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Ten Problems in Experimental Mathematics
, 2006
"... Challenge ” of Nick Trefethen, beautifully described in [12] (see also [13]). Indeed, these ten numeric challenge problems are also listed in [15, pp. 22–26], where they are followed by the ten symbolic/numeric challenge problems that are discussed in this article. Our intent in [15] was to present ..."
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Cited by 7 (6 self)
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Challenge ” of Nick Trefethen, beautifully described in [12] (see also [13]). Indeed, these ten numeric challenge problems are also listed in [15, pp. 22–26], where they are followed by the ten symbolic/numeric challenge problems that are discussed in this article. Our intent in [15] was to present ten problems that are characteristic of the sorts of problems
Evaluating the Complexity of Mathematical Problems. Part 1
, 2009
"... In this paper we provide a computational method for evaluating in a uniform way the complexity of a large class of mathematical problems. The method, which is inspired by NKS1, is based on the possibility to completely describe complex mathematical problems, like the Riemann hypothesis, in terms of ..."
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In this paper we provide a computational method for evaluating in a uniform way the complexity of a large class of mathematical problems. The method, which is inspired by NKS1, is based on the possibility to completely describe complex mathematical problems, like the Riemann hypothesis, in terms of (very) simple programs. The method is illustrated on a variety of examples coming from different areas of mathematics and its power and limits are studied.
SEASHELLS: THE PLAINNESS AND BEAUTY OF THEIR MATHEMATICAL DESCRIPTION
"... Abstract. One might at first tend to think that the growth of plants and animals, because of their elaborate forms, are ruled by highly complex laws. However, this is surprisingly not always true: many aspects of the growth of plants and animals may be described by remarkably simple mathematical law ..."
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Abstract. One might at first tend to think that the growth of plants and animals, because of their elaborate forms, are ruled by highly complex laws. However, this is surprisingly not always true: many aspects of the growth of plants and animals may be described by remarkably simple mathematical laws. An obvious example of this are the seashells and snails, as we show here: with a very simple model it is possible to describe and generate any of the many types of seashells of the classes of Gastropods, Bivalves, Cephalopods and Scaphopods that one may find classified in a good seashell bookguide. ≪Beauty of style and harmony and grace and good rhythm depends on simplicity.≫ — Plato ≪There is much beauty in nature’s clues, and we can all recognize it without any mathematical training. There is beauty, too, in the mathematical stories that start from the clues and deduce the underlying rules and regularities, but it is a different kind of beauty, applying to ideas rather than things. Mathematics is to nature as Sherlock Holmes is to evidence.≫ — I. Stewart [8] 1. How seashells grow The idea that mathematics is deeply implied in the natural forms goes back to the Ancient Greeks. Many aspects of the growth of plants and animals may be described by remarkably simple mathematical laws, in spite of their elaborate forms (cf., for instance, the classical book of D’Arcy Thompson [9] and the recent book of Stephen Wolfram [10]). An obvious example of this are the seashells and snails [6]. Why do so many shells form spirals? As far as the animal that lives in a shell grows he needs the shell to grow with him in order to keep accommodating him. The fact that the animal which lives at the open edge of the shell places new shell material always in that edge, and faster on one side than the other, makes the shell to grow in a spiral. The rates at which shell material is secreted at different points of the open edge are presumably determined by the anatomy of the animal. And, surprisingly, even fairly small changes in such rates can have quite tremendous effects on the overall shape of the shell, which is in the origin of the existence of a great diversity of shells.