Negative curvature

FIOUBE 16.B Schematic representation of the contact area between two partically sintered spheres (a) center-to-center distance is constant, (b) decreasing center-to-center distance. The sphere radius is a, x is the radius of the neck, 2h is the decrease in the center-to-center distance, and k is the radius of curvature (negative) for the neck. 

Methods have been devised that alter the potential energy hyperspace, which have been useful in locating the global minimum. Second-derivative information, discu.ssed above, indicates the curvature of the energy surface, which may be flattened or inflated, depending on whether the surface has a positive curvature (negative second derivative) or negative curvature (positive second derivative), respectively. - ... 

Deviations from Beer s law may be due to instrumental factors or to chemical factors. These deviations may result in an upward curvature (positive deviation) or in a downward curvature (negative deviation), as shown in Figure 7.5. A check on instrumental factors can be made by plotting absorbance versus cell length at a constant concentration this plot will be linear if the instrument is performing satisfactorily. Deviations arising from chemical factors are observed only when concentrations are changed. 

Swelling effect. Oil molecules are solubilized in the aggregate s core and expand its volume. In this case, as is almost constant. Since the volume of the surfactant s hydrophobic tail is increased, the surfactant mean curvature tends to be more positive (i.e., more convex toward water) in order to maintain Us constant. Thus, for the hydrophilic Ci2(EO)7 (whose curvature tends to be positive), the swelling effect will be dominant, leading to the Hi-Ii transition. However, when an aromatic oil like w-xylene is used instead of decane, it penetrates into the surfactant palisade layer and makes the curvature negative, leading to the Hi-L, transition [296]. 

Inflection point = position at which the curvature or slope changes sign = positive curvature = negative curvature. 

The localization of the oil, which is followed by a variation in the liquid crystal symmetry and periodicity, is directly related to the influence of the oil on the temperature of the zero spontaneous curvature of the water-oil-C,Ey system. If the oil tends to penetrate into the chain surfactant layer and make the curvature negative, the surfactant has to be more hydrophilic (large polar head) to maintain the zero spontaneous curvature. This explains why in this case the temperature of the zero spontaneous curvature is low. 

In field ionization (or field desorption), application of a large electric potential to a surface of high curvature allows a very intense electric field to be generated. Such positive or negative fields lead to electrons being stripped from or added to molecules lying on the surface. The positive or negative molecular ions so produced are mass measured by the mass spectrometer. 

The negative electron and light hole masses Hsted for HgTe are a consequence of its being a semimetal rather than semiconductor. The curvatures of these two bands are inverted with respect to the convention defined for semiconductors. 

The rise times of the elastic wave may be quite narrow in elastic single crystals, but in polycrystalline solids the times can be significant due to heterogeneities in physical and chemical composition and residual stresses. In materials such as fused quartz, negative curvature of the stress-volume relation can lead to dispersive waves with slowly rising profiles. 

The line = 0 can be considered as a borderline for applicability of the basic model, in which the Gaussian curvature is always negative. Recall that in the basic model the oil-water interface is saturated by the surfactant molecules by construction of the model. Hence, for equal oil and water volume fractions the Gaussian curvature must be negative, by the definition of the model. 

Positive curvature in a plot of k against 6, (at constant pH) indicates the presence of a rate term with an order higher than unity with respect to Negative curvature can be caused by complex formation between reactants. ... 

If we imagine a line drawn on the primitive surface dividing all parts of the surface which are convex downwards in all directions from those which are concave downwards in one or both directions of principal curvature, this curve will have the equation (26), and is known as the spinodal carve. It divides the surface into two parts, which represent respectively states of stable and unstable equilibrium. For on one side A is positive, and on the other it is negative. If we assume that the tie-line of corresponding points on the connodal curve is ultimately tangent to that the direction of equations ... 

The second derivative of a function, denoted d2y/dx2, is defined like the first derivative but is applied to the function obtained by taking the first derivative. For example, the second derivative of the function x1 is the derivative of the function lx, which is the constant 2. Likewise, the second derivative of sin ax is —a2 sin ax. The second derivative is an indication of the curvature of the function. Where d2y/dx2 is positive, the graph has a (j shape where it is negative, the graph has a m shape. The greater the magnitude of d2y/dx2, the sharper the curvature of the graph. 

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