Suppose τ is a cellular automaton over an amenable group and a finite alphabet. Celebrated Garden of Eden theorem states, that pre-injectivity of τ is equivalent to non-existence of Garden of Eden configuration. In this paper we will prove, that imposing some mild restrictions , we could add another equivalent assertion: non-existence of Garden of Eden configuration is equivalent to preservatio...

The methods measuring the departure between observation and the model were reviewed. The following statistics were applied on two experimental data sets: ChiSquared, Kolmogorov-Smirnov, Anderson-Darling, Wilks-Shapiro, and Jarque-Bera. Both investigated sets proved not to be normal distributed. The Grubbs’ test identified one outlier and after its removal the normality of the set of 205 chemica...

We define a new discrete version of scaled dimension and we find connections between the scaled dimension of a string and its Kolmogorov complexity and predictability. We give a new characterization of constructive scaled dimension by Kolmogorov complexity, and prove a new result about scaled dimension and prediction.

Heilbronn’s triangle problem asks for the least ∆ such that n points lying in the unit disc necessarily contain a triangle of area at most ∆. Heilbronn initially conjectured ∆ = O(1/n). As a result of concerted mathematical effort it is currently known that there are positive constants c and C such that c logn/n ≤ ∆ ≤ C/n for every constant ǫ > 0. We resolve Heilbronn’s problem in the expected ...

In this paper we answer the following well-known open question in computable model theory. Does there exist a computable not א0-categorical saturated structure with a unique computable isomorphism type? Our answer is affirmative and uses a construction based on Kolmogorov complexity. With a variation of this construction, we also provide an example of an א1-categorical but not א0-categorical sa...

We derive a forward partial integro-differential equation for prices of call options in a model where the dynamics of the underlying asset under the pricing measure is described by a -possibly discontinuoussemimartingale. This result generalizes Dupire’s forward equation to a large class of non-Markovian models with jumps and allows to retrieve various forward equations previously obtained for ...

The relation between relaxation, the time scale of Lyapunov instabilities, and the Kolmogorov-Sinai time in a one-dimensional gravitating sheet system is studied. Both the maximum Lyapunov exponent and the Kolmogorov-Sinai entropy decrease as proportional to N−1/5. The time scales determined by these quantities evidently differ from any type of relaxation time found in the previous investigatio...

The Kolmogorov-Sinai entropy in the sense of Tsallis under Bernoulli shifts was obtained by Mesón and Vericat [J. Math. Phys. 37, 4480(1996)]. In this paper, we propose a revised generalized Kolmogorov-Sinai-q entropy under Markov shifts. The form of this generalized entropy with factor q is nonextensive. The new generalized entropy contains the classical Kolmogorov-Sinai entropy and Renýı entr...

We show that there are computably enumerable (c.e.) sets with maximum initial segment Kolmogorov complexity amongst all c.e. sets (with respect to both the plain and the prefix-free version of Kolmogorov complexity). These c.e. sets belong to the weak truth table degree of the halting problem, but not every weak truth table complete set has maximum initial segment Kolmogorov complexity. Moreove...