Mathematical Background - Theory

The limitation of transfer function representation becomes plain obvious as we tackle more complex problems. For complex systems with multiple inputs and outputs, transfer function matrices can become very clumsy. In the so-called modem control, the method of choice is state space or state variables in time domain—essentially a matrix representation of the model equations. The formulation allows us to make use of theories in linear algebra and differential equations. It is always a mistake to tackle modem control without a firm background in these mathematical topics. For this reason, we will not overreach by doing both the mathematical background and the control together. Without a formal mathematical framework, we will put the explanation in examples as much as possible. The actual state space control has to be delayed until after tackling classical transfer function feedback systems. 

Particularly lucid explanations of many fundamental aspects of NMR theory are given in the third edition of Principles of Magnetic Resonance by C. P. Slichter.30 This book is directed toward physicists and assumes that the reader has a good mathematical background and a grounding in the density matrix formalism of the sort we provide in Chapter 11. However, it also includes among the equations excellent qualitative discussions of various topics. 

Theoretical interpretation of molecular vibration spectra is not a simple task. It requires knowledge of symmetry and mathematical group theory to assign all the vibration bands in a spectrum precisely. For applications of vibrational spectroscopy to materials characterization, we can still interpret the vibrational spectra with relatively simple methods without extensive theoretical background knowledge. Here, we introduce some simple methods of vibrational spectrum interpretations. 

The author would like to take this opportunity to express his gratitude to Professors Osvaldo Goscinski, Jan Linderberg, and Yngve Ohrn for some valuable discussions about the foundations for the special propagator theory during the 1983 Summer Institute at Uppsala University, Uppsala, Sweden. He would also like to thank Professor Brian Weiner for valuable comments as to the mathematical background of the theory and members of the Florida Quantum Theory Project for help with various details. 

Based on Biot s fundamental work Stoll (e.g. 1974, 1977, 1989) reformulated the mathematical background of this theory with a simplified uniform nomenclature. Here, only the main physical principles and equations are summarized. For a detailed description please refer to one of Stoll s publications or Biot s original papers. 

C. Johnson. Numerical solution of Partial Differential Equations by the Finite Element Method. Cambridge University Press, Cambridge, 1990. An excellent introductory book on the finite element method. The text assumes mathematical background of first year graduate students in applied mathematics and computer science. An excellent introduction to the theory of adaptive methods. 

The mathematical background and theory of the computational methods is beyond the scope of this chapter and can be found elsewhere [Young, 2001 Ramachandran et al, 2008],... 

Information theory is the mathematical background in the area of communication but can be also used to describe the output of a series of experiments. It deals with probabilities and is therefore not applicable to single experiments. 

Applications to Chemistry (1935), written jointly with Wilson, was addressed to chemists, experimental physicists, and beginning students of theoretical physics and did not presuppose much mathematical background on the part of its readers. The book became popular even among those for whom quantum theory was not unknown territory (Pauling and Wilson 1935). 

Forty years ago anyone interested in fineparticle characterization who had a science degree could cope with the theory of the techniques and was able to build relatively simple equipment to carry out the measurements. In the two most recent areas of fineparticle characterization — diffractometers (discussed earlier) and the method to be discussed in this section photon correlation spectroscopy (PCS), the relevant theories involve the use of concepts normally not studied until the post graduate level in an honours physics degree. The theory also involves electronic equipment and data processing concepts out of reach for people without a mathematical background. For these instruments the expensive electronic processing equipment and size evaluation equipment can truly be called a black box, from the point of view of the average operator. 

For a better understanding of the mathematical background method, a little excursion to the stability theory of differential equations is useful. To this end we consider the first order differential equation... 

Detailed description of the mathematical background of 2D correlation theory is provided in Appendix F of this book. Here we only briefly go over the correlation treatment of a set of discretely observed spectral data commonly encountered in practice. Let us assume spectral intensity x u, u) described as a function of two separate variables spectral index variable v of the probe and additional variable u reflecting the effect of the applied perturbation. Typically, spectral intensity x is sampled and stored as a function of variables v and u at finite and often constant increments. For convenience, here, we refer to the variable v as wavenumber v and the variable u as time t. For a set of m spectral data x(v, f,) with f = 1,2,. .., m, observed during the period between and we define the dynamic spectrum y(v, t ) as... 

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