Periods and Borders of Random Words

We investigate the behavior of the periods and border lengths of random words over a fixed alphabet. We show that the asymptotic probability that a random word has a given maximal border length k is a constant, depending only on k and the alphabet size l. We give a recurrence that allows us to determine these constants with any required precision. This also allows us to evaluate the expected period of a random word. For the binary case, the expected period is asymptotically about n-1.641. We also give explicit formulas for the probability that a random word is unbordered or has maximum border length one

Performance evaluation on optimisation of 200 dimensional numerical tests - results and issues

Abstract: Many tasks in science and technology require optimisation. Resolving such tasks could bring great benefits to community. Multidimensional problems where optimisation parameters are hundreds and more face unusual computational limitations. Algorithms, which perform well on low number of dimensions, when are applied to high dimensional space suffers insuperable difficulties. This article presents an investigation on 200 dimensional scalable, heterogeneous, real-value, numerical tests. For some of these tests optimal values are dependent on dimensions’ number and virtually unknown for variety of dimensions. Dependence on initialisation for successful identification of optimal values is analysed by comparison between experiments with start from random initial locations and start from one location. The aim is to: (1) assess dependence on initialisation in optimisation of 200 dimensional tests; (2) evaluate tests complexity and required for their resolving periods of time; (3) analyse adaptation to tasks with unknown solutions; (4) identify specific peculiarities which could support the performance on high dimensions (5) identify computational limitations which numerical methods could face on high dimensions. Presented and analysed experimental results can be used for further comparison and evaluation of real value methods

Performance evaluation on optimisation of 200 dimensional numerical tests - results and issues

Abstract: Many tasks in science and technology require optimisation. Resolving such tasks could bring great benefits to community. Multidimensional problems where optimisation parameters are hundreds and more face unusual computational limitations. Algorithms, which perform well on low number of dimensions, when are applied to high dimensional space suffers insuperable difficulties. This article presents an investigation on 200 dimensional scalable, heterogeneous, real-value, numerical tests. For some of these tests optimal values are dependent on dimensions’ number and virtually unknown for variety of dimensions. Dependence on initialisation for successful identification of optimal values is analysed by comparison between experiments with start from random initial locations and start from one location. The aim is to: (1) assess dependence on initialisation in optimisation of 200 dimensional tests; (2) evaluate tests complexity and required for their resolving periods of time; (3) analyse adaptation to tasks with unknown solutions; (4) identify specific peculiarities which could support the performance on high dimensions (5) identify computational limitations which numerical methods could face on high dimensions. Presented and analysed experimental results can be used for further comparison and evaluation of real value methods

Bouncing trimer: a random self-propelled particle, chaos and periodical motions

A trimer is an object composed of three centimetrical stainless steel beads equally distant and is predestined to show richer behaviours than the bouncing ball or the bouncing dimer. The rigid trimer has been placed on a plate of a electromagnetic shaker and has been vertically vibrated according to a sinusoidal signal. The horizontal translational and rotational motions of the trimer have been recorded for a range of frequencies between 25 and 100 Hz while the amplitude of the forcing vibration was tuned for obtaining maximal acceleration of the plate up to 10 times the gravity. Several modes have been detected like e.g. rotational and pure translational motions. These modes are found at determined accelerations of the plate and do not depend on the frequency. By recording the time delays between two successive contacts when the frequency and the amplitude are fixed, a mapping of the bouncing regime has been constructed and compared to that of the dimer and the bouncing ball. Period-2 and period-3 orbits have been experimentally observed. In these modes, according to observations, the contact between the trimer and the plate is persistent between two successive jumps. This persistence erases the memory of the jump preceding the contact. A model is proposed and allows to explain the values of the particular accelerations for which period-2 and period-3 modes are observed. Finally, numerical simulations allow to reproduce the experimental results. That allows to conclude that the friction between the beads and the plate is the major dissipative process.Comment: 22 pages, 10 figure

Computational Mechanics of Molecular Systems

New Yor