We derive the coincidence of Lutz’s constructive dimension and Kolmogorov complexity for sets of infinite strings from Levin’s early result on the existence of an optimal left computable cylindrical semi-measure M via simple calculations.

Degenerate parabolic equations of Kolmogorov type occur in many areas of analysis and applied mathematics. In their simplest form these equations were introduced by Kolmogorov in 1934 to describe the probability density of the positions and velocities of particles but the equations are also used as prototypes for evolution equations arising in the kinetic theory of gases. More recently equation...

Some explicit traveling wave solutions to a Kolmogorov-PetrovskiiPiskunov equation are presented through two ansätze. By a Cole-Hopf transformation, this Kolmogorov-Petrovskii-Piskunov equation is also written as a bilinear equation and further two solutions to describe nonlinear interaction of traveling waves are generated. Bäcklund transformations of the linear form and some special cases are...

1 From probability theory to Kolmogorov complexity 3 1.1 Randomness and Probability theory . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Intuition of finite random strings and Berry’s paradox . . . . . . . . . . . . 5 1.3 Kolmogorov complexity relative to a function . . . . . . . . . . . . . . . . . 6 1.4 Why binary programs? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5...

The construction of a quantal Kolmogorov entropy that tends to the classical Kolmogorov entropy in the limit h -t 0 is discussed. The approach is to use sets of functions rather than disjoint partitions and involves the use of pseudo-differential operators.

We have used the Kolmogorov complexities and the Kolmogorov complexity spectrum to quantify the randomness degree in river flow time series of seven rivers with different regimes in Bosnia and Herzegovina, representing their different type of courses, for the period 1965–1986. In particular, we have examined: (i) the Neretva, Bosnia and the Drina (mountain and lowland parts), (ii) the Miljacka ...

We introduce algorithmic information theory, also known as the theory of Kolmogorov complexity. We explain the main concepts of this quantitative approach to defining ‘information’. We discuss the extent to which Kolmogorov’s and Shannon’s information theory have a common purpose, and where they are fundamentally different. We indicate how recent developments within the theory allow one to form...

We show that the algorithms which behave well on average may have difficulty only for highly structured, non-random inputs, except in a finite number of cases. The formal framework is provided by the theory of Kolmogorov complexity. An experimental verification is done for graph S-colorability with Brhlaz’s algorithm.

For every total recursive time bound t, a constant fraction of all compressible (low Kolmogorov complexity) strings is t-bounded incompressible (high time-bounded Kolmogorov complexity); there are uncountably many infinite sequences of which every initial segment of length n is compressible to log n yet t-bounded incompressible below 1 4n − log n; and there are a countably infinite number of re...