Stability Poisson

One can see that if Xq is P (P )-stable, its trajectory is P (P )-stable too. Hence, we may generalize the notion of the Poisson stability over semitrajectories and whole trajectories. 

Modulus of elasticity (GN/m ) Poisson s Yield stress Ductile to , II-, Stress UTS brittle transition orys a isa ion relieving Hardness (VPN) Stability ... 

It is not uncommon for protons to be taken up or released upon formation of a biomolecular complex. Experimental data on such processes can be compared to computational results based on, for example, Poisson-Boltzmann calculations.25 There is a need for methods that automatically probe for the correct protonation state in free energy calculations. This problem is complicated by the fact that proteins adapt to and stabilize whatever protonation state is assigned to them during the course of a molecular dynamics simulation.19 When the change in protonation state is known, equations are available to account for the addition or removal of protons from the solvent in the overall calculation of the free energy change.11... 

One common characteristic of many advanced scientific techniques, as indicated in Table 2, is that they are applied at the measurement frontier, where the net signal (S) is comparable to the residual background or blank (B) effect. The problem is compounded because (a) one or a few measurements are generally relied upon to estimate the blank—especially when samples are costly or difficult to obtain, and (b) the uncertainty associated with the observed blank is assumed normal and random and calculated either from counting statistics or replication with just a few degrees of freedom. (The disastrous consequences which may follow such naive faith in the stability of the blank are nowhere better illustrated than in trace chemical analysis, where S B is often the rule [10].) For radioactivity (or mass spectrometric) counting techniques it can be shown that the smallest detectable non-Poisson random error component is approximately 6, where ... 

A maximum flexural stress of 9,500 psi is assumed for polycarbonate. This conservative stress value should account for degradation in ultraviolet stabilized polycarbonate exposed to long term solar exposure. While more research is required in this area, it is reasonable to expect at least a ten year useful life for ultraviolet stabilized polycarbonate. A Young s modulus of 345,000 psi and a Poisson s ratio of 0.38 are also assumed for polycarbonate. 

These workers also calculated the relative stability of the tautomers lOa-c in the gas phase by ab initio and density functional theory (DFT) methods and in solution using several continuum solvation models such as self-consistent reaction fields (SCRF) and the Poisson-Boltzmann method. These results showed good agreement between the experimental and theoretical approaches. 

When they subjected the allenylzinc reagent to the Hoffmann test for configurational stability,29 Poisson, Chemla and Normant found that at — 50 °C, racemization does not occur at a significant rate (equation 36)30,31. Accordingly, when the racemic allenylzinc reagent was added slowly to the /V-benzy limine of (R)-mandehc aldehyde at — 50 °C, a 1 1 mixture of the anti,syn and anti,anti adducts was isolated in 65% yield. However, when the addition process was reversed, a 3 1 mixture favoring the matched anti,anti adduct was formed in 53% yield, suggestive of a partial kinetic resolution. 

For studying the stability of colloidal particles in suspension (Chapter 13) or for determining the potential at the surface of particles (Chapter 12), one often needs expressions for potential distributions around small particles that have curved surfaces. Solving the Poisson-Boltzmann equation for curved geometries is not a simple matter, and one often needs elaborate numerical methods. The linearized Poisson-Boltzmann equation (i.e., the Poisson-Boltzmann equation in the Debye-Hiickel approximation) can, however, be solved for spherical electrical double layers relatively easily (see Section 12.3a), and one obtains, in place of Equation (37),... 

The question to be discussed is whether saturation of the electric field (asserted by Proposition 2.1) implies saturation of the interparticle force of interaction. Consider for definiteness repulsion between two symmetrically charged particles in a symmetric electrolyte solution. In the onedimensional case (for parallel plates) the answer is known—the force of repulsion per unit area of the plates saturates. (This follows from a direct integration of the Poisson-Boltzmann equation carried out in numerous works, primarily in the colloid stability context, e.g., [9]. Recall that again in vacuum, dielectrics, or an ionic system with a linear screening, the appropriate force grows without bound with the charging of the particles.)... 

Note that Eq. (4.15) is simply the linearized equation of motion for the classical upside-down barrier (55/5 = 0) for the new coordinate x. Therefore, while x = 0 corresponds to the instanton, the nonzero solution to (4.15) describes how the trajectory escapes from the instanton solution, when deviated from it. The parameter A, referred to as the stability angle [Gutzwiller, 1967 Rajaraman, 1975], generalizes the harmonic oscillator phase a>,/3, which would stand in (4.16), if w, were constant. The fact that A is real is a reminder of the aforementioned instability of the instanton in two dimensions. Guessing that the determinant det(-dj + w2) is a function of A only, and using the Poisson summation formula, we are able to write... 

The micellar surface has a high charge density and the stability of the aggregate is heavily dependent on the binding of counterions to the surface. From the solution of the Poisson-Boltzmann equation one finds that a large fraction (0.4—0.7) of the counterions is in the nearest vicinity of the micellar surface300. These ions could be associated with the Stern layer, but it seems simpler not to make a distinction between the ions of the Stern layer and those more diffusely bound. They are all part of the counterions and their distribution is primarily determined by electrostatic effects. 

Poisson, J.-F. Chemla, F. Normant, J. F. Configurational stability of propargyl zinc reagents. Synlett 2001, 305-307. 

The Poisson-Boltzmann approach [1,2] has the advantage of simplicity and is surprisingly accurate, at least for univalent ions in a certain range of electrolyte concentrations (1.0X10 3-5X10 2 M) and not too close to the interface. It was later employed [10] to explain the repulsion due to the overlap of two double layers, and the stability of colloids thereafter [11,12],... 

Whereas the corrections to the traditional Poisson— Boltzmann approach could explain many experimental results, there are systems, such as the vesicles formed by neutral lipid bilayers in water, for which an additional force is required to explain their stability.4 This force was related to the organization of water in the vicinity of hydrophilic surfaces therefore it was called hydration force .5... 

Another clear failure of the Poisson-Boltzmann approach was provided by the experiments regarding the force between neutral lipid bi layers [11], The repulsion required to explain their stability was determined to have an almost exponential dependence, with a decay length of about 2—3 A [11], and neither this decay length nor the magnitude of the interactions were dependent on the ionic strength. This interaction was initially attributed to the structuring of water near the surface (the hydration of the surfaces) and it is usually called hydration force [12]. The microscopic origins of this interaction are still under debate. 

In this chapter, mathematical procedures for the estimation of the electrical interactions between particles covered by an ion-penetrable membrane immersed in a general electrolyte solution is introduced. The treatment is similar to that for rigid particles, except that fixed charges are distributed over a finite volume in space, rather than over a rigid surface. This introduces some complexities. Several approximate methods for the resolution of the Poisson-Boltzmann equation are discussed. The basic thermodynamic properties of an electrical double layer, including Helmholtz free energy, amount of ion adsorption, and entropy are then estimated on the basis of the results obtained, followed by the evaluation of the critical coagulation concentration of counterions and the stability ratio of the system under consideration. 

A related approach integrating sequence information (conservation), geometric information (cleft detection), and data on local stability calculated by Poisson-Boltz-mann methods was reported by Ota et al. [54], The method was used for predicting catalytic residues (polar atoms only) in enzymes. A number of putative active sites for a series of hypothetical proteins were found and are discussed in the study. 

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