FYFD

Tag: mathematics

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    Seeking Randomness

    Securing information on the Internet requires a lot of random numbers, something computers are not good at creating on their own. This need for random input raises an important philosophical and practical question: what is randomness? How can we be sure that something truly is random, or is it enough for a system to be practically random? Joe explores these questions in this Be Smart video, which shows off how companies use systems — including fluid dynamical ones like lava lamps and wave machines — to generate random numbers for encryption. (Video and image credit: Be Smart)

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  • Proving Superdiffusion

    Proving Superdiffusion

    Turbulence is very good at spreading things out. Drop dye into a turbulent flow and it will quickly disperse. Add in particles — like rubber ducks — and they can spread apart, often at speeds quicker than one would expect, based on the background flow. This is (roughly speaking) a phenomenon known as “superdiffusion,” where turbulence makes particles that start out as neighbors part ways.

    Physicists conjectured that turbulence — including simplified and idealized versions of it that are simpler to deal with — had this superdiffusion property, but no one was able to show that in a mathematically rigorous way. But now a group of mathematicians has done so, using a technique known as homogenization. There’s a lot more on the story over at Quanta, or you can check out the original papers on arXiv. (Image credit: J. Richard; research credit: S. Armstrong et al. and S. Armstrong and T. Kuusi; see also Quanta)

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    Kirigami in the Flow

    Kirigami is a paper art that combines folding and cutting to create elaborate shapes. Here, researchers use cuts in thin sheets of plastic and explore how the sheets transform in a flow. Depending on the configuration of cuts, the sheets can stretch dramatically in the flow, creating complex, dynamic, and beautiful wakes. I feel like there must be some applications out there that would benefit from kirigami-induced mixing. (Video and image credit: A. Carleton and Y. Modarres-Sadeghi)

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    Why Nature Loves Fractals

    Trees, blood vessels, and rivers all follow branching patterns that make their pieces look very similar to their whole. We call this repeating, self-similar shape a fractal, and this Be Smart video explores why these branching patterns are so common, both in living and non-living systems. For trees, packing a large, leafy surface area onto the smallest amount of wood makes sense; the tree needs plenty of solar energy (and water and carbon dioxide) to photosynthesize, and it has to be efficient about how much it grows to get that energy. Similarly, our lungs and blood vessels need to pack a lot of surface area into a small space to support the diffusion that lets us move oxygen and waste through our bodies. Non-living systems, like the branches of viscous fingers or river deltas or the branching of cracks and lightning, rely on different physics but wind up with the same patterns because they, too, have to balance forces that scale with surface area and ones that scale with volume. (Video and image credit: Be Smart)

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  • The Real Butterfly Effect

    The Real Butterfly Effect

    The butterfly effect — that the flapping of a butterfly’s wings in Brazil can cause a tornado in Texas — expresses the sensitivity of a chaotic system to initial conditions. In essence, because we can’t possibly track every butterfly in Brazil, we’ll never perfectly predict tornadoes in Texas, even if the equations behind our weather forecast are deterministic.

    But this interpretation doesn’t fully capture the subtleties of the situation. With fluid dynamics, the small scales of a flow — like the turbulence in an individual cloud — are linked to the largest scales in the flow — for example, a hurricane. For short times, we’re actually quite good at predicting those large scales; our weather forecasts can distinguish sunny days and cloudy ones a week out. But at smaller scales, the forecast errors pile up quickly. No one can forecast that an individual cloud will form over your house three days from now. And because the small scales are linked to the larger scales, the uncertainties from the small scale cascade upward, limiting how far into the future we can reliably predict the weather.

    And, unfortunately, drilling down to capture smaller and smaller scales in our models can’t fix the problem, unless our initial uncertainties are identically zero. To get around this problem, weather forecasters instead use ensemble forecasting, where they run many simulations of the weather with slightly different initial conditions. Those differences in initial conditions let the forecasters play with those initial uncertainties — how accurate is the temperature reading from that station? How reliable is the instrument reporting that humidity? How old is the satellite data coming in? Once all the forecasts are run, they can see how many predicted sunny days versus rainy ones, which ones resulted in severe weather, and so on. Often the probabilities we see in our weather app — like 30% chance of rain — depend on factors including how many of the forecasts resulted in rain.

    Unfortunately, this butterfly effect permanently limits just how far into the future we can predict weather — at least until we fully understand the nature of the Navier-Stokes equations. For much more on this interesting aspect of chaos, check out this Physics Today article. (Image credit: NASA; see also T. Palmer at Physics Today)

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  • Universal Wingbeats

    Universal Wingbeats

    Eagles, butterflies, and whales don’t appear to have much in common, but a new study shows that they — along with over 400 other flying and swimming animals of all sizes — flap with a frequency determined by a simple equation. Their beat frequency is proportional to the square root of their mass divided by their wing area. As you can see in the graph below, this scaling collapses pretty much all of the data onto a single line:

    Illustration of the predicted relationship between size and wing freequency (black line) shown alongside various insects, birds, bats, penguins, and whales. The swimming animals also fall on the line, once adjustments are made for the difference in density between air and water.
    Illustration of the predicted relationship between size and wing frequency (black line) shown alongside various insects, birds, bats, penguins, and whales. The swimming animals also fall on the line, once adjustments are made for the difference in density between air and water.

    It’s surprising to find such a consistent relationship among animals of such vastly different sizes and types. The next big question for researchers will be unpicking exactly why and how animals evolved to use such a consistent pattern between their size and their wing(/fin) frequency. (Image credit: top – E. Ward, graph – J. Jensen et al.; research credit: J. Jensen et al.; via Physics World)

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    Visualizing Changes

    This rather mesmerizing video by Michiel de Boer uses a video editing technique to highlight movement and changes in video clips. From falling rain to rising mist to passing footsteps, the relatively simple technique visualizes all kinds of motion. De Boer calls it “motion extraction,” but it’s essentially a way to play with autocorrelation, a mathematical technique often used in fluid dynamics. It’s especially prevalent in turbulence, where it helps researchers identify parts of the flow that are closely related to one another. (Video and image credit: M. de Boer; via Colossal)

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    Filling Space

    While not directly fluid dynamical, this video from Steve Mould uses water to illustrate mathematical concepts like fractals and space-filling curves. Water, it turns out, does a great job of drawing our eyes to the way these one-dimensional curves fill up two- and three-dimensional space. Check out the full video for a mathematical dive into the concepts. (Video and image credit: S. Mould)

  • Abel Prize Winner Luis Caffarelli

    Abel Prize Winner Luis Caffarelli

    Tomorrow mathematician Luis Caffarelli will receive the Abel Prize — one of the highest honors in mathematics — in part for his work in fluid dynamics. Caffarelli is one of the authors of a partial proof of regularity for the Navier-Stokes equations, the equations governing fluid motion. A full proof of regularity and smoothness — essentially showing that the equations never break down or blow up to infinity — is one of the open Millennium Problems. Caffarelli is the first mathematician born and educated in South America to receive the Abel Prize. Congratulations to Professor Caffarelli! (Image credit: N. Zunk/University of Texas at Austin; via Nature; submitted by Kam-Yung Soh)

  • Optimal Bubble Clusters

    Optimal Bubble Clusters

    With a bubble wand, it’s quite easy to create clusters of two or more soap bubbles. These clusters seem to instantly find the lowest energy state, forming a shape that minimizes the cluster’s surface area (including interior walls) for the volume of air they enclose. But mathematicians have struggled for thousands of years to prove that this is actually the case.

    In 1995, mathematician John Sullivan had a breakthrough conjecture, at least for some types of bubble clusters. A proof for double bubble clusters quickly followed. But then progress stalled out, with the triple bubble version seemingly out of reach. But now a duo of mathematicians have published proofs for Sullivan’s bubble clusters in triple and quadruple clusters. Learn their story over at Quanta. (Image credit: N. Franz; via Quanta Magazine)