What's Happening in Mathematics?
page 6 (newest)
An occasionally updated series of links to mathematical news — new applications,
new discoveries, problems, personalities, prizes...
new discoveries, problems, personalities, prizes...
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Antonino Morassi and Alexandre Kawano present a theoretical mechanical model of how an orb-web spider works out where its prey has hit its web, from how the web vibrates. This is an 'inverse problem of source identification'. They use a continuous membrane model to represent the web's structural connectivity. Their equations
help identify the key parameters for the web's response — hence also the spider's. |
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A giant leap for bivalve-kind
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Alain Goriely and Régis Chirat (U Oxford) explain the interlocking mechanism of bivalve shells using a mathematical model of shell growth, especially the geometry of the shell halves and the mechanics involved when they come together. This ensures an almost perfect fit even when the shell is damaged and regrows. Edges in some species are wavy because the mantle grows faster than the shell edge, causing buckling. The interlocking pattern is constrained by the forces on the shells when closed.
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Big whorls have little whorls
Which feed on their velocity, And little whorls have lesser whorls And so on to viscosity. Lewis Fry Richardson |
Jacob Bedrossian, Samuel Punshon-Smith, and Alex Blumenthal (U Maryland) find first rigorous proof of Batchelor's law, fundamental to turbulent fluid flow. This previously empirical law states that the power spectrum of small and large whorls in a turbulent fluid scales with wavenumber to the power -1. The proof was presented at a SIAM meeting in December 2019.
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The Birch-Swinnerton-Dyer Conjecture about rational points on an elliptic curve is one of the major open problems in mathematics. It emerged from an early computer experiment. Now Laura Alessandretti, Andrea Baronchelli, and Yang-Hui He have employed state-of-the-art computing to gain new insights. Using the Cremona database they inspect over 2.5 million elliptic curves using machine-learning and topological data analysis, finding new patterns in elliptic curve data.
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Over 40 years ago (exact date unknown) Paul Erdös conjectured that any subset A of the positive integers with positive density contains B+C where B and C are infinite subsets. Using methods from ergodic theory, Joel Moreira, Florian Karl Richter, and Donald Robertson have now proved the conjecture.
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Mohammad Ghomi and Joel Spruck have proved that the total positive curvature of any closed hypersurface in a complete simply connected n-dimensional manifold of nonpositive curvature is bounded below by the volume of the unit sphere in n-dimensional Euclidean space. This yields the optimal isoperimetric inequality for bounded regions of finite perimeter, which proves the Cartan-Hadamard conjecture. The conjecture was proved for two dimensions in 1926, for four in 1984, and for three in 1992. The new work polishes off all other dimensions.
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Take any integer n > 0. If n is even, divide by 2; if odd, replace by 3n+1. Repeat indefinitely. The Collatz (or 3n+1) Conjecture states that the process always reaches n = 1, and then cycles 1, 4, 2, 1, 4, 2, ... . This conjecture has many names, and has long resisted efforts at a proof or disproof. Now Terry Tao has made significant progress using techniques from probability theory. Roughly, he proves that almost all n (in the sense of logarithmic density) eventually attain a value much smaller than n (less than f(n) for any f such that f(n) tends to infinity with n, such as log log log log n). A complete proof remains out of reach by these methods.
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42 is interesting after all...
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Which numbers are sums of three (positive or negative) cubes? A team led by the University of Bristol and MIT has solved a famous maths puzzle with an answer for the most elusive number of all: 42. The problem, set in 1954, looked for Solutions of the Diophantine Equation x^3+y^3+z^3=k, with k being all the numbers from 1 to 100. Until recently only 33 and 42 held out. Then 33 was solved, and finally 42. The numbers are x = -80538738812075974, y = 80435758145817515, z = 12602123297335631.
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Paul Krapivsky and Sidney Redner take a new look at an old optimisation problem to compare three strategies for parking your car. The "meek" strategy picks the first available. The "optimistic" strategy looks for a space next to the entrance, then backtracks to the closest vacancy. "Prudent" drivers drive past the first available space, betting that another space exists further in; then backtrack to the space a meek driver would have claimed initially if there isn't one. Which is best? There's also an unexpected connection to the microtubule "scaffolding" of living cells, which made it possible to find a solution.
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