# Deviations from the Average

Deviations from the Average [c.321]

Deviations from the Average [c.321]

Next compute the sum of squares of deviations from the average within the tth treatment [c.506]

The discontinuity disappears, since the boundary line is very sensitive to the local density variations, so that it is not only the function of the average film density, specified by B, but it also changes when the local density exhibits deviations from the average density of a given phase, (1) or (2), due to changes in the surface field. Thus, there will be a rounding of the transition region over a certain range of temperatures around Explicit calcula- [c.265]

The spectroscopic methods, NMR and mass spectrometry for predicting cetane numbers have been established from correlations of a large number of samples. The NMR of carbon 13 or proton (see Chapter 3) can be employed. In terms of ease of operation, analysis time (15 minutes), accuracy of prediction (1.4 points average deviation from the measured number), it is [c.220]

One approach has been to build on the DR equation by adding a Gaussian [228] or gamma function-type [224] distribution of B values. Another approach makes use of a modified Fowler-Guggenheim equation (see Eq. XVII-53) [229]. A rather different method for obtaining a micropore size distribution was proposed by Mikhail, Brunauer, and Bodor [230], often known as the MP method. The method is an extension of the t-curve procedure for obtaining surface areas (Section XVII-9) a plot of cubic centimeters STP adsorbed per gram v versus the value of t for the corresponding P/F (as given, for example, by Table XVII-4) should, according to Eq. XVII-84, give a straight line of slope proportional to the specific surface area E. As illustrated in Fig. XVII-30, such plots may bend over. This is now interpreted not as a deviation from the characteristic isotherm principle but rather as an indication that progressive reduction in surface area is occurring as micropores fill. The proposal of Mikhail et al. was that the slope at each point gave a correct surface area for the P/P and v value. The drop in surface area between successive points then gives the volume of micropores that filled at the average P/P of the two points, and the average t value, the size of the pores that filled. In this way a pore size distribution can be obtained. Figure XVII-31a shows adsorption isotherms obtained for an adsorbent consisting of a-FeOOH dispersed on carbon fibers, and Fig. XVII-31b, the corresponding distribution of micropore diameters [231]. [c.670]

If a sufficiently large number of iterations have been performed, the ensemble average of any given property should not change signihcantly with additional iterations. However, there will be fluctuations in any given property computable as a root-mean-square deviation from the ensemble average. These fluctuations can be related to thermodynamic derivatives. For example, fluctuations in energy can be used to compute a heat capacity for the fluid. Alter- [c.304]

The calculation of average energies and their deviations from the mean are useful in several aspects of molecular dynamics simulations, such as these [c.85]

Molecules are in continuous random motion, and as a result of this, small volume elements within the liquid continuously experience compression or rarefaction such that the local density deviates from the macroscopic average value. If we represent by 6p the difference in density between one such domain and the average, then it is apparent that, averaged over all such fluctuations, 6p = 0 Equal contributions of positive and negative 6 s occur. However, if we consider the average value of 6p, this quantity has a nonzero value. Of these domains of density fluctuation, the following statements can be made [c.679]

The first quantitative model, which appeared in 1971, also accounted for possible charge-transfer complex formation (45). Deviation from the terminal model for bulk polymerization was shown to be due to antepenultimate effects (46). Mote recent work with numerical computation and C-nmr spectroscopy data on SAN sequence distributions indicates that the penultimate model is the most appropriate for bulk SAN copolymerization (47,48). A kinetic model for azeotropic SAN copolymerization in toluene has been developed that successfully predicts conversion, rate, and average molecular weight for conversions up to 50% (49). [c.193]

In the absence of drying stress (ie, with small specimens and extremely slow drying), the degree of shrinkage from the green to ovendry condition is, as a first approximation, proportional to the specific gravity of the wood. The value of the slope of the linear relationship is equal to the average fiber saturation point of the wood. Serious deviations from the linear relationship may occur with species high in extractives. [c.323]

The value of n — 1 is used in the denominator because the deviations from the sample average must total zero, or [c.488]

Equation (16-39) gives ni = 1.3333 mol/kg and = 1.0000 mol/kg for an average deviation from the exact values of approximately 10 percent. Equation (16-40) gives nj = 1.4413 mol/kg and n = 0.8990 mol/kg for an average deviation of about 0.6 percent. [c.1508]

The mixing of miscible fluids is commonly practiced. This mixing can be for blending components having different physical properties or can be for ensuring the homogeneity of a single fluid. For the blending or mixing of miscible fluids a criteria describing the desired approach to homogeneity must be set. Examples of such criteria are maximum allowable deviation of a fluid property (i.e., concentration, conductivity, color, temperature, etc.) at a point from the average value at that point, maximum allowable deviation of a fluid property between two or more points, etc. In addition to homogeneity criteria, blending operations can be specified on the basis of an empirical batch mixing time, ratio of mixer discharge flow to stream flow rate (or for batch systems, tank volume divided by batch time). [c.466]

T>blc J5.5 Dimensions of some sulfur molecules. Average values are given except for S7 where deviations from the mean are more substantial (see text) [c.656]

Clearly, the assertion that the deviations of the observed binding energies from the calculated average binding energies reflect whether a ligand is particularly well or poorly fitted for binding is just a working hypothesis. Nevertheless, it has to be observed that even 20 years after publication of this method, average binding energies calculated by the Andrews scheme are stiU used for screening potential drug candidates. [c.327]

The use of an averaged dynamics structure for comparison with the X-ray geometry has the advantage that wc obtain an impression on how the protein is most likely to look over the simulation time. In contrast, comparing selected snapshots extracted from the trajectory may reflect conformationally extreme situations caused by local fluctuations or distortions, which arc not at all representative. In our test case (Figure 7-16), wc can sec clearly that the simulation reproduces the geometry of the X-ray starting structure veiy well, confirming the applicability of the protocol used. As already stated above, the average MD geometry can now serve as a geometrical reference for subsequent simulations with modifications, c.g., in the amino acid sequence or inducer structure. A closer look at Figure 7-16 reveals another interesting possibility of geometrically analyzing the two structures by calculating the RMS differences of the protein backbone for every residue pair, one should in principle be able to localize regions of low and high geometrical deviation. This information gives important hints about areas of different stability within the protein, which leads us to the final step our analysis. [c.371]

An average structure calculated from an MD trajectory shows us a representative geometry around which a protein is fluctuating during the course of the simulation. This structure may differ from the ciystallographic starting point, for example. but these deviations do not say anything about the intensity of the fluctuations and the mobility of the involved atoms or residues. Figure 7-17 shows a residue- [c.371]

In deciding the convergence of these averages, the RMS deviation of a value from its average (i.e, Dxi may be a very useful indicator. [c.317]

Values of the flux ratio n /N21 were also compared with the theoretical value (Mj/Mj ), and the average deviation of the experimental points from this value was found to be + 6.64 over the whole pressure range. [c.97]

This damping funetion s time seale parameter x is assumed to eharaeterize the average time between eollisions and thus should be inversely proportional to the eollision frequeney. Its magnitude is also related to the effeetiveness with whieh eollisions eause the dipole funetion to deviate from its unhindered rotational motion (i.e., related to the eollision strength). In effeet, the exponential damping eauses the time eorrelation funetion <( )j Eg [c.432]

Understanding how the force field was originally parameterized will aid in knowing how to create new parameters consistent with that force field. The original parameterization of a force field is, in essence, a massive curve fit of many parameters from different compounds in order to obtain the lowest standard deviation between computed and experimental results for the entire set of molecules. In some simple cases, this is done by using the average of the values from the experimental results. More often, this is a very complex iterative process. [c.240]

In deciding the convergence of these averages, the RMS deviation of a value from its average (i.e., Dx) may be a very useful indicator. [c.317]

In addition to being able to plot simple instantaneous values of a quantity x along a trajectory and reporting the average,

The t test can be applied to differences between pairs of observations. Perhaps only a single pair can be performed at one time, or possibly one wishes to compare two methods using samples of differing analytical content. It is still necessary that the two methods possess the same inherent standard deviation. An average difference d calculated, and individual deviations from d are used to evaluate the variance of the differences. [c.199]

A statistical measure of the average deviation of data from the data s mean value (s). [c.56]

To calculate the standard deviation for the analyte s concentration, we must determine the values for y and E(x - x). The former is just the average signal for the standards used to construct the calibration curve. From the data in Table 5.1, we easily calculate that y is 30.385. Calculating E(x - x) looks formidable, but we can simplify the calculation by recognizing that this sum of squares term is simply the numerator in a standard deviation equation thus, [c.123]

Single-operator characteristics are determined by analyzing a sample whose concentration of analyte is known to the analyst. The second step in verifying a method is the blind analysis of standard samples where the analyte s concentration remains unknown to the analyst. The standard sample is analyzed several times, and the average concentration of the analyte is determined. This value should be within three, and preferably two standard deviations (as determined from the single-operator characteristics) of the analyte s known concentration. [c.683]

Several derivatives of this ring system have been synthesized.These compounds exhibit properties indicating that the conjugated system is aromatic. They exhibit NMR shifts characteristic of a diamagnetic ring current. Typical aromatic substitution reactions can be carried out. An X-ray crystal structure (R = C2H5) shows that the bond lengths are in the aromatic range (1.39-1.40 A), and there is no strong alternation around the ring. The peripheral atoms are not precisely planar, but the maximum deviation from the average plane is only 0.23 A. The dimethyl derivative is essentially planar, with bond lengths between 1.38 and 1.40A. [c.520]

In real systems the microscopic configurations of the monolayers can differ from the average configuration. The surfactant density averaged over these configurations is a smooth function, non-vanishing in a region of width comparable to the standard deviation from the average position of the monolayer. In the model considered here within the MF approximation the surfactant-occupied interface is very diffuse close to the second-order transition. For example, in the lamellar phase with modulations of a period 27r/kb in the i direction, and for the surfactant surface located at 2 = 0, the density of oriented surfactant is given by 5 ecos(27T2/kb) close to the bifurcation (see (59), (71), and (65)). Hence it deviates from zero in an extended region. The projected surface corresponds to the average location of the interface between oil-rich and water-rich domains and in our case is given by 0(r) = 0. We expect that the average surface area of the mono-layer per unit volume S/V is larger than the projected surface area per unijt volume S, and that < a [c.729]

I h e calcti lation of average energies and their deviations from the mean are useful in several aspects of molectilar dynam ics simulation s, such as tb ese [c.85]

The astute reader will have noticed our use of the term reference bond length (sometimes called the natural bond length ) for the parameter Ig. This parameter is commonly called the equilibrium bond length, but to do so can be misleading. The reference bond length is the value that the bond adopts when all other terms in the force field are set to zero. The equilibrium bond length, by contrast, is the value that is adopted in a minimum energy structure, when all other terms in the force field contribute. The complex interplay between the various components in the force field means that the bond may well deviate slightly from its reference value in order to compensate for other contributions to the energy. It is also important to recognise that real molecules undergo vibrational motion (even at absolute zero, there is a zero-point energy due to vibrational motion). A true bond-stretching potential is not harmonic but has a shape similar to that in Figure 4.4, which means that the average length of the bond in a vibrating molecule will deviate from the equilibrium value for the hypothetical motionless state. The effects are usually small, but they are significant if one wishes to predict bond lengths to thousandths of an angstrom. When comparing the results of calculations with experimental data, one must also remember that different experimental techniques measure different equilibrium values, especially when the experiments are performed at different temperatures. The errors in experimentally determined bond lengths can be quite large for example, libration of a molecule in a crystal means that the bond len hs determined by X-ray methods at room temperature may have errors as large as 0.015 A. MM2 was parametrised to fit the values obtained by electron diffraction, which give the mean distances between atoms averaged over the vibrational motion at room temperature. [c.189]

At least one of the preparer giUs wiU incorporate autoleveling. After the thickness of the feed sUvers is sensed, adjustments to the speed of the dehvery toUets ate automaticaUy made to correct deviations from the desired average. The aim is to achieve as high a degree of uniformity as possible in the shvets at the entry to the comb. [c.346]

The PIMC data obtained for the imaginary-time correlations are shown in Fig. 8 for different densities at F = 1. The relative errors of the data are of the order 10 , which is necessary for the maximum entropy method to work when there is little previous knowledge, as in the present case. At low densities the average particle distances are large and, since the particle interaction is restricted to a square well region (d < r < 1.5i/, see Eq. (25)), the probability for particle interactions is small. Thus the particles occupy mainly eigenstates, resulting in a small correlation of the cF spins and a small value of G /3/2). In the limit of zero density the dynamics is given purely by the tunneling of the spins with frequency loq which can be described by the zeroth-order term in the virial expansion [296] (see Eq. (29)), which is shown in Fig. 8 for comparison. At higher densities the probabihty for interaction increases and the particles hybridize by leaving their ground states and occupying more and more eigenstates, and thus the value of G /3/2) increases. This effect finally even leads to a continuous phase transition [297-299] from a paramagnetic to a ferromagnetic phase at about p 0.53. In mean field approximation [296] the critical density is at Pc,UP and from Eq. (30) we see that for all P < P, MF the resulting correlation functions agree with the lowest-order virial expansion result, since the mean field value for m is zero. Thus the MF correlation functions increasingly deviate from the PIMC data with increasing density. Only for p > p mf is there reasonably good agreement. [c.103]

The first term in (44) is related to the deviation of the surfactant concentration at point r from the average concentraction, p. The second part is related to the orientational ordering and defines the vector field u. [c.722]

In the analysis of crystal growth, one is mainly interested in macroscopic features like crystal morphology and growth rate. Therefore, the time scale in question is rather slower than the time scale of phonon frequencies, and the deviation of atomic positions from the average crystalline lattice position can be neglected. A lattice model gives a sufiicient description for the crystal shapes and growth [3,34,35]. [c.858]

One potential problem associated with column coupling in reversed phase is relatively high back-pressure ( 2600 psi at 1 mL miir ). This will place a limit on the flow rate, which in turn limits the further reduction of analysis time. Also, compared to the new polar organic mode, the retention in reversed phase on coupled columns is deviated more from the average retention on the individual stationary phases. [c.40]

In the absence of anisotropy introduced by specific surfactant-surfactant interactions, a spherical droplet model is reasonable because it tends to minimize the surface energy. Deviations from spherical symmetry occur because of the finite size and anisotropy of surfactant molecules and the anisotropy of interactions. Many early experimental data were interjDreted on the assumption of spherical stmctures. In seminal Monte Carlo studies by Haan and Pratt [35], micelles simulating those of sodium octanoate were examined. They found that the chains adopted a spheroidal stmcture that was never close to perfectly spherical. An example packing configuration of the type observed is illustrated in figure C2.3.6 for the case of an assembly involving 30 monomers. The shaded headgroups are mostly situated at the micellar surface, but it is obvious that much of the surface is also composed of methylene and methyl groups. This stmcture also obviously departs significantly from spherical symmetry. Spherical packing just is not energetically feasible when surfactant tailgroups must fill space. The situation changes dramatically when another solvent is pennitted to fill the core region, as in microemulsions, and the surfactants can then pack in a more or less planar manner at the oil-water interface. Similar conclusions have been upheld by much more time-consuming molecular dynamics simulations, such as those of Jonsoon et al [36]. A molecular dynamics snapshot of a sodium octanoate micelle is illustrated in figure C2.3.7. This stmcture also shows that the micelle at a given instant is far from spherically symmetric. Of course, this stmcture is undergoing shape fluctuations as part of its dynamical equilibrium and it is constantly rotating in space. Such fluctuations and rotations tend to give an apparent spherical stmcture when averaged over time. This is why many stmctural studies based on neutron scattering. [c.2589]

They compared the PME method with equivalent simulations based on a 9 A residue-based cutoflF and found that for PME the averaged RMS deviations of the nonhydrogen atoms from the X-ray structure were considerably smaller than in the non-PME case. Also, the atomic fluctuations calculated from the PME dynamics simulation were in close agreement with those derived from the crystallographic temperature factors. In the case of DNA, which is highly charged, the application of PME electrostatics leads to more stable dynamics trajectories with geometries closer to experimental data [30]. A theoretical and numerical comparison of various particle mesh routines has been published by Desemo and Holm [31]. [c.369]

In addition to bein g able to plot sim pie in stan tan eous values of a quantity x along a trajectory and reporting the average,

See pages that mention the term

**Deviations from the Average**:

**[c.194] [c.359] [c.722] [c.639] [c.1821] [c.2497] [c.372] [c.137] [c.358] [c.397]**

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