Rectangle function

Fig. 8.32. Two-dimensional Fourier transformation applied to a rectangle function shown in original 3D representation (a) and 2D contour plot (b) and as Fourier transforms (c,d), (according to Danzer et al. [2001])...

If we convolve two single-peaked functions, the result is smoother than either component. Two rectangle functions convolved yield a triangle two triangle functions convolved (four rectangles convolved) produce a result that is astonishingly close to a gaussian (Fig. 1). 

Instead of the application of a low pass filter in the frequency domain itself (multiplication with 1 in the low-frequency range and above that with 0, i.e. multiplication of the frequency response with a rectangle function), one can use in the amplitude domain (i.e. for the measurement itself) the mathematically fully equivalent convolution with the Fourier transform of that rectangle function. 

Figure 6.11 Equatorial aberration coupled with the receiving slit width calculated by the proposed method (solid line) and as a convolution (open circles) of JFS with the rectangle function (dashed line) representing the receiving slit. The vertical line at 20° represents the Bragg angle to which the aberration function is related. (Reprinted from Ref. 53. Permission of the International Union of Crystallography.)...

If the rectangle function is approximated by a modified Gaussian function, the boundary conditions can be written as ... 

Figure 6. Illustration of the Convolution Theorem, a.) The two-rectangle function, p(x), representing a simplified electron density distribution from two scatterers and b.) its Fourier transform, F(u), the amplitude of scattered radiation c.) the autocorrelation function, p(x) o p(x) and d.) its Fourier transform, F(u). ...

Figure 3.8. The transformation of a rectangular into a normal distribution. The rectangle at the lower left shows the probability density (idealized observed frequency of events) for a random generator versus x in the range 0 < jc < 1. The curve at the upper left is the cumulative probability CP versus deviation z function introduced in Section 1.2.1. At right, a normal distribution probability density PD is shown. The dotted line marked with an open square indicates the transformation for a random number smaller or equal to 0.5, the dot-dashed line starting from the filled square is for a random number larger than 0.5.

Fig. 3.1.11 Relative effective intracrystalline diffusivities D(t)/D0 as function of JDot for n-hexane under single-component adsorption (circles) and for n-hexane (triangles) and tetrafluoromethane (rectangles) under two-...

Fig. 3. Secondary chemical shifts for 13C , 13CO, H , and 13C as a function of residue number in apomyoglobin at pH 4.1. Bars at the top of the figure indicate the presence of NOEs the smaller bars indicate that the NOE was ambiguous due to resonance overlap. Black rectangles at the base of the top panel indicate the locations of helices in the native holomyoglobin structure (Kuriyan et al, 1986). Hashed rectangles indicate putative boundaries for helical regions in the pH 4 intermediate, based on the chemical shift and NOE data. Reproduced from Eliezer et al (2000). Biochemistry 39, 2894-2901, with permission from the American Chemical Society.

Consider the function fix) = 10 - 10e 2x. Define x = a and x = b, and suppose it is desirable to compute the area between the curve and the coordinate axis y = 0 and bounded by Xi = a,xn = b. Obviously, by a sufficiently large number of rectangles this area could be approximated as closely as desired by the formula... 

The flat tire example is pictured using a fault tree logic diagram, shown in Figure 11-12. The circles denote basic events and the rectangles denote intermediate events. The fishlike symbol represents the OR logic function. It means that either of the input events will cause the output state to occur. As shown in Figure 11 -12, the flat tire is caused by either debris on the road or tire failure. Similarly, the tire failure is caused by either a defective tire or a worn tire. 

Figure 10.4 shows a BB tree, with the root node corresponding to the original rectangle, and each node onthe second level associated with one of these four partitions. Let ft(x) be the underestimating function for the partition associated with node i. The lower bounds shown next to each node are illustrative and are derived by minimizing ft(x) over its partition using any local solver, and the upper bounds... 

As rectangles are partitioned, the difference (w, — /,) decreases, so the successive underestimating functions become tighter approximations to/. 

A conceptualized cross section through a portion of the cell wall (rectangles), periplasmic space, and cell membrane (lipid bilayer with polar head groups in contact with cytoplasm and external medium, and hydrophobic hydrocarbon chains) of an aquatic microbe. Reactive functional groups (-SH, -COOH, -OH, -NH2) present on the wall consitutents and extracellular enzymes (depicted as shaded objects) attached by various means promote and catalyze chemical reactions extracellularly. 

Figure 4. Calculated HAB values as a function of Fe -Fe separation, based on the structural model given in Figure 1 and the diabatic wavefunctions I/a and f/B. Curves 1 and 2 are based on separate models in which the inner-shell ligands are represented, respectively, by a point charge crystal field model [Fe(H20)62 -Fe(HsO)63 ] and by explicit quantum mechanical inclusion of their valence electrons [Fe(HgO)s2 -Fe(H20)s3+] (as defined by the dashed rectangle in Figure 1). The corresponding values of Kei, the electronic transmission factor, are displayed for various Fe-Fe separations of interest.

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