# Solution existence

When the solute exists in only one form in each phase, then the partition coefficient and the distribution ratio are identical. If, however, the solute exists in more than one form in either phase, then Kd and D usually have different values. For example, if the solute exists in two forms in the aqueous phase, A and B, only one of which, A, partitions itself between the two phases, then [c.216]

Now we can prove the statement of solution existence to minimization problems. [c.30]

Further, in Section 3.1.4, an optimal control problem is analysed. The external forces u serve as a control. The solution existence of the optimal control problem with a cost functional describing the crack opening is proved. Finally, in Section 3.1.5, we prove C°°-regularity of the solution near crack points having a zero opening. [c.173]

Consider an approximate description of the nonpenetration condition between the crack faces which can be obtained by putting c = 0 in (3.45). Similar to the case c > 0, we can analyse the equilibrium problem of the plates and prove the solution existence of the optimal control problem of the plates with the same cost functional. We aim at the convergence proof of solutions of the optimal control problem as —> 0. In this subsection we assume that T, is a segment of a straight line parallel to the axis x. [c.194]

Observe that variational inequality (3.106) is valid for every function X G 82- It means that a solution % to problem (3.106) with 9 G Si coincides with the unique solution to problem (3.100) with the same 9] i.e. problems (3.100) and (3.106) are equivalent. For small 5, we write down an extra variational inequality for which a solution exists, and demonstrate that the solution coincides with the solution of variational inequality (3.98). [c.204]

We do not show the dependence of the solution to (5.147)-(5.151) on the parameters to simplify the notation. Our aim is first to prove the solution existence of the problem (5.147)-(5.151) and second to pass to the limit as —y 0, <5 —y 0. [c.323]

So the necessary estimates are obtained, and we can use the Galerkin method to prove the solvability of the parabolic boundary value problem (5.185)-(5.188) (see Lions, 1969). This proves that the solution exists in the following sense. [c.334]

The solution of the system may then be found by elimination or matrix methods if a solution exists (see Matrix Algebra and Matrix Computations ). [c.432]

The difficulties of the osmium determination are connected with its different degrees of oxidation in solutions, existence in various chemical forms, which can pass one into another. The analytical problem can be successfully solved in many cases by using of selective organic reagents. Among these reagents leading role belongs to the sulfur containing ones. [c.120]

When the solute exists in only one form in each phase, then the partition coefficient and the distribution ratio are identical. If, however, the solute exists in more than one form in either phase, then and D usually have different values. For example, if the solute exists in two forms in the aqueous phase, A and B, only one of which, A, partitions itself between the two phases, then [c.216]

After the addition of each plate volume of charge, a new concentration of solute exists in plate (1), and its contents will be eluted through the column in the normal manner. [c.198]

Solution Existing plant New plant [c.302]

Jl exists in this form only in solution, though stable derivatives of the aldehyde structure are known. The optical antipode of D-glucose in which the positions of every H and OH are transposed is L-glucose. [c.191]

Finally, we can mention the existence of additives acting specifically on the cloud point. These are polymers containing chemical groups resembling paraffins in order to associate with the paraffins and solubilizing functions to keep the associations in solution. The gains are more modest than those described above being on the order of 2 to 4°C for concentrations between 250 and 1000 ppm. These are, however, appreciable effects for the refiner, considering the difficulty encountered in meeting the cloud point specification. [c.217]

An oil reservoir which exists at initial conditions with an overlying gas cap must by definition be at the bubble point pressure at the interface between the gas and the oil, the gas-oil-contact (GOC). Gas existing in an initial gas cap is called free gas, while the gas in solution in the oil is called dissolved or solution gas. [c.104]

As solution gas is liberated, the oil shrinks. A particularly important relationship exists between the volume of oil at a given pressure and temperature and the volume of the oil at stock tank conditions. This is the oil formation volume factor (B, measured in rb/stb or rm /stm ). [c.110]

In a saturated oil reservoir containing an initial gas cap, the producing gas oil ratio (Rp) may be significantly higher than the solution gas oil ratio (Rg) of the oil, as free gas in the gas cap is produced through the wells via a coning or cusping mechanism. Free gas is the gas existing in the gas cap as a separate phase, as distinct from solution gas which is dissolved in the oil phase. [c.112]

The penetration of microwaves in various materials gives active microwave imaging a large potential for subsurface radar, civil engineering etc. Several inverse-scattering theories have been proposed in the scientific literature. Among them, the simplest Bom-type approach, which does not take into account multiple reflections, is valid for weakly scattering objects [1-2]. To improve the quality of reconstruction, the method based on the successive application of the perturbative algorithm was developed [3]. However, the inherent approximations of this approach are not overcome in the iterative scheme. Another class of algorithms aims to obtain the spatial distribution of permittivity by using numerical solutions of exact equations [4-6]. Unfortunately, a rate of convergence of the solution to the global minimum of cost function depends on actual contrast values, measurement error etc. That is why an importance of a priori knowledge about the object imder investigation is usually emphasized. In general, existing inversion algorithms suffer from serious problems when discontinuous profiles of high contrast, which are often encountered in practical applications, are to be reconstructed. Moreover, the frequency-swept imaging methods utilize usually reflection coefficient data measured in a very broad frequency band starting from zero frequency [1-2, 4-5]. Such methods are inappropriate from an application point of view. [c.127]

Here, each elementary contribution is refracted at the interface between the coupling medium and the piece. Instead of developing an exact treatment of the refraction at the piece boundary (decomposition of each elementary contribution into its two-dimensional spectrum of plane waves) which leads to a time consuming solution, an approximate solution using the geometrical optics approximation (denoted GO) is proposed. The GO approximation is nothing but an asymptotic solution of the exact solution around the path of stationary phase existing between the source and field points (Fermat s principle). This approximation, initially applied to treat the refraction into a solid isotropic medium has been recently extended to anisotropic materials [9] and is being implemented numerically. [c.736]

An interesting experimental technique is heat development of nuclei. The liquid is held at the desired temperature for a prescribed time, while nuclei accumulate they are then made visible as crystallites by quickly warming the solution to a temperature just below Tq, where no new nuclei form but existing ones grow rapidly. [c.337]

This results iu four equations and four unknowns. Siuee the equations are homogeneous, a nontrivial solution exists only if die detenuiuaut fonued by the eoeflfieieuts of A, B, C and D vanishes. The solution to this equation is [c.103]

Khludnev A.M. (1998) Regularization and solution existence in the equilibrium problem for elastoplastic plate. Siberian Math. J. 39 (3), 670-682. [c.379]

FlowInsta.hihty, In many flow situation it is found that a mathematically vaHd solution to the Navier-Stokes equations is closely verified by experiment over some ranges of the variables, but when the variables are changed new flow patterns that are not in keeping with that solution are observed. The change is often rather sudden, eg, the laminar—turbulent transition. Such behavior occurs because solutions to the Navier-Stokes equations are generally not unique. When more than one solution exists, it is possible to observe one flow pattern under one set of circumstances and a different pattern under another. At the present stage of development of fluid mechanics, an experiment must be performed to determine whether a given solution appHes. [c.98]

Various rational approaches to design do exist. We can exercise what really amounts to a brute-force approach, which is not a particularly good approach, but there is a way of making that approach almost rational. That is, we can try all the possible solutions for certain problems. We can sometimes define all the solutions, and then pick the best solution. Then, we know we have the optimum design because we have examined all possible designs. We will study an instance in which that brute-force approach is possible and even quite useful. Obviously, other instances exist in which we could not even imagine knowing all the solutions. For example, if we want to design a 150-passenger airplane to fly from New York to Los Angeles, then we must realize that an infinite number of solutions exists, and we cannot possibly try them all. But if we want a laminate that will carry a certain load, there is a finite number of solutions to that problem (if we ignore obviously inappropriate laminates such as all laminates thicker than necessary), and we actually can try them all. More rational optimization approaches involve more refined mathematical procedures, such as Monte Carlo techniques, dynamic programming, or nonlinear programming. Some very complicated issues arise with using those techniques for structures., [c.428]

Various rational approaches to design do exist. We can exercise what really amounts to a brute-force approach, which is not a particularly good approach, but there is a way of making that approach almost rational. That is, we can try all the possible solutions for certain problems. We can sometimes define all the solutions, and then pick the best solution. Then, we know we have the optimum design because we have examined all possible designs. We will study an instance in which that brute-force approach is possible and even quite useful. Obviously, other instances exist in which we could not even imagine knowing all the solutions. For example, if we want to design a 150-passenger airplane to fly from New York to Los Angeles, then we must realize that an infinite number of solutions exists, and we cannot possibly try them all. But if we want a laminate that will carry a certain load, there is a finite number of solutions to that problem (if we ignore obviously inappropriate laminates such as all laminates thicker than necessary), and we actually can try them all. More rational optimization approaches involve more refined mathematical procedures, such as Monte Carlo techniques, dynamic programming, or nonlinear programming. Some very complicated issues arise with using those techniques for structures., [c.428]

It has been discovered recently that the spectrum of solutions for growth in a channel is much richer than had previously been supposed. Parity-broken solutions were found [110] and studied numerically in detail [94,111]. A similar solution exists also in an unrestricted space which was called doublon for obvious reasons [94]. It consists of two fingers with a liquid channel along the axis of symmetry between them. It has a parabolic envelope with radius pt and in the center a liquid channel of thickness h. The Peclet number, P = vp /2D, depends on A according to the Ivantsov relation (82). The analytical solution of the selection problem for doublons [112] shows that this solution exists for isotropic systems (e = 0) even at arbitrary small undercooling A and obeys the following selection conditions [c.893]

The high resistance of gold to attack by a very wide range of corrosive media results from its very high nobility. It may be seen from Fig. 6.2 that gold is immune to attack over the whole range of pH values at redox potentials below about 0-4 V, and that the zone of immunity extends to higher potentials at the lower end of the pH range. Gold, however, is easily com-plexed, and its solubility in hydrochloric acid containing an oxidising agent (e.g. nitric acid) results from a combination of high redox potential and the formation of chlor-auro complex ions, AuCl4. The unstable Au ion and the easily reducible Au ion also readily form stable complexes. The Au form is not stable in aqueous solutions, existing only in the solid, insoluble sulphide form. Standard electrode potentials of gold are given in Table 6.7. [c.929]

By saccharic acid is usually meant D-gluco-saccharic acid, m.p. 125-126°C, obtained by the oxidation of glucose or starch. This exists in water solution in equilibrium with its two y lactones, both of which can be obtained crystalline, though the acid itself does not crystallize readily. [c.350]

S02NH) . Colourless crystalline solids formed by the action of ammonia on a solution of sulphuryl chloride in benzene free sulphimide exists only in the polymerized form. [c.376]

From this expectation curve, //there are hydrocarbons present (30% probability), then the low medium and high estimates of reserves are 20,48 and 100 MMstb. The NPV for the prospect for the low, medium and high reserves can be determined by estimating engineering costs and production forecasts for three cases. This should not be performed simply by scaling, but by tailoring an engineering solution to each case assuming that we would know the size of reserves before developing the field. For example, the low case reserves may be developed as a satellite development tied into existing facilities, whereas the high case reserves might be more economic to develop using a dedicated drilling and production facility. [c.328]

First, we want to offer G335SF in combination with STRUCTURIX SILVERFIX as a viable solution to customers who want to upgrade their existing equipment (installed base processors) in order to decrease the silver content of their rinsing water. [c.609]

See pages that mention the term

**Solution existence**:

**[c.268] [c.531] [c.34] [c.175] [c.71] [c.174] [c.322] [c.360] [c.433] [c.433] [c.25] [c.115] [c.291] [c.293] [c.434] [c.250] [c.170]**

See chapters in:

** Analysis of cracks in solids
-> Solution existence
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** Analysis of cracks in solids
-> Solution existence
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** Analysis of cracks in solids
-> Solution existence
**