Triangular functions

Fig. 44.23. Some common neighbourhood functions, used in Kohonen networks, (a) a block function, (b) a triangular function, (c) a Gaussian-bell function and (d) a Mexican-hat shaped function. In each of the diagrams is the winning unit situated at the centre of the abscissa. The horizontal axis represents the distance, r, to the winning unit. The vertical axis represents the value of the neighbourhood function. (Reprinted with permission from [70]).

In many cases, the profile a spectroscopist sees is just the instrumental profile, but not the profile emitted by the source. In the simplest case (geometric optics, matched slits), this is a triangular slit function, but diffraction effects by beam limiting apertures, lens (or mirror) aberrations, poor alignment of the spectroscopic apparatus, etc., do often significantly modify the triangular function, especially if high resolution is employed. 

Figure 16 Cut of S(QX, Qy, 0) along Qy for a KHCO3 singlecrystal at 20 K. Comparison of the measured profile with the MARI spectrometer (solid line with error bars) to the best fit (dash line) obtained with Eq. (16) convoluted with a triangular resolution function. Triangular functions ( ) were attributed to other scattering processes. uly = 1.26 x 10 2 A2...

Next, we construct a finite dimensional subspace f2h consisting of piece-wise linear functions. We define subintervals of length Az = Zj+ — Zj, j = 1,2,..., K. As parameters to describe how the function change over the subintervals, we choose the basis functions as the set of triangular functions defined as ... 

Explore the effect of other window functions, such as the triangular function (see exercise 7.5-1 under instruction 24), a trapezoidal function, a Gaussian, a Lorentzian, or whatever. 

Typical distance functions are based on triangular functions or the Mexican hat function (Figure 8.14). 

Important is the monotonicity of the membership function. The special form of the membership function has only a weak influence on the result of fuzzy operations. The parabola function in Eq. (8.37) can be approached, therefore, by a triangular function... 

The coefficients C and D can be found via the boundary conditions by the orthogonality properties of the triangular functions. 

Figure 13. Graphical constructions to show the convolution of two simple square pulses (e(t) and h(t) of Eq. 31) to yield a triangular function (f(t) of Eq. 31). Holding the first square function fixed, the second function is moved from left to right, and the two functions multiplied together, as shown in the shaded segments. Each point, a to e, of the convolution represents the area of the product shown at the lower left (see text).

Heat is also required to transform moisture from a liquid to gas (latent heat Cw = 2260 kJ kg ). The total heat depends on the moisture content of the material and the rate of change is determined by the evaporating rate. Evaporation can also be described by the equations of chemical kinetics. If the mass change of water during the heating process in known, the kinetic parameters can be estimated by the methods introduced previously. In Samanta et al. [4], a 1% mass of moisture content was assumed, while in Keller et al. [1] a 0.5% mass of moisture content was taken. In both cases, the effects of moisture evaporation on heat capacity was assumed roughly as a triangular function dependent on temperature without kinetic considerations. 

Figure 7 A triangular function centered about a spectral line reference value provides for uncertainty and error when matching a measured value to the reference value. The further the measured value is from the reference value, the less the match strength as given by its membership value, jj..

Fig. 7. Signal assignment in high field region of 300-MHz proton NMR spectrum of neurotoxin II in D O solution at pH 7.3 and 32°C. A - Normal spectrum. B - resolution enhanced spectrum by means of a triangular function(41).

Figure 26.1 7 (a) Theoretical predictions for the heat capacity C near the critical temperature, for two-dimensional systems. The Bragg-Williams mean-field lattice model of Chapter 25 leads to a triangular function, while the exact solution of the two-dimensional Ising model shows a sharp peak. Source R Kubo, in cooperation with H Ichimura, T Usui and N Hashitsome, Statistical Mechanics, Elsevier Pub. Co., New York (1 965). (b) Experimental data for helium on graphite closely resembles the Ising model prediction. Source RE Ecke and JC Dash, Phys Rev B 28, 3738 (1983). 

Now let s consider the part played by the slit width in determining the shape and intensity of an absorption band, i,e, its effect. We shall assume that the sample band is of the Lorentz curve shape and has a half band width of 8 cm (Ai ). We shall also assume that the slit width is 1 cm (Ay/) and follows a triangular function. In this case the band as seen by the instrument will have a half band width (Ayi/ ) approximately equal to Ay, and the band will be of the Lorentz shape. Here the spectrophotometer accurately sees the band shape (see Fig. 9). 

This process is illustrated in Eigure 3.5. Eigure 3.5a shows a discretely sampled sinc function sampled at a retardation of (2Vniax) and Eigure 3.5b shows its Eourier transform. As can be seen clearly, there is no overlap of the resulting spectra (triangular functions, in this case). If the sampling frequency were decreased so that... 

Figure D.3 (a) A triangular function, (b) a symmetric triangular function, and (c) the Fourier...

The relation between the exponential decay and the Lorentz profile and that between the cosine function and the delta function as the Fourier transform pairs are described, respectively, in Sections D.3.4 and D.3.5. The Fourier transform of the triangular function shown in Figure 6.6f is a sine function squared as described in Section D.3.6. 

MoM typical basis functions for approximating the distribution of current along the wire axis, (a) Piecewise triangular function and (b) piecewise sinusoidal function. 

If we now further denote the average reduction in capacity when the system is in a failed state by A, then, for our purposes, the function cp ( ) can be approximated by a triangular function. 

Figure 6.18 shows the membership function of the riskiness of an event on an arbitrary scale, which would later be used to defuzzify the fuzzy conclusion and rank the risk according to a priority number. The membership function used is a triangular function. Unlike the trapezoidal function, the membership value of 1 in the triangular function is limited to (mly one value of the variable on the x-axis. 

The membership function used (i.e. trapezoidal function) allows a membership value of 1 for a range of probabilities unlike the triangular function. This function is thought to model the probability of occurrence close to what it is in reality (Pillay (2001)). Figure 6.19 shows the membership function along with its ordinal scale. The limits and the centre point values of the ordinal scale are given by the dotted line and will be used to perform the fuzzy arithmetic. 

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