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| '''Hilbert's thirteenth problem''' is one of the 23 [[Hilbert problems]] set out in a celebrated list compiled in 1900 by [[David Hilbert]]. It entails proving whether or not a solution exists for all [[Septic equation|7th-degree equations]] using [[mathematical function|functions]] of two [[mathematical argument|arguments]]. It was first presented in the context of [[nomograph]]y, and in particular "nomographic construction" — a process whereby a function of several variables is constructed using functions of two variables. The actual question is more easily posed however in terms of [[continuous function]]s. Hilbert considered the general seventh-degree equation
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| :<math>x^7 + ax^3 + bx^2 + cx + 1 = 0</math>
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| and asked whether its solution, ''x'', a function of the three variables ''a'', ''b'' and ''c'', can be expressed using a finite number of two-variable functions.
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| A more general question is: can every continuous function of three variables be expressed as a [[function composition|composition]] of finitely many continuous functions of two variables? The affirmative answer to this general question was given in 1957 by [[Vladimir Arnold]], then only nineteen years old and a student of [[Andrey Kolmogorov]]. Kolmogorov had shown in the previous year that any function of several variables can be constructed with a finite number of three-variable functions. Arnold then expanded on this work to show that only two-variable functions were in fact required, thus answering Hilbert's question.
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| Arnold later returned to the question, jointly with [[Goro Shimura]] (V. I. Arnold and G. Shimura, ''Superposition of algebraic functions'' (1976), in ''Mathematical Developments Arising From Hilbert's Problems'').
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| ==References==
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| * G. G. Lorentz, ''Approximation of Functions'' (1966), Ch. 11
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| {{Hilbert's problems}}
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| [[Category:Polynomials]]
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| [[Category:Hilbert's problems|#13]]
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| [[Category:Disproved conjectures]]
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| {{mathanalysis-stub}}
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Revision as of 01:45, 27 February 2014
Oscar is how he's known as and he totally loves this name. Managing individuals is what I do and the wage has been truly fulfilling. Doing ceramics is what my family and I appreciate. South Dakota is where I've usually been living.
My website :: at home std test [related resource site]