One-parameter family

Thus, under such an approach a one-parameter family of fully conservative schemes is given by... 

There are, of course, two integration constants, but one of them is merely a shift in t and determines the phase. The solution curves in the (0, 0)-plane are therefore a one-parameter family. A careful study reveals that for a certain range of values of a, / there is one closed curve and that it is a limit cycle. 

As expected, this reduces to Eq. (69) for systems with density distributions from our earlier one-parameter family (67), where p0 can be expressed as a function of p, according to Eq. (68). With both p0 and p, retained as moment densities in the moment free energy, a number of linear terms in Eq. (78) can be dropped, giving the final result... 

The methods differ in the formula used to generate the sequence S, k=0,l,2,., and after Fletcher and Powell s 3 analysis of Davidon s method a whole spate of formulae were invented in the sixties. Broyden 4 introduced some rationalization by identifying a one-parameter family, and recommended a particular member, now commonly referred to as the BFGS (Broyden-Fletcher-Goldfarb-Shanno) formula. Huang 5 widened the family, but by the end of the sixties numerical experience was producing a consensus that the BFGS formula was the most robust of the formulae available. The formula is... 

We can construct the representation of the sphere with this parameterization. One must, however, be careful when 4 = 0 (resp. ly = 0) since then the corresponding angle 9 (resp. 0, ) is not defined. The two angles 0 < Qx,y < specify a 2-torus. To fully foliate the sphere, two distances 4,/y, with a linear relation, may be specified. A one-parameter family of tori of varying radii foliates the 3-sphere. Each trajectory is fully specified by a point on these tori. 

Although there are a number of techniques that have been proposed to solve the surface matching problem, they all suffer from the fact that an inherent uncertainty is replaced by ad hoc procedures. Even if the molecular surface concept is replaced by a one parameter family of isosurfaces," this does not lead to a unique matching technique. 

As mentioned above, statistical methods for abundance determinations assume that the nebulae under study form a one parameter family. This is why they work reasonably well in giant H II regions. They are not expected to work in planetary nebulae, where the effective temperatures range between 20 000 K and 200000 K. Still, it has been shown empirically that there is an upper envelope in the [O ill] A5007/H/ vs. O/H relation (Richer 1993), probably corresponding to PNe with the hottest central stars. 

Using the Bessel approximation as a start-up artifice always gives us a two-parameter family of solutions (or a one-parameter family for each initial abscissa xq) as is always the case with a second-order differential equation. The parameter x+ is directly related to the curvature yo or y" " of a given profile. However, to singularize a profile that passes through a point (x°, y°) three parameters are necessary (x°, y°, q>°), although solutions may not exist for some combination of parametric values (for example, if x° = 0, no profiles with finite nonzero slope at x° exist). In all cases, once x" " and y" " have been determined, we may proceed... 

Its general solution is a one-parameter family of curves y = y(x, c), but in physical problems a particular solution is needed and this is picked out by the initial condition... 

Fig. 4.1. Energy as a function of shear deformation in A1 (adapted from Mehl and Boyer (1991)). The lattice parameter a is fixed during the deformation and hence the energy characterizes a one-parameter family of deformations of the fee lattice, with the members of the family parameterized by b.

Aicardi F., Valentin P., Ferrand E. (2002). On the classification of generic phenomena in one-parameter families of thermodynamic binary mixtures. Phys. Chem. Chem. Phys., 4, 884-895. 

These flows are sketched in Fig. 9 9. Another interesting subset of the general case, (9-192), is 2D linear flows. In this case, the complete set of possible motions can be represented as a one-parameter family of the form (9-192) with... 

Consider a one-parameter family of coordinate systems (A),. .., variational principle of the energy functional to the coordinate parameter A. We give here the results in the semiclassical limit, as presented in Eqs. (8)-(l 1), for the fixed coordinates... 

Raptis and Simos33 have constructed the following one-parameter family of four-step sixth algebraic order methods ... 

Periodic waves. The system of equation (153), (154) has a one-parameter family of solutions... 

The NHIM has a special structure due to the conservation of the center actions, it is filled, or foliated, by invariant d — l)-dimensionaI tori. More precisely, for d = 3 DoFs, each value of Jz implicitly defines a value of h by the energy equation Kcnf(0,/2,/3) = E. For three DoFs, the NHIM is thus foliated by a one-parameter family of invariant 2-tori. The end points of the parameterization interval correspond to Jz = 0 (implying qz = Pi = 0) and /s = 0 (implying q3=pz = 0), respectively. At the end points, the 2-tori thus degenerate to periodic orbits, the so-called Lyapunov periodic orbits. 

In the case of a one-parameter family of operations, SaSp = e.g., translation (a, p stand for the translation... 

Hopf, 1942) to prove the existence of a one-parameter family of 27T-periodic... 

If we assume that we observe only one phase boundary when Vq and are specified in the different phases, it is plausible to employ the one-parameter (denoted as ) family of solutions of Fig. 3.1 in the isothermal case and Fig. 3.2 in the nonisothermal case when the initial data are appropriate. In these figures all the intermediate states (vl,v, Vr, etc.) are constants. The abbreviations F.W., B.W., C.D., and P.B. stand for forward wave, backward wave, contact discontinuity, and phase boundary, respectively. The forward and backward waves consist of a shock or a rarefaction wave. In the isothermal case this one-parameter family of solutions was discussed first time in [13]. We take to be Vl - Vq in the isothermal case and Ul - Ug in the nonisothermal case. Then the decay of entropy (the entropy rate) is given by... 

Theorem 3.1. Assume that there exists one-parameter family of solutions as in Fig. 3.1 and Fig. 3.2. Then, we have the following conclusions. 

The above results agree with the classical results. Namely, (i) agrees with the Maxwell construction of steady state solutions with coexistent phases and (ii) agrees with the fact that for any transformation occurring in an isolated system the entropy of the final state can never be less than that of the initial state. Therefore, in order to support the feasibility of the assumption of existence of one-parameter family of solution (Fig. 3.1 and 3.2), it is desirable to have nontrivial solutions which minimize locally the entropy rate among the solutions in Fig. 3.1 and 3.2. For this purpose, considering that the entropy rate also depends on Ug, u, and etc., we express as follows... 

Assume also that x (a) is a critical point of Eq. (29). The eigenvalues Pi( ) Piia),..., pjv(a) will now also depend on the parameter a. If for some values of a, say a < Gq, the critical point is stable, and if in addition a pair of complex conjugate eigenvalues pi(a), p2(o) cross the imaginary axes transversely (d Repi(a)/da a=ao5 0) then we say that a Hopf bifurcation takes place at the value a = Gq. If a Hopf bifurcation occurs in Eq. (29) then there exists a one-parameter family of periodic solutions for a in the neighborhood of ao with a period near 27r/ Im pi(ao)l- K the flow attracts to the critical point x (ao) when a = Gq, then x°(ao) is called a vague attractor. For this case the family of closed orbits is contained m a >Go and the orbits are of attracting type. ... 

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