what did the grasshopper need to tend to his farm
I am always amazed by the subtlety of probability. You can cite the Monty Hall trouble or The Fisher Yates Shuffle, but what about the Grasshopper problem? Like shooting fish in a barrel to state, but very difficult to solve and slightly unbelievable.
A new paper from Olga Goulko of the University of Massachusetts and Adrian Kent of the University of Cambridge produces some surprising results from a very simple problem.
The problem is linked to questions of quantum mechanics, specifically Bell's inequalities, but a slightly simplified version is easy to country and just every bit suprising:
"Yous are given a bag of grass seed from which you can grow a backyard of any shape (not necessarily connected) with unit of measurement area on a planar surface. A grasshopper lands at a random point on your lawn, and so jumps a given distance d in a random direction. What lawn shape should y'all choose to maximise the probability that the grasshopper remains on your lawn after jumping?"
Jumping in a random direction - surely that means all directions are the same then the solution must exist but a disk? Nevertheless, the showtime proof in the paper dismisses this intuition:
The disc of area 1 (radius π-1/ii) is not optimal for whatsoever d>π-1/2.
So, if the jump altitude is greater than the radius a disk, is not the answer to the question. The reason seems to exist the possiblity that the backyard isn't only connected. The proof relies on showing that removing a small disk from the center of the unit deejay increases the probability that the grasshopper will country on the lawn. Thus in that location is at least i pigsty in the backyard shape that is improve than the deejay.
Note this proof doesn't tell usa what that better shape is. It just provides an expanse that we can move to improve the probability - information technology doesn't say where the expanse should be deployed.
The outcome is afterwards generalized to all values of d > 0 - so your intuition is completely wrong.
The suggestion is that "knobbly" shapes are would practise ameliorate, but the problem is to hard to solve exactly. Instead the researchers recast the problem as a spin model, a detached approximation to the continuous grasshopper problem - a discrete grasshopper trouble?
Spin models are often studied in physics using simulation and this is what happened next. The solution to the original problem corresponds to the ground state, i.e. the lowest free energy land, of the spin system. This tin can be plant using imitation annealing, which models the system at a given "temperature". The temperature controls how much random movement there is in the organization and running the simulation for a long time lets information technology settle to equilibrium at that temperature. Slowly lowering the temperature lets the system settle into the lowest energy state with a loftier probability. This is, of course a simulation of what happens when yous allow something hot to cool downward - hence simulated annealing.
You can see the simulation in action in the following video:
For d<π-i/2 the best configurations were but connected "cogwheels". As d decreases the number of cogs decreases.
For values of d just greater than π-1/ii the shape changes from a cog to something disconnected and complicated:
This seems to exist a critical or phase change region for the problem.
For values of 0.65<d<0.87 the shape changes to another fairly stable 3 fan shape:
For d>0.87 the shape changes to a fan:
So all fascinating and counter intuitive, just are in that location any applications?
The paper suggests quite a few, simply the one that attracted my attention is:
Our numerical solutions strikingly illustrate the emergence of structures with discrete symmetries from an isotropic problem with continuous rotational symmetry. They also bear at least a passing resemblance to patterns seen elsewhere in nature, including the contours of flowers, the patterns seen on some seashells and the stripes on so me animals.
Turing'southward well-known theory of morphogenesis hypothesises that many such natural patterns arise as solutions to reaction-diffusion equations. This possibility has been demonstrated experimentally. Our results suggest that a rich variety of pattern formation can likewise a rise in systems with effectively stock-still-range interactions, including interactions associated with the sort of catalytic reaction described higher up. It may be worth looking for explanations of this type in any context where highly regular patterns naturally arise and are non otherwise easily explained.
At the end of the newspaper the subject returns to quantum mechanics and Bell's inequalities, only I cannot end without quoting the delightful last paragraph:
For projective measurements on pairs of qubits, such algorithms can be idea of as generalised grasshopper models, in which Alice has a unmarried lawn, Bob has a set of bachelor lawns, and Alice chooses which of Bob'southward lawns is used in each given run.
I bet you didn't see Bob and Alice making information technology into this story, permit alone their ownership of lawns.
Ah grasshopper....
More Information
The grasshopper problem
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Source: https://www.i-programmer.info/news/112-theory/11323-the-grasshopper-problem.html
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