Mathematical system theory

In mathematical system theory, the subject of model reduction has been studied for about 30 years. The focus is on model reduction of linear systems, in particular methods based on singular value decomposition. One of the best known of these methods is balanced truncation. It is used extensively for various engineering purposes, such as electronic chip design and the reduction of models of aerospace structures. This method does not require the type of a priori information about the system mentioned above. Only recently has it been tried out on biochemical systems [105, 106]. 

Persson, P.O. Strang, G. (2003). Smoothing by Savitzky-Golay and Legendre filters. In Mathematical Systems Theory in Biology, Communications, Computation and Finance, Rosenthal J. Gilliam D.S., pp. 301-315, IMA Vol. Math. Ap>pl., 134, Springer, ISBN 978-0387-40319-9, New York, USA... 

Knuth, D. E. (1968) Semantics of Context-Free Languages. Mathematical Systems Theory Wol 2, No 2, pp 127-145. 

Kalman, R.E., Falb, P.L. Arbib, M.A. 1969. Topics in Mathematical System Theory (Second ed.). New York McGraw-Hill Book Company. 

The notion of state was derived from the theory of mechanics and of thermodynamics and generalised by mathematical system theory. Following the tradition of the field theory of continua (Truesdell Noll, 1965) quantities of a theory can be classified into two categories state variables and constitutive quantities. State variables are functions such that their values... 

Polderman, J. W., Willems, J. C. (1998). Introduction to Mathematical Systems Theory A Behavioral Approach. Springer. 

In 1932 Gddel [3] proved that in any axiomatic mathematical system (theory), there are fuzzy propositions, that is, propositions which cannot be proved or disproved within the axioms of this system. 

Other important historical landmarks include the founding, in 1984, of the Santa Fe Institute, which is one of the leading interdisciplinary centers for complex systems theory research the first conference devoted solely to research in cellular automata (which is a prototypical mathematical model of complex systems), organized by Farmer, Toffoli and Wolfram at MIT in 1984 [farmer84] and the first artificial life conference, organized by Chri.s Langton at Los Alamos National Laboratory, in 1987 [lang89]. 

While the history of CA can be traced back to early Systems Theory and rigorous mathematical analyses conducted primarily by Russian researchers in the 1930s and 40s, their more recent incarnation as simple models of complexity in nature can arguably be traced to a single landmark review paper published by Wolfram in the Reviews of Modern Physics in 1983 [wolf83a]. 

From the last example, we may see why the primary mathematical tools in modem control are based on linear system theories and time domain analysis. Part of the confusion in learning these more advanced techniques is that the umbilical cord to Laplace transform is not entirely severed, and we need to appreciate the link between the two approaches. On the bright side, if we can convert a state space model to transfer function form, we can still make use of classical control techniques. A couple of examples in Chapter 9 will illustrate how classical and state space techniques can work together. 

This success, however, has a dark side. There are clear signs that MCA has heen oversold in a certain sense that some relative novices in systems mathematics (not all of whom are biologists) view MCA as the end all of metabolic systems theory that MCA is seen as the theory of everything for metabolic engineering. This is, sadly, far from true. 

Although the importance of a systemic perspective on metabolism has only recently attained widespread attention, a formal frameworks for systemic analysis has already been developed since the late 1960s. Biochemical Systems Theory (BST), put forward by Savageau and others [142, 144 147], seeks to provide a unified framework for the analysis of cellular reaction networks. Predating Metabolic Control Analysis, BST emphasizes three main aspects in the analysis of metabolism [319] (i) the importance of the interconnections, rather than the components, for cellular function (ii) the nonlinearity of biochemical rate equations (iii) the need for a unified mathematical treatment. Similar to MCA, the achievements associated with BST would warrant a more elaborate treatment, here we will focus on BST solely as a tool for the approximation and numerical simulation of complex biochemical reaction networks. 

Mathematical formulation of a new system theory - the three-step model -applicable to PBC. 

A new system theory - the three-step model - of packed-bed combustion is formulated. Some new quantities and efficiencies are deduced in the context of the three-step model, such as the conversion gas, the solid-fuel convertibles, the conversion efficiency and the combustion efficiency. Mathematical models to determine the efficiencies are formulated. 

The three-step model, which is a new system theory, identifies the least common functions (unit operations) of a packed-bed combustion system. The three steps are referred to as the conversion system, the combustion system, and the boiler system. Previously, PBC has been modelled in two steps [3,15]. The novel approach with the three-step model is the splitting of the combustion chamber (furnace) into a combustion system and a conversion system. The simple three-step model is a steady-state approach together with some other simplifications applied to the general three-step model. The simple three-step model implies a simplified approach to the mathematical analysis of the extremely differentiated and complex PBC process. 

General system theory views a system as possessing three basic elements - inputs, transforms, and outputs (see Figure 1.1). An example of a simple system is the mathematical relationship... 

The phenomena of ignition and extinction of a flame are typical examples of discontinuous change in a system under smooth variation of parameters. It is natural that they have played a substantial role in the formation of one of the branches of modern mathematics—catastrophe theory. In Ya.B. s work it is clearly shown that steady, time-independent solutions which arise asymptotically from non-steady solutions as the time goes to infinity are discontinuous. It is further shown that transition from one type of solution to the other occurs when the first ceases to exist. The interest which this set of problems stirred among mathematicians is illustrated by I. M. Gel fand s... 

Bottom-up systems biology does not rely that heavily on Omics. It predates top-down systems biology and it developed out of the endeavors associated with the construction of the first mathematical models of metabolism in the 1960s [10, 11], the development of enzyme kinetics [12-15], metabolic control analysis [16, 17], biochemical systems theory [18], nonequilibrium thermodynamics [6, 19, 20], and the pioneering work on emergent aspects of networks by researchers such as Jacob, Monod, and Koshland [21-23]. 

The orthodox or Copenhagen interpretation of quantum theory originated with three seminal papers published in 1925-26 by Heisenberg, Born and Jordan and an independent paper by Dirac (1926) all of these are available in English (translation) in a single volume [13]. A detailed summary was published by Heisenberg [9]. The primary aim of these studies was to formulate a mathematical system for the mechanics of atomic and electronic motion, based entirely on relations between experimentally observable quantities. An immediate consequence of this stipulation was that the motion of electrons could no longer be described in terms of the familiar concepts of space and time, but rather in terms of state functions constructed from matrix elements that relate to the Fourier sums over observed spectroscopic frequencies. The procedure became known as matrix mechanics. 

Hamiltonian dynamical system theory is the mathematical framework on which TST rests many textbooks, of various mathematical sophistication, describe this branch of pure/applied mathematics. Some of the various flavors are [20-24]. Very little of this vast information will be needed here, and we shall try to be as self-consistent as possible. 

Extension toward the fully nonlinear case is straightforward for 1-DOF Hamiltonians. The energy conservation relation H p,q) = E allows us to dehne (explicitly or implicitly) p = p q E), thereby reducing the ODE to a simple quadrature. In this procedure there is no problem of principle (unlike the n >2-DOE case). It works in practice also, and it is possible to adapt Eigs. 3-5 to the nonlinear regime. It must be underlined that besides that simple procedure, we present a theorem in dynamical system theory (containing Hamiltonian dynamics as a particular case). This theorem is valid for n DOEs (hence for n = 1) it relates the full dynamics to the linearized dynamics, called tangent dynamics in the mathematical literature. 

A scientific theory, like a mathematical system, never yields more than is built into it in the way of assumptions or postulates. The phenomenological approach presented in the preceding chapters could no more than characterize the kinetic behavior of systems in terms of the macroscopic variables used to describe them. We have obtained from this approach the kinetic quantities and rate constants or, in terms of the Arrhenius formulation, the frequency factors and the energies of activation. These quantities constitute our phenomenological category of kinetic language. If we are to relate them to the molecular properties of the reacting species, we must construct a new theory and a new nomenclature which starts with the molecule as the unit under consideration. 

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