Distance squared, defined

The distribution of distance h is characterized either by the most probable value /imax, defined by the condition dw(/i)/d/ = 0, or by the mean square of distance h, defined in the usual manner, h2 = J/i2w(/i) dh/ w(h) dh. The ratio (hmax)2/h2 gives the width of the distribution. The closer this value is to unity, the narrower the distribution. This ratio is equal to for the simplest model in statistics, the random walk . 

Balaban Index The Balaban index, J, has been introduced as the average-distance sum defined as the sum of the squares of the reciprocal nonzero distances in D (6,7] ... 

The constant k must be defined in such a way that the quantities on the two sides of the equation are expressed in the same units. Such dimensional analysis demands that k have the dimensions of a force times a distance squared. In terms of mass (M), length (L) and time (T), k has dimensions of ML3T-2. 

Another matter is the consistency of the molecular quantum similarity matrix Z. The MQSM produced by a specific alignment technique for a given molecular pair of the set of molecules that construct Z should not be contradictory with the computed MQSM for the other pairs of molecules. To illustrate this point, consider the Euclidean distance, as defined by the square root of Eq. [13] ... 

Some of the chemicals and placard ID numbers listed in the guidebook are highlighted in color. For the highlighted chemicals, the user is directed to another section listing initial isolation and protective action distances in miles and kilometers (or feet and meters). The initial isolation distance is defined by a circle, fhe center of which is at the spill site. The protective action distance is defined by a square fhaf has fhe cenfer of one end locafed af fhe spill site and the other side downwind. The square assumes that random changes in wind direction confine the vapor plume to an area within 30° on either side of the predominant wind direction, resulting in a crosswind protective action distance equal to the downwind protective action distance. This is shown in Fig. 17.1. 

Each rotational isomeric state for a chain molecule can be assigned a particular value of R. The mean-squared end-to-end distance is defined as ... 

For clustering purposes, a quantitative measure of similarity between the d-dimensional points forming the data set must be defined. Similarity can be measured by Euclidean distance, squared Euclidean distance, correlation coef-... 

The object we have thus created is usually addressed as the Brownian chain, and it is characterized by just one parameter, namely the mean squared end-to-end distance Although defined as a mathematical limiting structure, the Brownian chain may indeed serve as a representative of real polymer chains in a certain well-defined region, namely for length scales that are larger than the persistence length and smaller than the size Rq of the chain. The Brownian chain correctly represents the fractal properties and differs fi om the real chain in that it extends these fractal properties down to zero distances. 

Dietzel proposed classifying the oxides according to their ability to form or modify the glass network in view of their cation field (Dietzel, 1941, 1942, 1943, 1981, 1983). He defined the cation field intensity A as a charge-to-distance squared ratio ... 

By 1939 Pauling was able to present a table of covalent radii for the hydrogen atom and the sixteen elements in the square defined by the positions of C, Sn, F, and I in the periodic table [37]. These radii were based on the bond distances in the crystalline elements of the Group 14 metals the Group 15 elements P, As, and Sb and the Group 16 elements S, Se, and Te, all determined by X-ray diffraction the bond distances in H2 and the dihalogen molecules determined by spectroscopic methods and finally by the bond distances in the gaseous methyl derivatives of all the seventeen elements except Sb, Se, and Te determined by gas electron diffraction. 

In order to determine the matrix thresholds, we present an expression of the coefficients dispersion that is related to the flattening of the cloud of the points around the central axis of inertia. The aim is to measure the distance to the G barycentre in block 3. So, we define this measure Square of Mean Distance to the center of Gravity as follow ... 

Make the following approximate calculations for the surface energy per square centimeter of solid krypton (nearest-neighbor distance 3.97 A), and compare your results with those of Table VII-1. (a) Make the calculations for (100), (110), and (111) planes, considering only nearest-neighbor interactions, (b) Make the calculation for (100) planes, considering all interactions within a radius defined by the sum... 

One of the most important fiinctions in the application of light scattering is the ability to estimate the object dimensions. As we have discussed earlier for dilute solutions containing large molecules, equation (B 1.9.38) can be used to calculate tire radius of gyration , R, which is defined as the mean square distance from the centre of gravity [12]. The combined use of equation (B 1.9.3 8) equation (B 1.9.39) and equation (B 1.9.40) (tlie Zimm plot) will yield infonnation on R, A2 and molecular weight. 

Fig. 2. Distance classes j = 0,1, 2,... (left) are defined for an atom (central dot) by a set of radii Rj+i the right cnrves sketch the temporal evolntion of the tot il force acting on the selected atom originating from cill atoms in distance class j shown are the exact forces (solid line), their exact valnes to be computed within the multiple time step scheme (filled squares), linear force extrapolations (dotted lines), and resulting force estimates (open sqnares).

The dynamic viscosity, or coefficient of viscosity, 77 of a Newtonian fluid is defined as the force per unit area necessary to maintain a unit velocity gradient at right angles to the direction of flow between two parallel planes a unit distance apart. The SI unit is pascal-second or newton-second per meter squared [N s m ]. The c.g.s. unit of viscosity is the poise [P] 1 cP = 1 mN s m . The dynamic viscosity decreases with the temperature approximately according to the equation log rj = A + BIT. Values of A and B for a large number of liquids are given by Barrer, Trans. Faraday Soc. 39 48 (1943). 

A similarity measure is required for quantitative comparison of one strucmre with another, and as such it must be defined before the analysis can commence. Structural similarity is often measured by a root-mean-square distance (RMSD) between two conformations. In Cartesian coordinates the RMS distance dy between confonnation i and conformation j of a given molecule is defined as the minimum of the functional... 

The foregoing discussion of equivalent chains requires merely that its root-mean-square end-to-end distance shall equal that of the real chain. In order to define completely the equivalent chain, its contour lengths may also be required to coincide with that of the real chain. 

This is a relationship between unknown field g and two measured quantities, namely, the distance 5 and time t, provided that we neglect terms proportional to the square of the coefficient a and those of higher order. Besides, this equation contains three unknown parameters, namely, the position of the mass. so at the moment when we start to measure time, the initial velocity, vo, at this moment and, finally, the rate of change of the gravitational field, a, along the vertical. Thus, in order to solve our problem and find the field we have to perform measurements of the distance. s at four instants. If so is known, the number of these measurements is reduced by one. In modern devices the coefficient of the last term on the right hand side of Equation (3.14) has a value around 100 pGal and it is defined by calculations as a correction factor s(vo, g, t, a). In the case when we can let so equal to zero, it is sufficient to make measurements at two instances only. 

In Euclidean space we define squared distance from the origin of a point x by means of the scalar product of x with itself ... 

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