Convex potentials

The last term on the right hand side of the equation is diagonal in the time slices. For stable systems (convex potentials) such as the harmonic oscillator, U X) = k X — Xeq) ), it is negative. The first three terms form a tri-... 

A complex structural system, such as frame structures representing buildings, bridges or mechanical systems, can be assembled from components which are formulated as reciprocal structures. Reciprocal structures are those structures characterized by convex potential and dissipation functions (Stern, 1965). In this section, the concept of reciprocal structures is explained using simple spring-mass-damper-slider models shown in Figure 1. Mixed Lagrangian and Dissipation functions of such systems are derived for various structural components. 

A mathematical condition for core softening was proposed by Debenedetti etal. [86] and requires that A [r/(r)j < Ofor Ar < 0, in some interval n < r < rz, together with u" r) > 0 for r < ri and r > rz- The above conditions are satisfied if, in the Interval (ri, r2), the product r/(r) (rather than just f(r)) reduces with decreasing interparticle separation. This requirement can be met also by a strictly convex potential, yielding a repulsive force that everywhere increases for decreasing r, provided that in a range of interparticle distances, the increasing rate of f(r) be sufficiently small. Debenedetti condition is satisfied by the YK potential whenever a < 2.3. 

The stability of a stationer state against any perturbations could be determined from dissipation potentials, because for a strictly convex potential, the availability of global... 

Proof of the equivalence the local and global minimum of a strictly convex potential... 

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

The dependence of the interfacial tension on the potential is termed the electrocapillary curve. It is convex to the axis of potential and often reminiscent of a parabola (see Fig. 4.2). 

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)...

The tools for calculating the equilibrium point of a chemical reaction arise from the definition of the chemical potential. If temperature and pressure are fixed, the equilibrium point of a reaction is the point at which the Gibbs free energy function G is at its minimum (Fig. 3.1). As with any convex-upward function, finding the minimum G is a matter of determining the point at which its derivative vanishes. 

The term f rin p/p° is clearly the chemical potential of a surface of radius r with respect to a flat surface of the same material as standard state. It follows that the difference in chemical potential between two surfaces, p1 — p11, where surface I is convex of radius n, and the other surface II is concave of radius r2 is given by... 

In general, the first derivative of the Gibbs energy is sufficient to determine the conditions of equilibrium. To examine the stability of a chemical equilibrium, such as the one described above, higher order derivatives of G are needed. We will see in the following that the Gibbs energy versus the potential variable must be upwards convex for a stable equilibrium. Unstable equilibria, on the other hand, are... 

The use of the electron density and the number of electrons as a set of independent variables, in contrast to the canonical set, namely, the external potential and the number of electrons, is based on a series of papers by Nalewajski [21,22]. A.C. realized that this choice is problematic because one cannot change the number of electrons while the electron density remains constant. After several attempts, he found that the energy per particle possesses the convexity properties that are required by the Legendre transformations. When the Legendre transform was performed on the energy per particle, the shape function immediately appeared as the conjugate variable to the external potential, so that the electron density was split into two pieces that can be varied independent the number of electrons and their distribution in space. 

Kahler structures are easy to construct and flexible. For example, any complex submanifold of a Kahler manifold is again Kahler, and a Kahler metric is locally given by a Kahler potential, i.e. uj = / ddu for a strictly pseudo convex function u. However, hyper-Kahler structures are neither easy to construct nor flexible (even locally). A hypercomplex submanifold of a hyper-Kahler manifold must be totally geodesic, and there is no good notion of hyper-Kahler potential. The following quotient construction, which was introduced by Hitchin et al.[39] as an analogue of Marsden-Weinstein quotients for symplectic manifolds, is one of the most powerful tool for constructing new hyper-Kahler manifolds. 

Linear regions (constant slope, hence constant potentials) define the composition interval over which a two-phase assemblage is stable. Because the minimum Gibbs free energy curve of the system is never convex, the chemical potential of any component will always increase with the increase of its molar proportion in the system. 

The pairwise nature of the bond-boost makes this task easier since such traps would show up as a non-convexity of some of the biased effective pair potentials, which in the canonical ensemble can be taken to be the pairwise potential of mean force (PMF, denoted as V). Thus, assuming that V is approximately quadratic for lei safety condition can be enforced by setting Sa[Pg.92]

Field saturation. Consider a particle occupying a convex open domain CR3 (or ft2) with a smooth boundary du>, charged to the electric potential > 0, at equilibrium with an infinite solution of a symmetric electrolyte of a given average concentration. (Properties described below are directly generalizable to an arbitrary electrolyte or electrolyte mixture.)... 

Deep within the crystal, fiy = 0 and fiA = fi°A, and therefore < AL = n°A. The diffusion potential at the convex region of the surface is greater than that at the concave region, and atoms therefore diffuse to smooth the surface as indicated in Fig. 3.7. 

A qualitative and quantitative HPLC method for analysis of mixtures of 12 antioxidants was described Grosset et al. (121). For the identification of the components present, gradient elution with a convex profile from 35 65 water-methanol to pure methanol is used, on a Waters 5-/xm C18 column, with UV detector. Propyl gallate was not separated by this system. For quantitative analysis, with UV and electrochemical detectors in series, the water-methanol mixture or pure methanol was used as the eluent, under isocratic conditions, with lithium perchlorate as supporting electrolyte. An applied potential ranging from +0.8 to +1.7 V allows detection of all the antioxidants tested. Both modes of detection were very sensitive, with limits of detection as low as 61 pg. 

The packed bed breakthrough method for investigation of mass transfer phenomena in sorbent systems can in many instances offer certain advantages not found in other experimental methods. The method is especially useful when the adsorption isotherms for the principal sorbate exhibit favorable curvature (convex toward loading axis). In such a case, there is the potential for a portion of the sorption front to approach a stable wave form (shape of the front invariant with time). Given the existence of a stable or "steady-state" mass transfer zone (MTZ) and a detailed knowledge of the equilibrium loading characteristics within that zone, one can extract local values of the effective mass transfer resistance at any concentration in the zone. 

It should be noted, however, that we need to check whether Tkh(x) < 0 is quasi-convex at each iteration. If the quasi-convexity condition is not satisfied, then the obtained lower bound by OA/ER may not be a valid one that is, it may be above the global solution of the MINLP model. This may happen due to the potential invalid linearizations which may cut off part of the feasible region. 

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