Relations similarity

Similar relations hold for the higher distributions. The Gaussian property of the process implies that all statistical infonnation is contained in just and 1 2. 

Tensile Strength and Elongation. The tensile strength of latex mbber foam has been shown to depend on the density of the foam (149,177) and on the tensile strength of the parent mbber (177,178). At low densities the tensile modulus approximates a linear relation with density but kicreases with a higher power of density at higher densities. Similar relations hold for polyurethane and other flexible foams (156,179,180). 

A fuU listing of aU U.S. patents issued between Febmary 1970 through Febmary 1981 is given in Reference 26. Similar related material on membranes, ultrafiltration, and reverse osmosis can be found in References 46—49. 

With its sextet of 7T electrons, thiophene possesses the typical aromatic character of benzene and other similarly related heterocycles. Decreasing orders of aromaticity have been suggested to reflect the strength of this aromatic character benzene > thiophene > pyrrole > furan (9) and benzene > thiophene > selenophene > teUurophene > fuian (10). 

Similarity Variables The physical meaning of the term similarity relates to internal similitude, or self-similitude. Thus, similar solutions in boundaiy-layer flow over a horizontal flat plate are those for which the horizontal component of velocity u has the property that two velocity profiles located at different coordinates x differ only by a scale factor. The mathematical interpretation of the term similarity is a transformation of variables carried out so that a reduction in the number of independent variables is achieved. There are essentially two methods for finding similarity variables, separation of variables (not the classical concept) and the use of continuous transformation groups. The basic theoiy is available in Ames (see the references). 

Similar relations for other non-Newtouiau fluids may be found in Govier and Aziz and in Bird, Armstrong, and Hassager (Dynamics of Polymeric Liquids, vol. 1 Fluid Mechanics, Wiley, New York, 1977). 

Inelastic Constitutive Relations 169 spatial terms by using (A.26i), (A.46), and a similar relation for k... 

Codeine corresponding to 8-methylmorphimethine is neopine that similarly related to the S-methine is still unknown. 

However, though N2O can be made in this way it is not to be regarded as the anhydride of hyponitrous acid since H2N2O2 is not formed when N2O is dissolved in H2O (a similar relation exists between CO and formic acid). 

A happy consequence of nuclear systematics is that element 104 is in Group 4, element 105 is in Group 5, etc. This mnemonic holds to the end of the transition series at element 112 and presumably beyond it can be compared with the similar relation between the group numbers of the post-transition main-group elements (13-18) and the group numbers of the preceding transition elements (3-8). 

Consider now two solute species Bi and B3, between which no direct experimental comparison by the colorimetric method is possible, because their useful concentration ranges just fail to overlap. We can find an intermediate indicator solute B2, whose useful range partly overlaps that of B, and partly overlaps that of B3. Using B2, a relation between Bi and B3 may thus be established indirectly. In dilute solution this relation will be a simple one. We do not know enough about concentrated solutions to be in a position to say whether a similar relation should be expected. In the experiments to be described, the first aim was to obtain an answer to this question. 

However, Szabo-Rethy293) showed that Flory s assumption leads to wrong values of rate constants and that the error can be as high as 15 to 35%. This author gave a method of calculation which takes into account the elimination of water. Similar relations have been used recently by Lin and Hsieh270) but in our opinion erroneously. 

Exercise By analogy with the above, obtain the cut-set matrix and show that a similar relation holds between the circuit matrix and the cut-set matrix. Note that the cut-sets that correspond to the rows of this matrix are ... 

Similar relations are obtained for Z2X and XZ2 from Eq. (7-20c). When they are substituted in Eq. (7-21), it is found that... 

For mass transfer to a surface, a similar relation to equation 12.117 can be derived for equimolecular counterdiffusion except that the Prandtl number is replaced by the Schmidt number. It follows that ... 

Thus the ratio of the actual increase in humidity produced in the saturator to the maximum possible increase in humidity (that is, the production of saturated gas) is equal to (1 — e 0flWG/), and complete saturation of the gas is reached exponentially. A similar relation exists for the change in the temperature of the gas stream ... 

From these we obtain similar relations in the F s. The following special cases are needed in evaluating the matrix elements in Eqs. (6), (7) and (8) ... 

For dimensional reasons a similar relation should also hold when a hard solid sphere slides on a soft mbber surface. 

A similarity-related approach is k-nearest neighbor (KNN) analysis, based on the premise that similar compounds have similar properties. Compounds are distributed in multidimensional space according to their values of a number of selected properties the toxicity of a compound of interest is then taken as the mean of the toxicides of a number (k) of nearest neighbors. Cronin et al. [65] used KNN to model the toxicity of 91 heterogeneous organic chemicals to the alga Chlorella vulgaris, but found it no better than MLR. 

For instance, if we would like to measure a field with an accuracy of 1 mgal, that is, dg/g = 10 , it is necessary to know the period 1 s with an accuracy dT — 0.5 x 10 s. In this light, it is useful to obtain similar relation for the ballistic gravimeter. In accordance with the solution of the equation of motion we have... 

C° is related to C byjlie effective equilibrium curve, and Cw is similarly related toC. C is related to C by the operating line. 

The addition of similar relations for the x- and -components of these angular momentum veetors leads to the result that [J, L S] = 0, so that J and L S commute. Furthermore, we may easily show that commutes with L S because eaeh term m = I + S +2 u S commutes with L S. Thus, J and commute with H in equation (7.33) and J and are eonstants of motion. 

A similar relation applies to the inverse of die product example, define the product of three matrices by ABC = W W exists, it is given by W l — (ABC)-1. Now consider tin... 

Similar relations between different scaling exponents were also developed by Stauffer [37] by combining two of the scaling relations at a time to eliminate Ip - Pel-... 

Relation [1] Is the frequency-dependent analogue of a formula proposed by Chasset and Thirion (2, 3) which has since been applied very frequently to relaxation measurements on cured rubbers. The next three equations are Inspired by similar relations In dielectrics (they are not derived from these) Equation [2] by the Cole-Cole and Equation [3] by the Davidson-Cole relation (15, 16). Both are special cases of the most general Equation [4] which contains five parameters (17). 

However, its was found possible to infer all four microscopic tensor coefficients from macroscopic crystalline values and this impossibility could be related to the molecular unit anisotropy. It can be shown that the molecular unit anisotropy imposes structural relations between coefficients of macroscopic nonlinearities, in addition to the usual relations resulting from crystal symmetry. Such additional relations appear for crystal point group 2,ra and 3. For the monoclinic point group 2, this relation has been tested in the case of MAP crystals, and excellent agreement has been found, triten taking into account crystal structure data (24), and nonlinear optical measurements on single crystal (19). This approach has been extended to the electrooptic tensor (4) and should lead to similar relations, trtten the electrooptic effect is primarily of electronic origin. 

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