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Rearrangement proof of the Pythagorean theorem. (The area of the white space remains constant throughout the translation rearrangement of the triangles. At all moments in time, the area is always c 2. And likewise, at all moments in time, the area is always a 2 + b 2.)
In mathematics, the Riemann series theorem, also called the Riemann rearrangement theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, and rearranged such that the new series diverges.
Supposing , we have that + + +. Define = (,,) and = (+, +, +). By the rearrangement inequality, the dot product of the two sequences is maximized when the terms are arranged to be both increasing or both decreasing.
The rearrangement inequality can be regarded as intuitive in the following way. Imagine there is a heap of $10 bills, a heap of $20 bills and one more heap of $100 bills. You are allowed to take 7 bills from a heap of your choice and then the heap disappears.
In the one-dimensional case, the inequality is first proved when the functions , and are characteristic functions of a finite unions of intervals. Then the inequality can be extended to characteristic functions of measurable sets, to measurable functions taking a finite number of values and finally to nonnegative measurable functions.
An alternative proof is simply obtained with the rearrangement inequality, writing that
Another proof that uses triangles considers the area enclosed by a circle to be made up of an infinite number of triangles (i.e. the triangles each have an angle of d𝜃 at the centre of the circle), each with an area of 1 / 2 · r 2 · d𝜃 (derived from the expression for the area of a triangle: 1 / 2 · a · b · sin𝜃 ...
The proof goes by restating the problem as a minimization of the Rayleigh quotient. [1] The isoperimetric inequality can be deduced from the Pólya–Szegő inequality with =. The optimal constant in the Sobolev inequality can be obtained by combining the Pólya–Szegő inequality with some integral inequalities. [2] [3]