Kantorovich, Leonid Vitalyevich
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[2] Kantorovich was one of the first to use linear programming as a tool in economics. His most famous work is [1]. The characteristic of Kantorovich's work is a combination of theoretical and applied research. His first works concerned delicate problems of set theory. Later he became one of the first Soviet specialists on functional analysis. In the 1930s he laid down the foundations of the theory of semi-ordered spaces which constitutes now a vast chapter of functional analysis bordering algebra and measure theory. At the same time he anticipated the ideas of the future theory of generalized functions which became current only in the 1950s. Kantorovich obtained beautiful
results on approximation theory. The approach to Sobolev's embedding theorem suggested by Kantorovich (based on his estimations of integral operators) is well known. See also: L i n e a r p r o g r a m m i n g ; H i s t o r y of optimization.
References [1] KANTOROVICH, L.V.: The best use of economic resources, 1959. [2] KANTOROVICH, L.V.: 'My way in mathematics', Uspekhi Mat. Nauk 42, no. 2 (1987). [3] LEIFMAN, L.J. (ed.): Functional analysis, optimization and mathematical economics (dedicated to the memory of L. V. Kantorovich), Oxford Univ. Press, 1990. [4] LINBECK, A. (ed.): Nobel Lectures Economic Sciences 1969-1980, World Sci., 1992. [5] M~LER, K.G. (ed.): Nobel Lectures Economic Sciences 1981-1990, World Sci., 1992. Panos M. Pardalos
Center for Applied Optim. Dept. Industrial and Systems Engin. Univ. Florida Gainesville, FL 32611, USA E-mail address: pardalos~ufl.edu
MSC 2000:01A99 Key words and phrases: Kantorovich, linear programming, economics, functional analysis.
algorithmic complexity, algorithmic information, algorithmic entropy, Solomonoff-Kolmogorov-Chaitin complexity, descriptional complexity, shortest program length, algorithmic randomness KOLMOGOROV
COMPLEXITY,
In the mid1960s R. Solomonoff [16], A.N. Kolmogorov [11] and G. Chaitin [4] independently invented the field now generally known as Kolmogorov complexity. It is also known variously as
Kolmogorov complexity algorithmic complexity, algorithmic in/ormation, algorithmic entropy, Solomonoff-KolmogorovChaitin complexity, descriptional complexity, shortest program length, algorithmic randomness and others. An extensive history of the field can be found in [14]. The Kolmogorov complexity formalizes the notion of amount of information necessary to uniquely describe a digital object. A digital object means one that can be represented as a finite binary string, for example, a genome, an Ising microstate, or an appropriately coarse-grained representation of a point in some continuum state space. In particular, the Kolmogorov complexity of a string of bits is the length of the shortest computer program that prints that string and stops running. The Kolmogorov complexity of an object is a form of absolute information of the individual object. This is not possible to do by Shannon's information theory. Unlike Kolmogorov complexity, information theory is only concerned with th
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