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Patterns appear widely throughout nature and math, from the Fibonacci spirals of sea shells to the periodicity of crystals. But certain math problems can sometimes trick the human solver into seeing a pattern, but then, out of the blue, the pattern suddenly disappears. These illusive patterns crop up in many areas of math, with one example coming from certain calculus integrals that have deceived the intuition of even the best mathematicians.
Now in a new study, two physicists have approached these integrals using the physics concept of random walks. Whereas solving these integrals usually requires a great deal of effort and ingenuity, the physicists have shown that the new approach can find solutions intuitively and sometimes even without the need for explicit calculations.
Physicists - Satya - N - Majumdar - Emmanuel
Physicists Satya N. Majumdar and Emmanuel Trizac at the University of Paris-Sud, CNRS, in France, have published a paper on using random walkers to solve integrals in a recent issue of Physical Review Letters.
"We have shown that physics insight allows us to obtain in a calculation-free way a wealth of curious integrals, and in addition, to obtain previously unknown identities (either integrals, or equalities between discrete sums and integrals)," Trizac told Phys.org. "Our work reveals that when mathematical intuition is deceived, physical intuition may save the day."
Integrals - Question - Figure - Borwein - Integrals
The integrals in question (see figure) are "Borwein integrals," named after David and Jonathan Borwein (father and son), who noticed unusual patterns in them in 2001. The Borwein integrals involve the product of sinc (cardinal sine) functions, which have widespread applications, such as in optics, signal processing, and other areas. These two particular integrals can be used to compute the volumes of hypercubes.
Solving the Borwein integrals involves substituting numbers in for the variable n. Each number gives a different solution value, allowing mathematicians to observe patterns in the resulting sequence of values. For...
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