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Preliminary Concepts
Published in Hillel Rubin, Joseph Atkinson, Environmental Fluid Mechanics, 2001
where η is the displacement of the free surface and c is the wave velocity. This is a linear hyperbolic differential equation in a two-dimensional space. For the analysis and calculation of many wave phenomena, it is sufficient to consider a one-dimensional space. For this case, Eq. (8.2.8) collapses to
Fixed points for several classes of mappings in variable Lebesgue spaces
Published in Optimization, 2021
T. Domínguez Benavides, S. M. Moshtaghioun, A. Sadeghi Hafshejani
According to Lemma 3.6, to compute , we only need to consider finite not singleton equidistant sets. Let be such a set, if h is the equidistant Chebyshev centre of K, we choose real positive numbers , with such that . We can apply Lemma 3.7 to the one-dimensional space where x is any point in Ω such that . Hence, we have The integration of this inequality on Ω gives us which implies that
Segmentation and recognition of continuous gesture based on chaotic theory
Published in Behaviour & Information Technology, 2020
The optimal embedding dimension is computed using the false nearest neighbour (FNN) method proposed by Kennel (1992). From the geometric point of view, a chaotic time series is a projection of high-dimensional phase space onto one-dimensional space with distorted trajectories. The false neighbour points are those independent points that turn out to be adjacent after projection in high-dimensional space. Those points will be gradually removed with an increase of embedding dimension , and the trajectory of chaotic series will be restored to a regular one. This reveals the fact that if points are sufficiently close in an RPS, then they shall remain closed during a forward iteration, a phase space point that does not fulfil this criterion has a false neighbour. In a m-dimensional phase space, every point denoted as has a nearest neighbour , their distance isWhen dimension increases from to , their distance will change to
Modified progressive random walk with chaotic PRNG
Published in International Journal of Parallel, Emergent and Distributed Systems, 2018
Adam Viktorin, Roman Senkerik, Michal Pluhacek, Tomas Kadavy
The PRW algorithm produces a biased sequence of steps of random length from a specified range. Whereas, with the classical random walk, the starting position x0 is random, starting position in PRW is generated in one of the starting zones. Starting zones are non-overlapping sections of the search space, which divides it into 2n parts. This means that every dimension is split into halves and the starting zone can be then represented by a binary string sZone of length n, where 0 denotes the lower half of the dimension and 1 the other half. Moreover, the starting position x0 is generated on the edge of a multi-dimensional space in its corresponding starting zone. This edge is represented in one-dimensional space by a point, in two-dimensional space by line and in three-dimensional space by plane. The direction of a walk is biased towards the opposite starting zone and if the edge of a multi-dimensional space is reached, the direction in this particular dimension j is changed (sZonej is changed from 1 to 0 or vice versa). If the starting position is generated in the lower half of the j-th dimension (sZonej = 0), the randomly generated offset rndj will be added to the value in j-th dimension of the current position xi,j, otherwise (sZonej = 1) it will be subtracted, as can be seen in (2).