We introduce a new class of strongly degenerate nonlinear parabolic PDEs ( p − 2 ) Δ ∞ , X N + u Y t m ⋅ ∇ ∂ = 0 ∈ R × 1 combining the classical PDE Kolmogorov and normalized -Laplace operator. characterize solutions in terms an asymptotic mean value property results are connected to analysis certain tug-of-war games with noise. The functions for introduced approximate stated when parameter tha...

We introduce quantum-K ($QK$), a measure of the descriptive complexity density matrices using classical prefix-free Turing machines and show that initial segments weak Solovay random quantum Schnorr states are incompressible in sense $QK$. Many properties enjoyed by Kolmogorov ($K$) have analogous versions for $QK$; notably counting condition. Several connections between randomness $K$, includi...

‖f ‖ ≤ Crq ‖f‖ 1−q/r ‖f ‖ was proved by Kolmogorov in [11] for f ∈W r ∞(R). Here ‖ · ‖ means ‖ · ‖L∞(R) and the sharp constant Crq = Kr−qK −1+q/r r is given in terms of the Favard constants Kp := 4 π ∑∞ n=0 [ (−1) 2n+1 p+1 . For the definition of the Sobolev space W r ∞(R) see e.g. [4, Ch. 1.5]. The first results of this type (the case q = 1, r = 2) were obtained by Hadamard in [9] and by Landa...

An extractor is a function that receives some randomness and either “improves” it or produces “new” randomness. There are statistical and algorithmical specifications of this notion. We study an algorithmical one called Kolmogorov extractors and modify it to resourcebounded version of Kolmogorov complexity. Following Zimand we prove the existence of such objects with certain parameters. The uti...

The concepts of cumulative distribution function and empirical distribution function are investigated for fuzzy random variables. Some limit theorems related to such functions are established. As an application of the obtained results, a method of handling fuzziness upon the usual method of Kolmogorov–Smirnov one-sample test is proposed. We transact the α-level set of imprecise observations in ...

We consider stationary solutions of the Extended Fisher-Kolmogorov (EFK) equation , a fourth-order model equation for bi-stable systems. We show that as long as the stable equilibrium points are real saddles, the paths in the (u; u 0)-plane of two bounded solutions do not cross. As a consequence we derive that the bounded solutions of the EFK equation correspond exactly to those of the classica...