Curvilinear coefficient

The same idea can be developed in the case of a non-Euclidean metric such as the city-block metric or L,-norm (Section 31.6.1). Here we find that the trajectories, traced out by the variable coefficient kj are curvilinear, rather than linear. Markers between equidistant values on the original scales of the columns of X are usually not equidistant on the corresponding curvilinear trajectories of the nonlinear biplot (Fig. 31.17b). Although the curvilinear trajectories intersect at the origin of space, the latter does not necessarily coincide with the centroid of the row-points of X. We briefly describe here the basic steps of the algorithm and we refer to the original work of Gower [53,54] for a formal proof. 

Electromagnetism can therefore be defined geometrically in curvilinear coordinates, and has vacuum properties such as scalar curvature, metric coefficient, affine connection, and Ricci tensor that manifest themselves fully on the 0(3) level ... 

Numerically, correlation coefficients can range from -1 to +1. A value of -1 indicates a perfect negative linear relationship as one variable increases, the other decreases in a precise linear fashion. A value of 0 indicates a complete lack of linear association between two variables. A value of +1 indicates a perfect positive linear relationship as one variable increases, the other increases in a precisely linear fashion. (The technique of correlation cannot meaningfully describe nonlinear patterns of association between two variables, such as curvilinear and exponential relationships.)... 

The mobility of the whole chain, free to move along the curvilinear axis of the tube is N/Nd smaller than the mobility of one bead, as the friction on the full necklace is the friction on one bead times the number of beads. The diffusion coefficient of the chain along the tube... 

In the curvilinear case where all hydrodynamic interactions between the two particles are considered, only those small particles in the shaded area denoted as Ac can come into contact with the larger particle. The area Ac can be determined numerically. The result is that the actual collision rate is less than the rectilinear rate, and the actual mass-transport coefficient is equal to t(Ac/Ar)PDS(ij)]. Han and Lawler (24) calculated reductions in the rectilinear transport rate by differential sedimentation ranging from about 0.3 to 0.001, so the effects of hydrodynamic interactions on this transport process can be substantial in many cases. 

Figure 9. Effect of particle size on physical aspects of particle collisions by fluid shear. The ratio of rectilinear to curvilinear transport coefficients is plotted as a function of the ratio of the size of the smaller particle (dj) to a larger one (dj). The circled A, B, and C refer to cases discussed in the text. (Reproduced with permission from reference 25. Copyright 1992.)...

At times longer than the Rouse time tr, all monomers move coherently with the chain. The chain diffuses along the tube, with a curvilinear diffusion coefficient given by the Rouse model Dg... 

This curvilinear motion continues up to the reptation time trep where the chain has curvilinearly diffused the complete length of the tube, of order aNjN. At times longer than the reptation time (/ > Trep) the mean-square displacement of a monomer is approximately the same as the centre of mass of the chain and is a simple diffusion with diffusion coefficient D [Eq. (9.12)]. 

The primary mode of motion of a linear chain along its confining tube is reptation, first proposed by de Gennes. Reptation is a snake-like diffusion of a chain, as a whole, along the contour of its tube, with a Rouse curvilinear diffusion coefficient. The relaxation time of the melt is the time it takes the chain to reptate out of its original tube, called the reptation time Trep. The reptation time and the viscosity of entangled polymers are... 

Consider a segment 5 of the tube of an entangled A-mer at time t = 0. Assume that the chain moves along its tube by simple diffusion with curvilinear diffusion coefficient D. ... 

Appendix D The Calculation of Drift and Diffusion Coefficients in Curvilinear Coordinates... 

APPENDIX D THE CALCULATION OF DRIFT AND DIFFUSION COEFFICIENTS IN CURVILINEAR COORDINATES... 

Film condensation on a horizontal tube. For a curvilinear surface, in particular, for a horizontal circular cylinder along which a condensate film flows, the angle 6 is a nonconstant variable. By taking into account the fact that 6(6) d, where d is the diameter of a circular cylinder, and proceeding by analogy with (5.7.7), one can readily obtain the following formula for the heat transfer coefficient averaged over the external surface of the tube provided that the flow in the condensate film is laminar [200] ... 

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