A basis in the space of solutions of a convolution equation by Napalkov V. V.


By Napalkov V. V.

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An old parlor entertainment involves passing a pair of scissors around and around a circle of players. " Those acquainted with the secret rule tell a player when he says the wrong word, and the joke continues until everyone has guessed the rule inductively. The scissors' blades are a red herring; a player should say "Crossed if and only if his legs are crossed. Familiar games such as Battleship and Jotto have slightly stronger analogies with scientific method, but the first full- fledged induction game was Eleusis, a card game invented by Robert Abbott and first explained in m y Scientific American column for June 1959.

Slice a sphere with a plane and the cross section is a circle. Slice a 4-sphere with a 3-space hyperplane and the cross section is a 3-sphere. (You can't divide a 4-sphere into two pieces with a 2-plane. A hyperapple. ) Imagine a +sphere moving slowly through our space. We see it first as a point and then as a tiny sphere that slowly grows in size to its maximum cross section, then slowly diminishes and disappears. A sphere of any dimension, made of sufficiently flexible material, can be turned inside out through the next-highest space.

Hence it lies on the circumcenter of an equilateral triangle, the side of which is 2R. Then ( R r ) is the hypotenuse of a right triangle, the altitude of which is ( R - r ) and the base of which is 2 R f l / 3 . So + The answer to Leo Moser's paradox of the hypercubic chessboard in four-dimensional space is that no portion of a white cell remains unenclosed by the hyperspheres surrounding each black cell. The radius of each hypersphere is fior 2. Since the hypercubic cells have edges of length 2, we see at once that each of the eight hyperspheres around a white cell will extend all the way to the center of that cell.

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