Dilative rotation

Combination of these two operations represents a continuous dilative rotation that transforms the general point (r, 9) into (a r, 6 +1), as t changes continuously. The locus of the transform of (a, 0) is the equiangular spiral... 

The distinctive property of logarithmic spirals is the constant dilatation for equal rotations. A dilative rotation of 2n transforms any point on the spiral into a homothetic point, which is similarly placed and directed. The spiral, said to be homothetic to itself, therefore has the property of self-similarity at all scales. We note that the origin (r, 0 = 0) transforms into the homothetic points e" " after n rotations. We propose that this property, described by the three fundamental constants e, n, and r, is related to the general curvature of space-time, which is responsible for the observed cosmic self-similarity e for growth, n for rotation, and X for dilatation. 

Ex. 2 The optical rotation of sucrose in presence of dil. HCl at various intervals is given in the following table ... 

Amorphous solid. Practically insol in water, dil acids, di] and coned alkalies, alcohol and other organic solvents sol in coned HCI, H2S04. 78-97% H3P04, anhydr HCOOH. There are substantial variations in solubility, molecular weight, acetyl valnes, specific rotation among chit ins of different origins and prepared by different methods. 

As can be seen in Figure 14-la. the visible absorption spectrum for 1.2,4.5-ictrazine vapor shows the line structure that is due lo the numerous rotational and vibrational levels associated wiih the excited electronic states of this aromatic molecule. In the gaseous state, the individual tetrazine molecules are suflicicntly separated from one another to vibrate and rotate freely, and the many individual absorption lines appear as a rcsuli of the large number ol vibrational and rotational energy states. In the condensed slate or in solution, however, the lelrazine molecules have little freedom to rotate, so lines due to dil ferencos in rotational cnergv... 

Fig. 5.132 Anisotropy decay of a Dil molecule in a PMMA matrix. The molecule has a limited rotational degree of freedom. A fit delivers a final anisotropy of 0.683+0.003, a cone of wobbling of 12.4 0.3°, and a rotational correlation time of 2.7T0.4 ns [419]...

We fix a local Cartesian coordinate system (e, e, e ) to every contour point of an elastic wire of finite thickness. In so doing, the C axis is taken in the direction of the unit tangent vector u of the chain centroid (or contour), and the and axes in the directions of the principal axes of inertia for the cross-section of the wire at the same point. When the wire is deformed (bent and twisted), the local coordinate system at contour point 5 4- is related to that at contour point s by an infinitesimal rotation dQ. The deformed state of the wire is then represented by a vector 0 (0, 0, 0 ) as a function of s, where 0 is defined by dil/ds. 

Preferential formation of 1,2-nitrosochlorides. A soln. of coned. HCl in isopropanol and a coned, aq. soln. of NaNO added simultaneously below 10° to a soln. of (+)-limonene in isopropanol, stirred for an additional 15 min., and allowed to stand 1 hr. in the refrigerator limonene nitrosochloride (Y 80.7% the amount of acid, 33-50 ml. per 0.1 mole olefin, is critical) refluxed 30 min. with dimethylformamide in isopropanol until the rotation is constant carv-oxime (Y 83.5%) melted in a separatory funnel by means of an infrared lamp, added dropwise to boiling aq. H2SO4 at pH 0.8, the product steam-distilled out of the reaction mixture as rapidly as it is formed and isolated in a continuous separator while the condensate water is returned to the still pot and dil. H2SO4 added portionwise to maintain the pH at 0.7-0.9 carvon (Y 83%). R. H. Reitsema, J. Org. Chem. 23, 2038 (1958). 

As illustrated in Fig. 13, the critical state (liquefied) shear strength ratio varies uniquely with state parameter for individual sands. However, for positive values of state parameter (i.e., contractive and contractive then dilative responses), liquefied shear strength ratios vary over similar ranges for many sands with varying gradations and particle shapes. Olson and Mattson (2008) collected a database of 386 laboratory triaxial compression, direct simple shear, rotational shear, and triaxial extension test results to examine yield and liquefied strength ratio concepts. 

Residual Strength of Liquefied Soils, Fig. 14 Summary of liquefied strength ratio data for (a) triaxial compression, (b) direct simple shear and rotational shear, and (c) triaxial extension tests. Type A = contractive response, Type B1 = contractive then dilative response where minimum shear resistance is maintained over range of shear strain greater than 3 %, Type... 

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