Algebra. Laws

The algebra of sets-BooIean algebra-governs tlie way in wliich sets can be manipulated to form equivalent sets. The principal Boolean algebra laws used for tliis purpose are as follows. 

Let us consider now the gas-liquid system near the critical point (Fig. 1.3). At T Tc both phases coexist, their densities til (liquid) and nG (gas) could be formally written as nL = nc + 5n/2, nG = nc - Sn/2, where nc is density at the critical point. Note that in the physics of critical phenomena the order parameter is often defined subtracting the background value of nc, i.e., as the order parameter the difference of densities, Sn = n — nG, could be used rather than these individual densities themselves. Such an order parameter is zero at T > Tc and becomes nonzero at T < Tc. Another distinctive feature of the order parameter is that for all simple systems the algebraic law 6n a (Tc — T) holds, where j3 is constant. 

It is useful to check whether this kind of relations is valid for other systems like ferromagnetics and ferroelectrics too. Here the order parameters are the magnetization M and the polarization P, respectively. At high temperatures (T > Tc), and zero external field these values are M = 0 (paramagnetic phase) and P = 0 (paraelectric phase) respectively. At lower temperatures close to the phase transition point, however, spontaneous magnetization and polarization arise following both the algebraic law M, P oc (Tc - Tf. 

The advantage of defining a transfer function in terms of Laplace transforms of input and output is that the differential equations developed to describe the unsteady-state behaviour of the system are reduced to simple algebraic relationships (e.g. cf. equations 7.17 and 7.19). Such relationships are much easier to deal with, and normal algebraic laws can be used to relate the various transfer functions of each component in the control loop (see Section 7.9). Furthermore, the output (or response) of the system to a variety of inputs may be obtained without classical integration. 

EXAMPLE 4 We cannot add 5 hours (time) to 20 miles/hour (speed) since time and speed have different physical significance. If we are to add 21b (mass) and 4 kg (mass), we must first convert lb to kg or kg to lb. Quantities of various types, however, can be combined in multiplication or division, in which the units as well as the numbers obey the algebraic laws of multiplication, squaring, division, and cancellation. Keeping these concepts in mind ... 

Most commonly situations of this kind are attacked by variants of the method of finite differences. A numerical model of the electrochemical system is set up within a computer, and the model is allowed to evolve by a set of algebraic laws derived from the differential equations. In effect, one carries out a simulation of the experiment, and one can extract from it numeric representations of current functions, concentration profiles, potential transients, and so on. 

Sets and Boolean algebra laws play an important role in probabilistic safety analysis, and both are presented here [10]. 

The book is composed of 11 chapters. Chapter 1 presents the various introductory aspects of patient safety including patient safety-related facts and figures, terms and definitions, and sources for obtaining useful information on patient safety. Chapter 2 reviews mathematical concepts considered useful to understand subsequent chapters and covers topics such as mode, median, mean deviation. Boolean algebra laws, probability definition and properties, Laplace transforms, and probability distributions. 

Note Generalized (s-domain) impedances Z(s) and admittances Y(s) obey the same algebraic laws of series and parallel combination as do resistors, thereby simphfying circuit analysis.)... 

Boolean algebra plays an important role in various types of transportation system reliability and safety studies and is named after George Boole (1813-1864), a mathematician. Some of fhe Boolean algebra laws are as follows [3,4] ... 

More recently, another intermediate timescale was explored by NMR relaxom-etry [102]. The spin-lattice relaxation times of water molecules were determined in the range of 20 ns to 20 ps by varying the magnetic field frequency from 10 kHz to 20 MHz and depending on the water content. This technique is well suited for the study of ionomer membranes because of its extreme sensitivity to water-polymer interactions, but it requires a structural and dynamic model to extract characteristic features. The effect of confinement is predominant in polyimides even at high water content (algebraic law with a slope of —0.5 characteristic of porous materials), whereas the diffusion quickly reaches a bulk behavior in Nafion (a plateau is observed at low magnetic fields). 

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