Convergence of non-autonomous attractors for subquintic weakly damped wave equation

We study the non-autonomous weakly damped wave equation with subquintic growth condition on the nonlinearity. Our main focus is the class of Shatah--Struwe solutions, which satisfy the Strichartz estimates and are coincide with the class of solutions obtained by the Galerkin method. For this class we show the existence and smoothness of pullback, uniform, and cocycle attractors and the relations between them. We also prove that these non-autonomous attractors converge upper-semicontinuously to the global attractor for the limit autonomous problem if the time-dependent nonlinearity tends to time independent function in an appropriate way

Lefschetz fixed-point theorem

Celem niniejszej pracy jest przedstawienie i udowodnienie twierdzenia Lefschetaza o punkcie stałym. Pierwsze dwa rozdziały zawierają definicje i fakty dotyczące kompleksów symplicjalnych oraz grup homologii, które są potrzebne do zdefiniowania liczby Lefschetza. Trzeci rozdział zawiera wypowiedź i dowód tytułowego twierdzenia oraz dowody twierdzeń pomocniczych związanych z śladami odwzorowań na grupach homologii, czwarty natomiast proste przykłady jego zastosowania.The main goal of this paper is to present and to prove the Lefschetz fixed point theorem. First two chapters contain definitions and general informations about simplicial complexes and homology groups, later needed in order to define the Lefschetz number. In chapter three there has been written the main theorem and it's proof and also proofs of auxiliary theorems. The last chapter contains simple examples of the Lefschetz theorem applications

Flows generated by combinatorial dynamical systems

W pracy przedstawiona zostaje teoria ciągłych i kombinatorycznych układów dynamicznych.W obu tych teoriach można zdefiniować grafy Conleya-Morse'a, które mogą być użyte jako narzędzie do porównywania dyskretnych i kombinatorycznych układów dynamicznych. Przedstawionazostaje konstrukcja semipotoku dla kombinatorycznego pola wektorowego, którego to dynamika porównując grafy Conleya-Morse'a wygląda tak samo jak dynamika wyjściowego kombinatorycznegopola wektorowego.This work contains a description of the theory of continuous and combinatorial dynamical systems.In both cases, it is possible to define Conley-Morse graphs which can be used as a tool for comparing continuous and combinatorial dynamical systems. There is described the construction of semiflow for combinatorial vectors field which dynamics look the same by comparing Conley-Morse graphs as dynamic of given combinatorial vector field

On numerical broadening of particle-size spectra : a condensational growth study using PyMPDATA 1.0

This work discusses the numerical aspects of representing the condensational growth of particles in models of aerosol systems such as atmospheric clouds. It focuses on the Eulerian modelling approach, in which fixed-bin discretisation is used for the probability density function describing the particle-size spectrum. Numerical diffusion is inherent to the employment of the fixed-bin discretisation for solving the arising transport problem (advection equation describing size spectrum evolution). The focus of this work is on a technique for reducing the numerical diffusion in solutions based on the upwind scheme: the multidimensional positive definite advection transport algorithm (MPDATA). Several MPDATA variants are explored including infinite-gauge, non-oscillatory, third-order terms and recursive antidiffusive correction (double-pass donor cell, DPDC) options. Methodologies for handling coordinate transformations associated with both particle-size spectrum coordinate choice and with numerical grid layout choice are expounded. Analysis of the performance of the scheme for different discretisation parameters and different settings of the algorithm is performed using (i) an analytically solvable box-model test case and (ii) the single-column kinematic driver (“KiD”) test case in which the size-spectral advection due to condensation is solved simultaneously with the advection in the vertical spatial coordinate, and in which the supersaturation evolution is coupled with the droplet growth through water mass budget. The box-model problem covers size-spectral dynamics only; no spatial dimension is considered. The single-column test case involves a numerical solution of a two-dimensional advection problem (spectral and spatial dimensions). The discussion presented in the paper covers size-spectral, spatial and temporal convergence as well as computational cost, conservativeness and quantification of the numerical broadening of the particle-size spectrum. The box-model simulations demonstrate that, compared with upwind solutions, even a 10-fold decrease in the spurious numerical spectral broadening can be obtained by an apt choice of the MPDATA variant (maintaining the same spatial and temporal resolution), yet at an increased computational cost. Analyses using the single-column test case reveal that the width of the droplet size spectrum is affected by numerical diffusion pertinent to both spatial and spectral advection. Application of even a single corrective iteration of MPDATA robustly decreases the relative dispersion of the droplet spectrum, roughly by a factor of 2 at the levels of maximal liquid water content. Presented simulations are carried out using PyMPDATA – a new open-source Python implementation of MPDATA based on the Numba just-in-time compilation infrastructure