Marginally stable solution

If 5v //v /coex is not small, the simple description Eq. (14) in terms of bulk and surface terms no longer holds. But one can find AF from Eq. (5) by looking for a marginally stable non-uniform spherically symmetric solution v /(p) which leads to an extremum of Eq. (5) and satisfies the boundary condition v /(p oo) = v(/ . Near the spinodal curve i = v /sp = Vcoex /a/3 (at this stability limit of the metastable states both and S(0) diverge) one finds "... 

In the plane E E" this is a circular attractor, but there is no motion along it. Each point is a stationary solution, marginally stable in the tangential direction. 

Mullins and Sekerka (88, 89) analyzed the stability of a planar solidification interface to small disturbances by a rigorous solution of the equations for species and heat transport in melt and crystal and the constraint of equilibrium thermodynamics at the interface. For two-dimensional solidification samples in a constant-temperature gradient, the results predict the onset of a sinusoidal interfacial instability with a wavelength (X) corresponding to the disturbance that is just marginally stable as either G is decreased... 

As pointed out above in regard to the hyperpolarizability effect, the lability of a biopattern is closely related to its stability. Consider, for example, the lability of the Fucus egg polarity in its early development. In the absence of an applied gradient, the equations describing the dynamics of the egg are invarient under a rotation of coordinates about the center o the egg. This implies that if we have a solution of tt e form Y(r,t) then fo any ota ion operation R taking a vector r to a new vector r = Rr, r = Jr, then Y(r, t) is also a solution (state) of the Fucus system. This implies that the state Y(r,t) is only marginally stable as follows. Assume Y obeys the equation... 

Thus the condition for finding marginally stable modes that correspond to a spatial pattern is that (90,92) have nontrivial solutions. This happens if the determinant of the infinite dimensional matrix fi formed by writing (92) in the general form... 

There are only a few marginally stable Npvn and Puvn compounds, which formally contain the dioxo unit An02+ in alkaline solution. 

Uncomplexed 2 is marginally stable in aqueous solution, but the disproportionation equilibrium can be upset by any ligand for which the Hg11 compound is more stable. Thus addition of... 

The modem era starts with the seminal paper by Suzuki [2] with the systematic observations on the stability of ovalbumin, the main protein component of egg white, and hemoglobin. These proteins show an elliptic stability phase diagram in the temperature-pressure plane. Nowadays it is well known that proteins in solution are marginally stable under conditions of high temperature and pressure. By contrast there is the observation that certain bacteria live under extreme temperature conditions and it is well known that bacteria can survive in the deepest parts of the ocean. 

We finally mention that the stability analysis of front solutions mentioned before (van Saarloos, 2003 Ebert and van Saarloos, 2000) reveals that the selected velocity (c or v for the pulled or the pushed situation respectively) arising from well localized initial conditions is the marginally stable one in the sense that it is the smallest velocity... 

As with hexacenes, the i-Pr group (TIPS) was too small to stabilize the heptacene. Even the t-Bu group (TTBS) resulted in a heptacene derivative that was described as being only marginally stable. Heptacene 40b decomposed within a day in a rigorously oxygen-free solution, and even faster when the solution was... 

Figure 4.5 Illustration of the positions of the five stable solution regimes ideal, dilute, semi-dilute, marginal, and concentrated, and the phase separated region, hatched, for polymer solutions. The dependencies of the various crossovers on N, p, and v and the dependence of the correlation lengths on the concentration cp are indicated. (G. J. Fleer, M. A. Cohen Stuart, J. M. H. M. Soheutjens, T. Cosgrove, and B. Vincent, Polymers at Interfaces, Chapman and Hall, London, 1993).

This chapter provides a brief overview of analytical methods commonly used to assess the structure, purity, safety, stability, and potency of recombinant protein pharmaceuticals. These methods are technically challenging to develop because the protein products are large in size, complex in structure and function, and marginally stable in aqueous solution. The basic biochemical, biophysical, and cell biology principles behind the methods are described in the chapter. 

The objective of this note is to make this last statement mathematically more precise and in the course of the analysis delineate and solve problems which do not appear in the classical treatment. In trying to examine the morphological stability of planar fronts one must first find a special planar solution and show it is marginally stable among all planar solutions before one can use it as the base of a perturbation expansion. In section 2 we examine the existence, uniqueness and asymptotic behavior of... 

Driven nonlinear systems often tend to develop spatially periodic patterns. The underlying mathematical models usually permit a continuous set of linearly stable solutions. As a possible mechanism of selecting a specific pattern the principle of marginal stability is presented, being applicable to situations, where a propagating front leaves a periodic structure behind. We restrict our discussion to patterns on interfaces which are more easily accessible than three-dimensional structures, for example in hydrodynamic flow. As a concrete system a recently analyzed model for dendritic solidification is discussed. 

Experiment 2. At the threshold value k = 1.5 to the transition to another type of solution, namely that of solutions with sustained cycles, there occurs a critical reduction of the damping effect. Although the origin is still the only singular point and is still - marginally - stable, the oscillatory relaxation takes much more time than in the previous case. 

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