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Abstract Let T f : [ 0 , 1 ] → [ 0 , 1 ] be an expanding Lorenz map, this means T f x : = f (x) (mod 1) where f : [ 0 , 1 ] → [ 0 , 2 ] is a strictly increasing map satisfying inf ⁡ f ′ > 1. Then T f has two pieces of monotonicity. In this paper, sufficient conditions when T f is topologically mixing are provided. For the special case f (x) = β x + α with β ≥ 2 3 a full characterization of parameters (β , α) leading to mixing is given. Furthermore relations between renormalizability and T f being locally eventually onto are considered, and some gaps in classical results on the dynamics of Lorenz maps are corrected. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to establish the residual and the wild bootstrap procedures for periodically autoregressive models. We use the least squares estimators of model's parameters and generate their bootstrap equivalents. We prove that the bootstrap procedures for causal periodic autoregressive time series with finite fourth moments are weakly consistent. Finally, we confirm our theoretical considerations by simulations. [ABSTRACT FROM AUTHOR]

Modern construction materials, including steels, have to combine strength with good formability. In metallic materials, these features are obtained for heterogeneous multiphase microstructures. Design of such microstructures requires advanced numerical models. It has been shown in our earlier works that models based on stochastic internal variables meet this requirement. The focus of the present paper is on deterministic and stochastic approaches to modelling hot deformation of multiphase steels. The main aim was to survey recent advances in describing the evolution of dislocations and grain size accounting for the stochastic character of the recrystallization. To present a path leading to this objective, we reviewed several papers dedicated to the application of internal variables and statistical approaches to modelling recrystallization. Following this, the idea of the model with dislocation density and grain size being the stochastic internal variables is described. Experiments composed of hot compression of cylindrical samples are also included for better presentation of the utility of this approach. Firstly, an empirical data describing the loads as a function of time during compression and data needed to create histograms of the austenite grain size after the tests were collected. Using the measured data, identification and validation of the models were performed. To present possible applications of the model, it was used to produce a simulation imitating industrial hot-forming processes. Finally, calculations of the dislocation density and the grain size distribution were utilized as inputs in simulations of phase transformations during cooling. Distributions of the ferrite volume fraction and the ferrite grain size after cooling recapitulate the paper. This should give readers good overview on the application of collected equations in practice. [ABSTRACT FROM AUTHOR]