Convex Convexity

Ring roller presses differ from more conventional roll presses because of the convex-concave relationship of the ring and roller in the former case, compared with the convex-convex relationship of two rolls in contact in the latter case. This means that pressure build-up and release are both more gradual in the ring roller press. The slower pressure release is claimed to avoid crack formation in the agglomerated product as it expands more slowly after passing the nip of a ring roller press. 

A function f(x) is convex (convex from below) over an interval [x, X2] if and only if... 

SO that the system is stable. Equations (16.79) may also be shown to imply that at this point the surface g is convex downwards. It is said to be convex-convex since all sections made by planes normal to the surface at this point have their radii of curvature on the same side. ... 

Thus on the surface g(x2, x ), the spinodal is the boundary which separates that part of the surface which is convex-convex from that which is convex-concave. 

In fig. 16.19 the line aKb represents the spinodal curve whose presence indicates the existence of a fold iii the surface g. Any system whose representative point is situated on the convex-concave surface is unstable and breaks into two stable phases, each of which will be represented by a point on the convex-convex surface. It can be shown... 

This criterion resumes all the a priori knowledge that we are able to convey concerning the physical aspect of the flawed region. Unfortunately, neither the weak membrane model (U2 (f)) nor the Beta law Ui (f)) energies are convex functions. Consequently, we need to implement a global optimization technique to reach the solution. Simulated annealing (SA) cannot be used here because it leads to a prohibitive cost for calculations [9]. We have adopted a continuation method like the GNC [2]. 

Similarly, the identical expression holds for a liquid that completely fails to wet the capillary walls, where there will be an angle of contact between the liquid and the wall of 180°, a convex meniscus and a capillary depression of depth h. 

The method has been extended to mixtures of hard spheres, to hard convex molecules and to hard spherocylinders that model a nematic liquid crystal. For mixtures m. subscript) of hard convex molecules of the same shape but different sizes. Gibbons [38] has shown that the pressure is given by... 

Given the pair and surface potentials, the weights are then constructed by solving the convex bound constrained quadratic program... 

The simplest form of apparatus consists of a small porcelain evaporating dish covered with a filter paper which has been perforated with a number of small holes a watch glass of the same size, convex side uppermost, is placed on the filter paper. The substance is placed inside the dish, and the latter heated with a minute flame on a wire gauze or sand bath. The sublimate collects in the Fig. II, 45, 1. watch glass, and the filter paper below prevents the sublimate from falling into the residue. The watch glass may be kept cool by covering it with several pieces of damp filter... 

A convex hull is a molecular surface that is determined by running a planar probe over a molecule. This gives the smallest convex region containing the molecule. It also serves as the maximum volume a molecule can be expected to reach. 

Unique chemistry is associated with the cyclopentenone all five carbon atoms can be functionalized, and the endo-methyl groups of the acetonide assure clean stereoselective addition of the alkenylcopper reagent from the convex side. The use of the acetonide group to control enolate regioselectivity and to mask alcohols should be generally applicable. 

It is these kinds of uncertainties that have led to the development of mercury porosimetry, in which, since the meniscus is convex, the mercury has to be forced into the pores under pressure. Mercury porosimetry is the subject of Section 3.9. 

Frequently, however, the DR plot deviates from linearity, and in a number of ways. Sometimes the plot is convex to the log (p°/p) axis, as in Fig. 4.19(a), and sometimes concave, as in Fig. 4.19(b). The question then arises as to whether one should extropolate from the low-pressure. 

For a second active carbon, AG, the DR plot was convex to the logio(p7p) This carbon was believed from X-ray results to have a wider distribution of pores. It was found that the isotherms of both benzene and cyclohexane could be interpreted by postulating that the micropore system consisted of two sub-systems each with its own Wq and and with m = 2 ... 

Both Type III and Type V isotherms are characterized by convexity towards the relative pressure axis, commencing at the origin. In Ty )e III isotherms the convexity persists throughout their course (Fig. 5.1(a), whereas in Type V isotherms there is a point of inflection at fairly high relative pressure, often 0-5 or even higher, so that the isotherm bends over and reaches a plateau DE in the multilayer region of the isotherm (cf. Fig. 5.1 (b)) sometimes there is a final upward sweep near saturation pressure (see DE in Fig. 5.1(b)) attributable to adsorption in coarse mesopores and macropores. 

The weakness of the adsorbent-adsorbate forces will cause the uptake at low relative pressures to be small but once a molecule has become adsorbed, the adsorbate-adsorbate forces will promote the adsorption of further molecules—a cooperative process—so that the isotherms will become convex to the pressure axis. 

I (curve D). Thus the micropores had been able to enhance the adsorbent-adsorbate interaction sufficiently to replace monolayer-multilayer formation by micropore filling and thereby change the isotherm from being convex to being concave to the pressure axis. 

When Fj = 1/f2, the copolymer composition curve will be either convex or concave when viewed from the Fj axis, depending on whether Fj is greater or less than unity. The further removed from unity rj is, the farther the composition curve will be displaced from the 45° line. This situation is called ideal copolymerization. The example below explores the origin of this terminology. 

In this chapter, principal relations of solid mechanics, elements of convex analysis and calculus of variations, and methods of approximation are considered. 

Submitting the main topic, we deal with models of solids with cracks. These models of mechanics and geophysics describe the stationary and quasi-stationary deformation of elastic and inelastic solid bodies having cracks and cuts. The corresponding mathematical models are reduced to boundary value problems for domains with singular boundaries. We shall use, if it is possible, a variational formulation of the problems to apply methods of convex analysis. It is of importance to note the significance of restrictions stated a priori at the crack surfaces. We assume that nonpenetration conditions of inequality type at the crack surfaces are fulfilled, which improves the accuracy of these models for contact problems. We also include the modelling of problems with friction between the crack surfaces. 

The function is assumed to be convex and continuous. Thus, we have the static elastoplastic model... 

Let the functional J be convex and differentiable. We can prove the validity of the inequality... 

We have pointed out the following fact. If if c E is a convex set then the variational inequality... 

Theorem 1.1. The inequality (1.62) gives necessary and sujficient conditions of the minimum over the set K for a convex and differentiable functional J. 

Lemma 1.1. For every convex and differentiable functional J the function... 

The previous considerations can be specified. Namely, if J is a strictly convex functional then... 

Convex functionals have a convenient description in terms of their derivatives. We briefly discuss this question. 

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