The quotient cohomology of tiling spaces is a topological invariant that relates a tiling space to one of its factors, viewed as topological dynamical systems. In particular, it is a relative version of the tiling cohomology that distinguishes factors of tiling spaces. In this work, the quotient cohomologies within certain families of substitution tiling spaces in 1 and 2 dimensions are determined. Specifically, the quotient cohomologies for the family of the generalised Thue-Morse sequences and generalised chair tilings are presented.
The Fourier-based diffraction approach is an established method to extract order and symmetry properties from a given point set. We want to investigate a different method for planar sets which works in direct space and relies on reduction of the point set information to its angular component relative to a chosen reference frame. The object of interest is the distribution of the spacings of these angular components, which can for instance be encoded as a density function on ℝ_+. In fact, this radial projection method is not entirely new, and the most natural choice of a point set, the integer lattice ℤ², is already well understood. We focus on the radial projection of aperiodic point sets and study the relation between the resulting distribution and properties of the underlying tiling, like symmetry, order and the algebraic type of the inflation multiplier.
In 1989, Godrèche and Luck introduced the concept of local mixtures of primitive substitution rules along the example of the well-known Fibonacci substitution and foreshadowed heuristic results on the topological entropy and the spectral type of the diffraction measure of associated point sets. In this contribution, we present a generalisation of this concept by regarding the so-called "noble means families", each consisting of finitely many primitive substitution rules that individually all define the same two-sided discrete dynamical hull. We report about results in the randomised case on topological entropy, ergodicity of the two-sided discrete hull, and the spectral type of the diffraction measure of related point sets.
In previous papers formulas have been derived describing distribution of a random variable whose values are positions of an oscillator at the moment t, which, in the interval [0, t], underwent the influence of stochastic impulses with a given distribution. In this paper we present reasoning leading to an opposite inference thanks to which, knowing the course of the oscillator, we can find the approximation of distribution of stochastic impulses acting on it. It turns out that in the case of an oscillator with damping the stochastic process ξ_{t} of its deviations at the moment t is a stationary and ergodic process for large t. Thanks to this, time average of almost every trajectory of the process, which is the n-th power of ξ_{t} is very close to the mean value of ξ_{t}^{n} in space for sufficiently large t. Thus, having a course of a real oscillator and theoretical formulae for the characteristic function ξ_{t} we are able to calculate the approximate distribution of stochastic impulses forcing the oscillator.
In our previous works we introduced and applied a mathematical model that allowed us to calculate the approximate distribution of the values of stochastic impulses η_{i} forcing vibrations of an oscillator with damping from the trajectory of its movement. The mathematical model describes correctly the functioning of a physical RLC system if the coefficient of damping is large and the intensity λ of impulses is small. It is so because the inflow of energy is small and behaviour of RLC is stable. In this paper we are going to present some experiments which characterize the behaviour of an oscillator RLC in relation to the intensity parameter λ, precisely to λ E(η). The parameter λ is a constant in the exponential distribution of random variables τ_{i}, where τ_{i} = t_{i} - t_{i - 1}, i = 1, 2, ... are intervals between successive impulses.
Solving a stochastic problem for systems subjected to random series of pulses is, in the present case, aimed at determining of an approximate distribution of amplitudes of random pulses forcing vibrations of an oscillator with damping. The applied model of investigations indicated the source of difficulties connected with interpretation of the obtained results. Another issue discussed in the paper is how a change of the damping coefficient b of the system may result in a decrease of the difference between the actual distribution of random pulses and that determined from the waveform.
In the paper, the energy harvesting from vibration of two-degree-of-freedom mechanical system is analyzed. The considered system consists of two mass linked in series by means of springs and dampers. The kinematic excitation of the system was assumed. The energy conversion system was placed in the suspension of lower mass. As a result of the analysis, the methods to increase the energy harvesting from vibration were proposed. The laboratory stand has been built and a series of measurements performed. Results of numerical simulations and measurements are presented in graphs.
We look at the topology of the tiling space of locally random Fibonacci substitution, which is defined as a ↦ ba with probability p, a ↦ ab with probability 1-p and b ↦ a for 0
The paper presents another phase of the study aimed at determining distributions of random pulses forcing vibration of an oscillator with damping. At this stage, the impact of the pulses amplitudes on distributions determined in a finite time interval is discussed. Application of a mathematical model in simulations allows to determine the differences between the distributions generated in MATLAB environment and those determined by a function. The experiment was designed so that the qualitative analysis of the issue was possible.
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