Academic literature on the topic 'Exponential Generating functions'

This chapter talks about the behaviour of 'playing on the computer' that primarily involved sharing jokes in the 1990s. It mentions folklorist Paul Smith who dubbed the computer as the 'Joke Machine' and predicted the exponential growth of its humour-generating function. It also discusses how the internet brandishes the expressive, interactive features or cultural functions of folklore that frequently lodge as commentary on popular culture. The chapter contends how joking became associated with digital transmission and how it serves emotionally and psychologically to respond to anxieties concerning diminished human control and competency for users. It cites Jewish Orthodox groups that claim that the internet and computerized devices should be used for work, and not play.

This chapter studies a deterministic approach to transient trajectory generation and control as applied to the forced Van der Pol oscillatory system. This type of system tends towards a strongly nonlinear system, which can be considered chaotic. A classical tuning method, targeted exponential weighting, and isolated trajectory fractionalization trajectory generation methods are examined. Illustrating the given deterministic approach via the Van der Pol system highlights the potentially iterative nature of deterministic methods, and that traditional optimal linear time-invariant control techniques are unable to perform as desired whereas even an idealized nonlinear feedforward control significantly outperforms at the steady-state. It will be shown that utilizing a-priori knowledge of the system dynamics will enable the isolated trajectory fractionalization method to minimize the nonlinear transient effects due to miss-modeled or unmodeled plant dynamics, and that this benefit can be coupled with the targeted exponential weighting approach for greatly decreased trajectory tracking error on the order of a 92% reduction of the objective cost function in the presented case study based on the forced Van der Pol system.

Over the past two decades, electronic technology and miniaturization of electronic devices continue to grow exponentially, and heat dissipation becomes a critical issue for electronic devices due to larger heat generation. So, the need to cool down electronic components has led to the development of multiple cooling methods and microscale heat sinks. This chapter reviewed recent advances in developing an efficient heat sink, including (1) geometry parameters, (2) flow parameters that affect the hydraulic–thermal performance of the heat sink. Also, the main goal of this chapter is to address the current gap between academic research and industry. Furthermore, commercialized electronic cooling devices for various applications are highlighted, and their operating functions are discussed, which has not been presented before.

In this paper, explicit friction factor formulae (Fff), which is a function of Reynolds number (Re) and relative roughness (Rr) were updated and evaluated. Fff were obtained from archive and conduct performance evaluation. Performance evaluation of the Fff were conducted using relative error; model of selection (MSC) and the Akaike Information Criterion (AIC) using Colebrook-White's friction factors as the reference Ff. The study revealed that there are 47 Fff in use. The growth of the Fff can be grouped into four subgroups based on the pattern and into three subgroups based on number of authors and into three subgroups, based on the accuracy. The growth rates were combinations of linear and exponential based on the pattern. The study revealed that Fff provided the lowest relative error of less than 1.00%, the highest MSC of greater than 6.64 and the lowest AIC of less than -34324.17. The study concluded that the recent and third generation Fff are the best and using Microsoft Excel Solver for calculating Ff in the pipe flow systems is a good tool for engineers.

Systems such as robotic systems and systems with large input-output data tend to be difficult to model using mathematical techniques. These systems have typically high dimensionality and have degrees of uncertainty in many parameters. Artificial intelligence techniques such as neural networks, fuzzy logic, genetic algorithms and evolutionary algorithms have created new opportunities to solve complex systems. Application of fuzzy logic [Bai, Y., Zhuang H. and Wang, D. (2006)] in particular, to model and solve industrial problems is now wide spread and has universal acceptance. Fuzzy modelling or fuzzy identification has numerous practical applications in control, prediction and inference. It has been found useful when the system is either difficult to predict and or difficult to model by conventional methods. Fuzzy set theory provides a means for representing uncertainties. The underlying power of fuzzy logic is its ability to represent imprecise values in an understandable form. The majority of fuzzy logic systems to date have been static and based upon knowledge derived from imprecise heuristic knowledge of experienced operators, and where applicable also upon physical laws that governs the dynamics of the process. Although its application to industrial problems has often produced results superior to classical control, the design procedures are limited by the heuristic rules of the system. It is simply assumed that the rules for the system are readily available or can be obtained. This implicit assumption limits the application of fuzzy logic to the cases of the system with a few parameters. The number of parameters of a system could be large. The number of fuzzy rules of a system is directly dependent on these parameters. As the number of parameters increase, the number of fuzzy rules of the system grows exponentially. Genetic Algorithms can be used as a tool for the generation of fuzzy rules for a fuzzy logic system. This automatic generation of fuzzy rules, via genetic algorithms, can be categorised into two learning techniques, supervised and unsupervised. In this paper unsupervised learning of fuzzy rules of hierarchical and multi-layer fuzzy logic control systems are considered. In unsupervised learning there is no external teacher or critic to oversee the learning process. In other words, there are no specific examples of the function to be learned by the system. Rather, provision is made for a task-independent measure of the quality or representation that the system is required to learn. That is the system learns statistical regularities of the input data and it develops the ability to learn the feature of the input data and thereby create new classes automatically [Mohammadian, M., Nainar, I. and Kingham, M. (1997)]. To perform unsupervised learning, a competitive learning strategy may be used. The individual strings of genetic algorithms compete with each other for the “opportunity” to respond to features contained in the input data. In its simplest form, the system operates in accordance with the strategy that ‘the fittest wins and survives’. That is the individual chromosome in a population with greatest fitness ‘wins’ the competition and gets selected for the genetic algorithms operations (cross-over and mutation). The other individuals in the population then have to compete with fit individual to survive. The diversity of the learning tasks shown in this paper indicates genetic algorithm’s universality for concept learning in unsupervised manner. A hybrid integrated architecture incorporating fuzzy logic and genetic algorithm can generate fuzzy rules for problems requiring supervised or unsupervised learning. In this paper only unsupervised learning of fuzzy logic systems is considered. The learning of fuzzy rules and internal parameters in an unsupervised manner is performed using genetic algorithms. Simulations results have shown that the proposed system is capable of learning the control rules for hierarchical and multi-layer fuzzy logic systems. Application areas considered are, hierarchical control of a network of traffic light control and robotic systems. A first step in the construction of a fuzzy logic system is to determine which variables are fundamentally important. Any number of these decision variables may appear, but the more that are used, the larger the rule set that must be found. It is known [Raju, S., Zhou J. and Kisner, R. A. (1990), Raju G. V. S. and Zhou, J. (1993), Kingham, M., Mohammadian, M, and Stonier, R. J. (1998)], that the total number of rules in a system is an exponential function of the number of system variables. In order to design a fuzzy system with the required accuracy, the number of rules increases exponentially with the number of input variables and its associated fuzzy sets for the fuzzy logic system. A way to avoid the explosion of fuzzy rule bases in fuzzy logic systems is to consider Hierarchical Fuzzy Logic Control (HFLC) [Raju G. V. S. and Zhou, J. (1993)]. A learning approach based on genetic algorithms [Goldberg, D. (1989)] is discussed in this paper for the determination of the rule bases of hierarchical fuzzy logic systems.