Most probable

Schrodinger wave equation The fundamental equation of wave mechanics which relates energy to field. The equation which gives the most probable positions of any particle, when it is behaving in a wave form, in terms of the field. 

The uncertainty may be addressed by constructing a base case which represents the most probable outcome, and then performing sensitivities around this case to determine which of the inputs the project is most vulnerable to. The most influential parameters may then be studied more carefully. Typical sensitivities are considered in Section 13.7, Sensitivity Analysis . 

In the Maximum Entropy Method (MEM) which proceeds the maximization of the conditional probability P(fl p ) (6) yielding the most probable solution, the probability P(p) introducing the a priory knowledge is issued from so called ergodic situations in many applications for image restoration [1]. That means, that the a priori probabilities of all microscopic configurations p are all the same. It yields to the well known form of the functional 5(/2 ) [9] ... 

The comparisons between measured and simulated data lead to the same conclusion as in case 2. The simulated data show more details on the curves, especially in the slot edges zones. This is linked most probably to the measured data resolution. 

Emulsion A has a droplet size distribution that obeys the ordinary Gaussian error curve. The most probable droplet size is 5 iim. Make a plot of p/p(max), where p(max) is the maximum probability, versus size if the width at p/p(max) = j corresponds to... 

The microcanonical ensemble is a certain model for the repetition of experiments in every repetition, the system has exactly the same energy, Wand F but otherwise there is no experimental control over its microstate. Because the microcanonical ensemble distribution depends only on the total energy, which is a constant of motion, it is time independent and mean values calculated with it are also time independent. This is as it should be for an equilibrium system. Besides the ensemble average value (il), another coimnonly used average is the most probable value, which is the value of tS(p, q) that is possessed by the largest number of systems in the ensemble. The ensemble average and the most probable value are nearly equal if the mean square fluctuation is small, i.e. if... 

Among all possible partitions in the above expression, the equilibrium partition eorresponds to the most probable partition, for whieh dP = 0. Evaluating tiiis differential yields the following relation ... 

Sinee E and A are independent variables, tlieir variations are arbitrary. Henee, for the above equality to be satisfied, eaeh of the two braeketed expressions must vanish when the (E, N) partition is most probable. The vanishing of the eoeffieient of dE implies the equality of temperatures of I and II, eonsistent with thennal equilibrium ... 

The result that the eoeffieient of dA is zero for the most probable partition is the eonsequenee of the ehemieal equilibrium between the system and the reservoir. It leads us to identify the ehemieal potential p as... 

Millikan R A A new modification of the cloud method of determining the elementary electrical charge and the most probable value of that charge Phil. Mag. 19 209-28... 

A saddle point approximation to the above integral provides the definition for optimal trajectories. The computations of most probable trajectories were discussed at length [1]. We consider the optimization of a discrete version of the action. 

If the above assumption is reasonable, then the modeling of most probable trajectories and of ensembles of trajectories is possible. We further discussed the calculations of the state conditional probability and the connection of the conditional probability to rate constants and phenomenological models. 

The Maxwell-Boltzmann velocity distribution function resembles the Gaussian distribution function because molecular and atomic velocities are randomly distributed about their mean. For a hypothetical particle constrained to move on the A -axis, or for the A -component of velocities of a real collection of particles moving freely in 3-space, the peak in the velocity distribution is at the mean, Vj. = 0. This leads to an apparent contradiction. As we know from the kinetic theor y of gases, at T > 0 all molecules are in motion. How can all particles be moving when the most probable velocity is = 0 ... 

The most probable value of the speed v p can be obtained by differentiation of the distribution function and setting dG(v)/dv = 0 (Kauzmann, 1966 Atkins 1990) to obtain... 

In so doing, we obtain the condition of maximum probability (or, more properly, minimum probable prediction error) for the entire distribution of events, that is, the most probable distribution. The minimization condition [condition (3-4)] requires that the sum of squares of the differences between p and all of the values xi be simultaneously as small as possible. We cannot change the xi, which are experimental measurements, so the problem becomes one of selecting the value of p that best satisfies condition (3-4). It is reasonable to suppose that p, subject to the minimization condition, will be the arithmetic mean, x = )/ > provided that... 

This method, because it involves minimizing the sum of squares of the deviations xi — p, is called the method of least squares. We have encountered the principle before in our discussion of the most probable velocity of an individual particle (atom or molecule), given a Gaussian distr ibution of particle velocities. It is ver y powerful, and we shall use it in a number of different settings to obtain the best approximation to a data set of scalars (arithmetic mean), the best approximation to a straight line, and the best approximation to parabolic and higher-order data sets of two or more dimensions. 

The reactivity of dibenzyl is similar to those of alkylbenzenes, and it is therefore most probable that the nitrations of the latter substances were also influenced by mixing. ... 

The "zip-reaction (U. Kramer, 1978, 1979) leads to giant macrocycles. Potassium 3- ami-nopropyl)amide = KAPA ( superbase ) in 1,3-diaminopropane is used to deprotonate amines. The amide anions are highly nucleophilic and may, for example, be used to transam-idate carboxylic amides. If N- 39-atnino-4,8,12,16,20,24,28,32,36-nonaazanonatriacontyl)do-decanolactam is treated with KAPA, the amino groups may be deprotonated and react with the macrocyclic lactam. The most probable reaction is the intramolecular formation of the six-membered ring intermediate indicated below. This intermediate opens spontaneously to produce the azalactam with seventeen atoms in the cycle. This reaction is repeated nine times in the presence of excess KAPA, and the 53-membered macrocycle is formed in reasonable yield. 

Many monomeric heterocyclic anhydrobases can be isolated now using specific methods (44), but application of these methods to thiazole ring did not succeed however, appropriate conditions lead to the separation of a dimer, the structure of which has been established by its NMR Spectra and chemical reactivity (26). The most probable mechanism of its formation appears identical with the one previously described in the benzothiazolium series (24). A second molecule of quaternary salt A3... 

The conclusion of all these thermodynamic studies is the existence of thiazole-solvent and thiazole-thiazole associations. The most probable mode of association is of the n-rr type from the lone pair of the nitrogen of one molecule to the various other atoms of the other. These associations are confirmed by the results of viscosimetnc studies on thiazole and binary mixtures of thiazole and CCU or QHij. In the case of CCU, there is association of two thiazole molecules with one solvent molecule, whereas cyclohexane seems to destroy some thiazole self-associations (aggregates) existing in the pure liquid (312-314). The same conclusions are drawn from the study of the self-diffusion of thiazole (labeled with C) in thiazole-cyclohexane solutions (114). 

F. 1-26. (a) ir-Bond order of the C-S bonds in the ground state, (fc) ir-Bond order of the C-S bonds in the first excited state, (c) Free-valence number of the intermediate diradicaf. (bicyclic intermediate resulting from the ring closure of the diradicai. 

The standardized variable (the z statistic) requires only the probability level to be specified. It measures the deviation from the population mean in units of standard deviation. Y is 0.399 for the most probable value, /x. In the absence of any other information, the normal distribution is assumed to apply whenever repetitive measurements are made on a sample, or a similar measurement is made on different samples. 

Equation (2.28), being statistical in nature, requires a large number of particles to be measured, especially if the spread of particle size is wide. The possibility of error from this source is stressed by Arnell and Henneberry who found that in a particular sample of finely ground quartz, two particles in a total of 335 had a diameter about twenty times the most probable diameter, and that if these were overlooked the calculated value of A would be nearly doubled. 

The intercept on the adsorption axis, and also the value of c, diminishes as the amount of retained nonane increases (Table 4.7). The very high value of c (>10 ) for the starting material could in principle be explained by adsorption either in micropores or on active sites such as exposed Ti cations produced by dehydration but, as shown in earlier work, the latter kind of adsorption would result in isotherms of quite different shape, and can be ruled out. The negative intercept obtained with the 25°C-outgassed sample (Fig. 4.14 curve (D)) is a mathematical consequence of the reduced adsorption at low relative pressure which in expressed in the low c-value (c = 13). It is most probably accounted for by the presence of adsorbed nonane on the external surface which was not removed at 25°C but only at I50°C. (The Frenkel-Halsey-Hill exponent (p. 90) for the multilayer region of the 25°C-outgassed sample was only 1 -9 as compared with 2-61 for the standard rutile, and 2-38 for the 150°C-outgassed sample). 

Earlier we introduced the confidence interval as a way to report the most probable value for a population s mean, p, when the population s standard deviation, O, is known. Since is an unbiased estimator of O, it should be possible to construct confidence intervals for samples by replacing O in equations 4.10 and 4.11 with s. Two complications arise, however. The first is that we cannot define for a single member of a population. Consequently, equation 4.10 cannot be extended to situations in which is used as an estimator of O. In other words, when O is unknown, we cannot construct a confidence interval for p, by sampling only a single member of the population. 

A solution can be diluted by a factor of 200 using readily available pipets (f-mL to fOO-mL) and volumetric flasks (fO-mL to fOOO-mL) in either one, two, or three steps. Limiting yourself to glassware listed in Table 4.2, determine the proper combination of glassware to accomplish each dilution, and rank them in order of their most probable uncertainties. 

Determine the uncertainty for the gravimetric analysis described in Example 8.1. (a) How does your result compare with the expected accuracy of 0.1-0.2% for precipitation gravimetry (b) What sources of error might account for any discrepancy between the most probable measurement error and the expected accuracy ... 

Spike recoveries for samples are used to detect systematic errors due to the sample matrix or the stability of the sample after its collection. Ideally, samples should be spiked in the field at a concentration between 1 and 10 times the expected concentration of the analyte or 5 to 50 times the method s detection limit, whichever is larger. If the recovery for a field spike is unacceptable, then a sample is spiked in the laboratory and analyzed immediately. If the recovery for the laboratory spike is acceptable, then the poor recovery for the field spike may be due to the sample s deterioration during storage. When the recovery for the laboratory spike also is unacceptable, the most probable cause is a matrix-dependent relationship between the analytical signal and the concentration of the analyte. In this case the samples should be analyzed by the method of standard additions. Typical limits for acceptable spike recoveries for the analysis of waters and wastewaters are shown in Table 15.1. ... 

In addition to and r nis ai other way of characterizing coil dimensions is to consider which end-to-end distance has the greatest probability of occurring for specified n and 1 values. Derive an expression for this most probable value of r, r, from Eq. (1.44). Compare the ratio r ms/ m the ratio from the kinetic molecular theory of gases (consult, say,... 

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