By John Casti
"Casti is without doubt one of the nice technology writers." -San Francisco Examiner"Casti's reward is so as to permit the nonmathematical reader proportion in his realizing of the great thing about an exceptional theory." -Christian technology MonitorFollowing up the acclaimed 5 Golden ideas, one other quintet of sparkling math discoveriesWith 5 extra Golden ideas, readers are handled to a different attention-grabbing set of theoretical gemstones from acclaimed renowned technology writer John Casti. Injecting all-new elements into his trademark recipe of real-world examples, historic anecdotes, and simple causes, Casti once more brings math to exciting existence. All who loved the original pleasures of the unique will love this follow-up survey highlighting the creme de l. a. creme of math within the final century.Explores how knot conception informs the vintage story of Alexander the good and the Gordian Knot* Considers how the Shannon Coding idea applies to interpreting the human genomeJohn L. Casti, PhD (Santa Fe, NM), a resident member of the Santa Fe Institute, is a professor on the Technical collage of Vienna and the writer of Would-Be Worlds (Wiley) and Cambridge Quintet.
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Extra info for Five More Golden Rules: Knots, Codes, Chaos, and Other Great Theories of 20th-Century Mathematics
Feedback Control and Dynamic Programming The key ingredient in the dynamic programming approach to optimal control is Bellman's 35 36 Principle of Optimality. Assume that u*(t) is the optimal control function and that x*(t) is the associated optimal state trajectory. Let v = u*(0) denote the optimal action to be taken at the initial moment, with the initial state being x(0) = c = x*(0). Then the Principle of Optimality states that the part of the optimal trajectory starting at time t = ∆ from the state c + f(c, v, 0) ∆ is also the optimal trajectory for a problem that begins not at time t = 0 in the state c, but at time t = ∆ in the state c + f (c, v, 0) ∆.
A mathematical framework is then developed within which the particular example of a point in space is seen to be just a very special case of a much broader structure, say a point in three-dimensional space. Further generalizations then show this new structure itself to be only a special case of an even broader framework, the notion of a point in a space of n dimensions. And so it goes, one generalization piled atop another, each element leading to a deeper understanding of how the original object fits into a bigger picture.
Thus, the function u(t) accounts for our uncertainty about the true dynamics and for whatever stochastic effects may be influencing the state. It is clear that, on the one hand, we want to choose u(t) so that the state follows a trajectory reasonably close to that dictated by what we think the state dynamics actually are (given by the vector field f). On the other hand, we don't want our estimate of the state to depart too wildly from what has actually been observed (the function y(t)). So we need to choose the control function to optimally trade off these two costs.
Five More Golden Rules: Knots, Codes, Chaos, and Other Great Theories of 20th-Century Mathematics by John Casti