledge came to head in the French Revolution, and how that inspired Lambert Adolphe Jacques Quetelet, a Belgian astronomer, mathematician, statistician, and sociologist, to introduce a radically new way of thinking about human beings:
… God had been forcefully excluded from astronomy during the French Revolution, when Pierre-Simon Laplace rewrote Newton’s ideas to create his deterministic cosmos, in which scientific laws govern every movement of every planet with no need for divine intervention. Inspired by this success, a Belgian astronomer called Alphonse Queteler decided that human societies are also controlled by laws. Each country has its own statistical patterns that remain constant from year to year–suicide and crime rates, for instance–and so Quetelet suggested that an ‘average man’ can consistently encapsulate a nation’s characteristics. Politicians should, Quetelet prescribed, operate like social physicists and try to improve average behaviour rather than worry about extreme anomalies. For him, variations from the statistical mean were–like planetary wobbles–imperfections to be smoothed out so that overall progress could be ensured.
Quetelet had introduced a radically new way of thinking about human beings. As one of his admirers put it, ‘Man is seen to be an enigma only as an individual, in mass, he is a mathematical problem.’ Quetelet’s successors took his ideas in many different directions. For one thing, his work was valuable politically because it could be interpreted in different ways. While conservatives insisted that little could be done to alter the current system, radicals accused governments of impeding the natural course of progress, and Utopians–such as Karl Marx–envisaged harmonious societies governed by nature’s own laws guaranteeing improvement. Data collection projects proliferated, and statisticians searched for laws governing every aspect of life, ranging from the weather to the growth of civilization, from stock market fluctuations to the incidence of disease. Many scientists took their ideas from Quetelet rather than from abstract textbooks–but they added their own twist. Whereas Quetelet regarded individual deviations from the norm as errors to be eliminated, scientists set out to study how variations occur…
An excerpt from Fara’s Science: A Four Thousand Year History, via the invaluable Delanceyplace.com (@delanceyplace): “God, Science, and Data.”
* Fyodor Dostoevsky, The Idiot
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As we focus on frames, we might spare a thought for a man who kept his eye on the individual, Wilhelm Reich. A medical doctor and psychoanalyst, he was a member of the second generation of analysts after Sigmund Freud. Reich developed a system of psychoanalysis concentrating on overall character structure, rather than on individual neurotic symptoms. His early work on psychoanalytic technique was overshadowed by his involvement in the sexual-politics movement and by “orgonomy,” a pseudoscientific system he developed. He also built a device he called a cloud buster, with which he claimed he could manipulate the weather by manipulating the “orgone” in the atmosphere. Reich’s claims aroused much controversy; and he was taken to court for fraud by the Food and Drug Administration (FDA). The court ordered his books and research burned and his equipment destroyed. Reich was sentenced to prison where he died of heart failure on this date in 1957.
“Why, sometimes I’ve believed as many as six impossible things before breakfast”*…
Imaginary numbers were long dismissed as mathematical “bookkeeping.” But now, as Karmela Padavic-Callaghan explains, physicists are proving that they describe the hidden shape of nature…
Many science students may imagine a ball rolling down a hill or a car skidding because of friction as prototypical examples of the systems physicists care about. But much of modern physics consists of searching for objects and phenomena that are virtually invisible: the tiny electrons of quantum physics and the particles hidden within strange metals of materials science along with their highly energetic counterparts that only exist briefly within giant particle colliders.
In their quest to grasp these hidden building blocks of reality scientists have looked to mathematical theories and formalism. Ideally, an unexpected experimental observation leads a physicist to a new mathematical theory, and then mathematical work on said theory leads them to new experiments and new observations. Some part of this process inevitably happens in the physicist’s mind, where symbols and numbers help make invisible theoretical ideas visible in the tangible, measurable physical world.
Sometimes, however, as in the case of imaginary numbers – that is, numbers with negative square values – mathematics manages to stay ahead of experiments for a long time. Though imaginary numbers have been integral to quantum theory since its very beginnings in the 1920s, scientists have only recently been able to find their physical signatures in experiments and empirically prove their necessity…
Learn more at “Imaginary numbers are real,” from @Ironmely in @aeonmag.
* The Red Queen, in Lewis Carroll’s Through the Looking Glass
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As we get real, we might spare a thought for two great mathematicians…
Georg Friedrich Bernhard Riemann died on this date in 1866. A mathematician who made contributions to analysis, number theory, and differential geometry, he is remembered (among other things) for his 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, regarded as one of the most influential papers in analytic number theory.
Andrey (Andrei) Andreyevich Markov died on this date in 1922. A Russian mathematician, he helped to develop the theory of stochastic processes, especially those now called Markov chains: sequences of random variables in which the future variable is determined by the present variable but is independent of the way in which the present state arose from its predecessors. (For example, the probability of winning at the game of Monopoly can be determined using Markov chains.) His work on the study of the probability of mutually-dependent events has been developed and widely applied to the biological, physical, and social sciences, and is widely used in Monte Carlo simulations and Bayesian analyses.
“Torture the data, and it will confess to anything”*…
Add movement to a bar chart, and you’ve got yourself an audience-pleaser. These so-called “bar chart races” are not popular with data visualization experts– but what do experts know?…
I’m not a betting man. But I do enjoy a good bar chart race — a popular way to visually display and compare changing data over time. Bars lengthen and shorten as time ticks away; contenders accordingly hop over each other to switch places in the ranking. Will your favorite keep their lead? Look at that surprise challenger rush to the front! Meanwhile, furious battles are waged for the middle and even the lower spots on the list.
Bar chart races are a spectacular way to animate certain types of information, but the so-called dataviz community is skeptical. Many data visualization specialists complain that bar chart races are like a sugar rush: a lot of entertainment, but very little analysis. Big on grabbing attention, small on conveying causality. Instead of good seats at the data ballet, you get standing room only at the information dog track.
Well, all that may be true. But when is the last time you’ve been glued to a statistic about global coffee production? Bar chart races are fun to watch, not least because you can pick a favorite early on and get to see them win — or lose. In other words, you’re emotionally invested in the animation in a way that’s lacking from static stats.
Bar chart races are used for just about any dataset that can be quantified over time: best-selling game consoles, most trusted brands, highest grossing movies…
Any dataset that can be quantified over time can be turned into a contest that is both exciting and (a little bit) enlightening: from @VeryStrangeMaps, 10 examples of “Bar chart races: short on analysis, but fun to watch,” for example…
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As we ruminate on representation, we might check our watches: it was on this date in 1918 that the Standard Time Act (AKA, the Calder Act) became effective. Passed by Congress earlier in the year, it implemented across the U.S. both Standard time (the creation of time zones anchored in UTC, the successor to GMT) and Daylight Saving Time.
“No structure, even an artificial one, enjoys the process of entropy. It is the ultimate fate of everything, and everything resists it.”*…
A 19th-century thought experiment that motivates physicists– and information scientists– still…
The universe bets on disorder. Imagine, for example, dropping a thimbleful of red dye into a swimming pool. All of those dye molecules are going to slowly spread throughout the water.
Physicists quantify this tendency to spread by counting the number of possible ways the dye molecules can be arranged. There’s one possible state where the molecules are crowded into the thimble. There’s another where, say, the molecules settle in a tidy clump at the pool’s bottom. But there are uncountable billions of permutations where the molecules spread out in different ways throughout the water. If the universe chooses from all the possible states at random, you can bet that it’s going to end up with one of the vast set of disordered possibilities.
Seen in this way, the inexorable rise in entropy, or disorder, as quantified by the second law of thermodynamics, takes on an almost mathematical certainty. So of course physicists are constantly trying to break it.
One almost did. A thought experiment devised by the Scottish physicist James Clerk Maxwell in 1867 stumped scientists for 115 years. And even after a solution was found, physicists have continued to use “Maxwell’s demon” to push the laws of the universe to their limits…
A thorny thought experiment has been turned into a real experiment—one that physicists use to probe the physics of information: “How Maxwell’s Demon Continues to Startle Scientists,” from Jonathan O’Callaghan (@Astro_Jonny)
* Philip K. Dick
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As we reconsider the random, we might send carefully-calculated birthday greetings to Félix Édouard Justin Émile Borel; he was born on this date in 1871. A mathematician (and politician, who served as French Minister of the Navy), he is remembered for his foundational work in measure theory and probability. He published a number of research papers on game theory and was the first to define games of strategy.
But Borel may be best remembered for a thought experiment he introduced in one of his books, proposing that a monkey hitting keys at random on a typewriter keyboard will – with absolute certainty – eventually type every book in France’s Bibliothèque Nationale de France. This is now popularly known as the infinite monkey theorem.
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