Bell-shape

The density of states increases rapidly with energy but the Boltzmann factor decrease exponentially, meaning that Pcanon(T, E) is bell-shaped, with values that can vary by mar orders of magnitude as the energy changes. In the multicanonical method the simulatic... 

The plot of y against I gives a curve of the well known bell-shaped form (Fig. 1.13). The sharpness of the peak is determined by the value of <7 , the peak becoming narrower as the value of o decreases. Curve I of Fig. 1.13, with its sharper peak, corresponds to a more uniform size distribution than does Curve II. 

Bell-shaped probability distribution curve for measurements and results showing the effect of random error. 

The proof that these expressions are equivalent to Eq. (1.35) under suitable conditions is found in statistics textbooks. We shall have occasion to use the Poisson approximation to the binomial in discussing crystallization of polymers in Chap. 4, and the distribution of molecular weights of certain polymers in Chap. 6. The normal distribution is the familiar bell-shaped distribution that is known in academic circles as the curve. We shall use it in discussing diffusion in Chap. 9. 

P(x, t) dx has the familiar bell shape of a normal distribution function [Eq. (1.39)], the width of which is measured by the standard deviation o. In Eq. (9.83), t takes the place of o. It makes sense that the distribution of matter depends in this way on time, with the width increasing with t. 

Normal Distribution of Observations Many types of data follow what is called the gaussian, or bell-shaped, curve this is especially true of averages. Basically, the gaussian curve is a purely mathematical function which has very specif properties. However, owing to some mathematically intractable aspects primary use of the function is restricted to tabulated values. 

A normal distribution cur ve is bell-shaped (see Sec. 3). The cur ve obeys the relationship... 

Plots of E(L) and/(L) for various values of n appear in Fig. 23-9. The bell shapes of E t ) are more distinctive. The experimental curves of Figs. 23-10 and 23-11 clearly are of that family. 

Procedures for curve fitting by polynomials are widely available. Bell-shaped curves, however, are fitted better and with fewer constants by ratios of polynomials. For figuring chemical conversions, the... 

The solution of this partial differential equation is recorded in the literature (Otake and Kunigata, Kngaku Kogaku, 22, 144 [1958]). The plots of E(t ) against t are bell-shaped, resembling the corresponding Erlang plots. A relation is cited later between the Peclet number,... 

In the above ealeulations of the mean, varianee and standard deviation, we make no prior assumption about the shape of the population distribution. Many of the data distributions eneountered in engineering have a bell-shaped form similar to that showed in Figure 1. In sueh eases, the Normal or Gaussian eontinuous distribution ean be used to model the data using the mean and standard deviation properties. 

The crater surfaces obtained in the LA-TOF-MS experiment on the TiN-TiAlN-Fe sample were remarkably smooth and clearly demonstrated the Gaussian intensity distribution of the laser beam. Fig. 4.45 shows an SEM image of the crater after 100 laser pulses (fluence 0.35 J cm ). The crater is symmetrical and bell-shaped. There is no significant distortion of the single layers. Fig. 4.45 is an excellent demonstration of the potential of femtosecond laser ablation, if the laser beam had a flat-top, rather than Gaussian, intensity profile. 

Fuzzy sets represented by symmetrical triangles are commonly used because they give good results and computation is simple. Other arrangements include non-symmetrical triangles, trapezoids, Gaussian and bell shaped curves. 

A frequently encountered pH-rate profile exhibits a bell-like shape or hump, with two inflection points. This graphical feature is essentially two sigmoid curves back-to-back. By analogy with the earlier analysis of the sigmoid pH-rate curve, where the shape was ascribed to an acid-base equilibrium of the substrate, we find that the bell-shaped curve can usually be accounted for in terms of two acid-base dissociations of the substrate. The substrate can be regarded, for this analysis, as a dibasic acid H2S, where the charge type is irrelevant we take the neutral molecule as an example. The acid dissociation constants are... 

The intersection of the curves Fhjs and Fhs occurs at pH = pA"i, as can be found by setting Eqs. (6-73) and (6-74) equal. Also, when Fhs = Fs, pH = pA 2. These are general relationships. We note, at this point, that the function Fhs has the previously mentioned bell shape, and it is this function that will be of later kinetic interest. 

Equation (6-78) has the same form as Eq. (6-74) for Fhs, so the simple scheme embodied in Eq. (6-77) can account for a bell-shaped curve when k is plotted against pH. 

Figure 6-18 shows a bell-shaped pH-rate profile for the hydrolysis of monomethyl dihydrogen phosphate. Other examples are the hydrolysis of o-carboxyphenyl hydrogen succinate and the hydration of fumaric acid. ... 

Kinetic schemes other than that embodied in Eq. (6-77) can give rise to a bellshaped curve. As in Eq. (6-77), however most of these involve two ionizations. Thus Scheme V, where HS is a monoprotic acid and B is a base (or the kinetic equivalent of S + BH ) yields a bell-shaped curve. 

A less obvious scheme that can lead to a bell-shaped curve has been recognized by Zemer and Bender. The substrate is a weak acid or base, but possesses only one ionizable group, and no other ionizable reactant is involved (other than water). The second inflection in the curve is ascribed to an ionizable group created in an intermediate. An example has been discovered in the hydrolysis of o-carboxy-phthalimide ... 

A bell-shaped pH-rate profile can also be produced in a two-step reaction involving a single ionizable group if the rate-determining step changes when the pH is altered. An example, the oximation of acetone, is shown in Fig. 5-12. 

In tlie case of a random sample of observations on a continuous random variable assumed to have a so-called nonnal pdf, tlie graph of which is a bell-shaped curve, tlie following statements give a more precise interpretation of tlie sample standard deviation S as a measure of spread or dispersion. 

The bell-shaped curve in part (c) of the figure combines both of these behaviors, so that activity first increases, then decreases, as pH is increased. This is consistent with the involvement of two ionizable groups—one with a low pA that acts as a base above its pK, and a second group with a higher pK that acts as an acid below its pK. ... 

Kinetic studies with pepsin have produced bell-shaped curves for a variety of substrate peptides see below, (a). 

Bell-shaped activity versus pH profiles arise from two separate active-site ionizations, (a) Enzyme activity increases upon deprotonation of (b) Enzyme activity decreases upon deprotonation of A-H. (c) Enzyme activity is maximal in the pH range where one ionizable group is deprotonated (as B ) and the odier group is protonated (as A-H). 

The most advanced among today s projects is the future American reusable launch system known as Venture Star. It utilizes two new concepts that are expected to boost efficiency and cut down the cost of a payload. The first is the single stage-to-orbit concept—the launch vehicle carries the entire propellant load and does not have any expendable parts. The latter is planned to be achieved by a revolutionary rocket engine design called Aerospike. The bell-shaped noz-... 

Although there are many styles and dimensions of caps in use, the round bell shaped bubble cap is quite practical and efficient. It is recommended as a good basis for the contacting requirements. This selection does not infer that other contacting caps are not acceptable, in fact many are in use in the chemical and petroleum industry. Their design criteria is limited to the proprietary knowledge of the manufacturer and not available to the designer. 

Good quality steel is used and electrozinc is preferred for washing machines. Steel is pretreated with iron phosphate for economy electrozinc with a fine crystal zinc phosphate. No primer is normally used 25-40/im of finish is applied direct to metal. The required properties are best obtained with a thermosetting acrylic or polyester/melamine-formaldehyde finish. Self-reactive acrylics are usually preferred these resins contain about 15 Vo 7V-butoxymethyl acrylamide (CH2=CH —CO —NH —CHj—O —C4H,) monomer and cure in a manner similar to butylated melamine-formaldehyde resins. Resistance or anti-corrosive properties may be upgraded by the inclusion of small amounts of epoxy resin. Application is usually by electrostatic spray application from disc or bell. Shapes are complex enough to require convected hot-air curing. Schedules of 20 min at 150-175°C are... 

If a large number of replicate readings, at least 50, are taken of a continuous variable, e.g. a titrimetric end-point, the results attained will usually be distributed about the mean in a roughly symmetrical manner. The mathematical model that best satisfies such a distribution of random errors is called the Normal (or Gaussian) distribution. This is a bell-shaped curve that is symmetrical about the mean as shown in Fig. 4.1. 

A number of studies have focused on D-A systems in which D and A are either embedded in a rigid matrix [103-110] or separated by a rigid spacer with covalent bonds [111-118], Miller etal. [114, 115] gave the first experimental evidence for the bell-shape energy gap dependence in charge shift type ET reactions [114,115], Many studies have been reported on the photoinduced ET across the interfaces of some organized assemblies such as surfactant micelles [4] and vesicles [5], wherein some particular D and A species are expected to be separated by a phase boundary. However, owing to the dynamic nature of such interfacial systems, D and A are not always statically fixed at specific locations. 

Kabanov etal.123 hydrolyzed 44 (PNPA) and 56 (NABA) in the presence of QPVP. They found that the hydrolysis rate decreased with increasing degree of quatemization in water. A bell-shaped dependence was observed for 56 (NABA) in alcohol-aqueous media (see Fig. 13). Recently, Cho and Morawetz discussed the hydrolyses of a polymer substrate by a pyridine-containing polymer124. ... 

The three-dimensional structure of the sodium channel (from electric eel) was determined at 19-A resolution using cryo-electron microscopy and single-particle image analysis. The sodium channel has a bell-shaped outer surface of 135 A in height, 100 A in side length at the square bottom, and 65 A in diameter of the spherical top. An interesting finding is that there are several inner cavities connected to outer orifices. 

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