Curvature positive

Within the elastic regime, the conservation relations for shock profiles can be directly applied to the loading pulse, and for most solids, positive curvature to the stress volume will lead to the increase in shock speed required to propagate a shock. The resulting stress-volume relations determined for elastic solids can be used to determine higher-order elastic constants. The division between the elastic and elastic-plastic regimes is ideally marked by the Hugoniot elastic limit of the solid. 

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. ... 

A saddle point in the density with 2 negative and 1 positive curvatures, called... 

A local minimum in the density with 3 positive curvatures, called a pit. 

These necessary conditions for local optimality can be strengthened to sufficient conditions by making the inequality in (3-87) strict (i.e., positive curvature in all directions). Equivalently, the sufficient (necessary) curvature conditions can be stated as follows V /(x ) has all positive (nonnegative) eigenvalues and is therefore defined as a positive (semidefinite) definite matrix. 

Second-Order Conditions As with unconstrained optimization, nonnegative (positive) curvature is necessary (sufficient) in all the allowable (i.e., constrained) nonzero directions p. The necessary second-order conditions can be stated as... 

Redlight pretreatment decreases the phototropic sensitivity for the first positive curvature by a factor of 10. Con and Briggs37) demonstrated by action spectroscopy that this decrease is mediated by phytochrome. It appears that there is an interaction... 

Zimmerman and Briggs explain their dosage response curves on the basis of three independent pigment systems. However, for several reasons it appears more reasonable to ascribe their complicated patterns to different secondary rather than to distinct primary processes. First, the first and second positive curvatures show essentially the same action spectra (Fig. 3 4 and 5). Second, the Bunsen-Roscoe law holds only for the first 100 s of irradiation. After that time factors other than photochemical ones clearly govern phototropism. Third, the dosage response curves are not real kinetics, i.e. they do not represent continuous traces of bending in time, as the authors assume for their calculations. However, curvature was allowed to develop for 100 min in darkness, measured and plotted as a function of dosage. 

Fig. 3. Comparison of different blue-light action spectra for directional responses (2) positive phototopotaxis of Euglena4 (2) Phototropism of Philobolus121, (3) phototropism of Phycomyces40 (4) second positive curvature of the avena coleoptile58), (5) first positive curvature of the avena coleotpile39)...

Nevertheless, the avena coleoptile exhibits a curvature to unilateral UV-illumina-tion with a satisfactory log-linear response/time relationship38) (the bending mode is similar to that observed for the second positive curvature which develops from the coleoptile base cf. 2.2). Fig. 5 338) shows that the double-peaked action spectrum does not match neither flavin (Fig. 5 5,16S)) nor carotenoid absorption (Fig. 5 4,183)), most likely excluding both as photoreceptors. The growth hormone auxin (cf. 2.4 and Scheme 1) has been discussed to be a possible photoreceptor. However, in this case, this is not supported by the action spectrum either. 

Unfortunately, no data are available comparing phototropic quantum efficiency in the far UV and the visible region. This is mainly due to the qualitatively different responses in both regions (tip or base curvature 1st and 2nd positive curvature). 

Fig. 5. Absorption and action spectra in the UV-region. Absorption spectra of (2) auxin178), (4) carotene183), (5) flavin165). Action spectra of (1) the negative phototropism of Phycomyces l, (S) second positive curvature of the avena coleotile38)...

If only the extreme apex of the oat coleoptile is irradiated unilaterally, curvature nevertheless develops normally, migrating down to the light shielded region (first positive curvature). This observation led to the demonstration by Boysen-Jensen in 191018), showing that the principle causing curvature could be transmitted across... 

Fig. 13. Electrical and curvature responses of avena coleoptiles to unilateral irradiation of two minutes. (A) intensity chosen to produce positive curvature (B) intensity chosen to produce negative curvature. Clearly the convex side of the coleoptile is electrically positive, regardless of the type of curvature. This indicates a strong correlation of bending and electrical potential gradient3)...

Here r, 9, 4> are dimensionless co-moving coordinates attached to fundamental observers and R(t) a scale factor with a dimension of length depending only on cosmic time t. k is the curvature constant, which with suitable choice of units takes one of the three values +1 (closed world model with positive curvature), 0 (flat, open model) or —1 (open model with negative curvature). Some consequences of Eq. (4.7) are the relation between redshift and scale factor Eq. (4.2) and the variation of temperature... 

The penalty term of an augmented Lagrangian method is designed to add positive curvature so that the Hessian of the augmented function is positive-definite. 

Figure 13.14 The PPC traces of the thermal expansion coefficient a (deg-1) as a function of temperature for chymotrypsinogen and RNase A. The data show that both native (low-temperature region) proteins exhibit a strong negative temperature coefficient as well as a large positive curvature. (Permission to use the figure granted by MicroCal, LLC.)...

For positive cooperative systems, all the curves with S > 2 start with positive curvature [Eq. (4.3.6)] and then, atx = x the curvature becomes negative, where x, is defined as the point for which the slope is maximal, i.e.. 

Pales two curvatures are positive, and, at rc, p is a minimum in the plane defined by the axes corresponding to the positive curvatures p is a maximum at rc along the third axis which is perpendicular to this plane. The (3, + 1) critical points are found at the center of a ring of bonded atoms. 

Figure 3-2 Concentration curves for (a) positive curvature and (b) negative curvature. Diffusion will increase the concentration along the curve for case (a), and will decrease the concentration for case (b).

Surfactants having a positive curvature, above a given concentration usually called the critical micellar concentration, cmc, self-assemble to form oil-in-water aggregates called normal micelles. The surfactant most often used is sodium dodecyl sulfate, Na(DS) or SDS. To make particles, the counterion of the surfactant is replaced by ions which participate in the chemical reaction. These are called functionalized surfactants. 

One can describe the state of affairs without reference to the fourth dimension as follows. In the case of the point spectrum the geometry of Riemann (constant positive curvature) reigns in momentum space, while in the case of the continuous spectrum the geometry of Lobatschewski (constant negative curvature) applies. 

---