Approximate reasoning

Reasoning is the most important human brain operation that leads to creative ideas, methodologies, algorithms and conclusions in addition to a continuous process of research and development. Reasoning stage can be reached provided that 

The qualitative jump of consciousness to a higher level results in transcending the fuzziness. As far as consciousness is of a holistic characteristic of human, and perhaps not just of human, but also nature and not only a product of mind, but its growth and transformation are possible when the factors responsible for the integrity of all three inseparable constituents of human individuality, which are body, mind and soul, become simultaneously activated. This simultaneous activation ( firing or triggering ) is referred to as a consciousness resonance and hence The fuzziness of understanding can be transcended when the consciousness resonance occurs. 

The consciousness resonance does not eliminate fuzziness, which is an eternal companion to any process of thinking and knowing. At the same time, when the consciousness resonance helps one to transcend the fuzziness related to a problem that dissolves, it opens space for new problems to emerge bringing with them new 

The last statement Zadeh calls the principle of incompatibility. Its message is that the closer one looks at a real-world problem, the fnzzier becomes its solution. 

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These quartic equations are solved in an iterative maimer and, as such, are susceptible to convergence difficulties. In any such iterative process, it is important to start with an approximation reasonably close to the final result. In CC theory, this is often achieved by neglecting all of tlie temis tliat are nonlinear in the t amplitudes (because the ts are assumed to be less than unity in magnitude) and ignoring factors that couple different doubly-excited CSFs (i.e. the sum over i, f, m and n ). This gives t amplitudes that are equal to the... 

The minimum recommended saturated suction temperature on series multistage reciprocating-centrifugal applications, column (K), represents an approximate reasonable limit on suction temperature. The temperatures shown correspond to suction pressures below atmospheric. 

The rheological properties of a particular suspension may be approximated reasonably well by either a power-law or a Bingham-plastic model over the shear rate range of 10 to 50 s. If the consistency coefficient k is 10 N s, /m-2 and the flow behaviour index n is 0.2 in the power law model, what will be the approximate values of the yield stress and of the plastic viscosity in the Bingham-plastic model ... 

The NEMCA time constant, t, is defined1,4 as the time required for the rate increase Ar to reach 63% of its steady-state value during a galvanostatic transient, such as the one shown in Fig. 4.13 and 4.14. Such rate transients can usually be approximated reasonably well by ... 

From this, however, it should not be concluded that the statistical model of the atom is a very good one. As Fano (1963) has pointed out, I appears only as a logarithm and an error Si in the computation of I shows up as a relative error in the stopping power as (l/5)<5l II. Besides, it is an average quantity and can be approximated reasonably well without knowing the details of the distribution. 

The reasons for the effect of pH on the catalytic properties of enzymes are numerous and will not be discussed here. For most enzymes, however, there is a pH at which they are optimally effective changing the pH to lower (more acidic) levels or to higher (more basic) levels will decrease the overall rate at which the associated chemical reaction occurs. In the region of the optimum pH, the reaction rate vs. pH response surface can usually be approximated reasonably well by a second-order, parabolic relationship. 

Williamson RC, Downs T. 1990. Probabilistic arithmetic I numerical methods for calculating convolutions and dependency bounds. Int J Approximate Reasoning 4 89-158. 

For pressures below one atmosphere, a number of simplifications can be made. The volume change on formation of vapor can be approximated reasonably by the volume of the vapor. The vapor is assumed to behave like an ideal gas. In particular, AV = VM = RT/ P. Substituting this value for AV into Equation (1) yields the Clausius-Clapeyron equation ... 

As for AsF3 and AsBr3 a rather good fit of the curves in Fig. 5 to a polynomial series in Tb as in equation (8) is possible, with parameters listed in Table 4. Again N( rb) is approximated reasonably well by a linear function, while for Zsg(rb), Zse(rb) and E (rb) at least one additional an harmonic term a4rb is needed in addition to a2. 7h, in contrast to the assumptions in equations (10) and (11). 

Have the required statistical mechanical considerations been taken into account In many cases, converting potential energies to free energies can be done approximately reasonably easily, and it is often free energies that are more relevant to experiment. For very large systems with multiple conformers, this aspect can be challenging—and a simpler truncated model may be more informative. 

Fuzzy methods are suitable for approximate reasoning (Isukapalli Georgopoulos, 2001), especially for analysis of systems where uncertainty arises due to vagueness or fuzziness or... 

According to eqn.(3.29) retention (fc) varies exponentially with the polarity of the solute ( ). It can easily be seen from eqn.(3.29) that as soon as the solubility parameter product (Sm + Ss — 2Si)(Sm — Ss) becomes significantly positive or negative, the capacity factor will either be impractically high or uselessly low. Hence, to a first approximation reasonable values for k will be obtained when the above product is about equal to zero. This can be achieved in two ways ... 

Epistemically ideal conditions , of course, are like frictionless planes we cannot really attain epistemically ideal conditions, or even be absolutely certain that we have come sufficiently close to them. But frictionless planes cannot really be attained either, and yet talk of frictionless planes has cash value because we can approximate them to a very high degree of approximation. (Reason, Truth and History, 55)... 

The values (B) and (C), derived from analyses of potassium bromate and iodate, are obviously worthless. The results (A), (D), and (F) depend mainly on the careful work of Stas, and their arithmetic mean is 39-139. The results (E) and (G) are based on Marignac s work on potassium chloride and iodide, and approximate reasonably to those of Stas. Excluding (B) and (C), the arithmetic mean of the results is... 

As an example, consider the cnfissivity function plotted in Fig. 12-24. It. seems like this function can be approximated reasonably well by a step function c -the form... 

Reasoning based on fuzzy propositions of the four types, possibly quantified by various fuzzy quantifiers, is usually referred to as approximate reasoning. Although approximate reasoning is currently a subject of intensive research, its basic principles are already well established. For example, the most common inference rules of classical logic, such as modus ponens,... 

The structure of approximate reasoning is not simple. Consider the Born-Oppenheim approximation (separability of electronic and nuclear motions due to extreme mass difference), which in application produces "fixed nuclei" Hamiltonians for individual molecules. In assuming a nuclear skeleton, the idealization neatly corresponds to classical conceptions of a molecule containing localized bonds and definite structure. All early quantum calculations, and the vast majority to date, invoke the approximation. In 1978, following decades of quiet assumption, Cambridge chemist R. G. Woolley asserted ... 

Shafer, G., Perspectives on the Theory and Practice of Belief Functions, International Journal of Approximate Reasoning, 3, 1, 1990. 

Finally, it is worth noting that Tx inax, which is the value of Tx where the isothermal crystallite growth rate is fastest, can be approximated reasonably well by using Equation 6.46 where the temperatures are in degrees Kelvin [64],... 

It is possible to formulate rational reasons for centering on scientific grounds. Basically, centering should be performed only if there are common offsets in the data or if removing such offsets provides an approximately reasonable model. Centering should make a difference and this difference can manifest itself as follows ... 

We may expect the behaviour at intermediate values of valve pressure ratio to be approximated reasonably well by a linear combination of the two functions, with the weighting of each dependent on the distance of the valve pressure ratio from the ends of the range. Hence we postulate that the throat pressure ratio may... 

Fuzzy logic for fuzzy control and for approximate reasoning. 

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