Symmetric solution

J. L. Ging, Countercurrent Flow in a Semi-Infinite Gas Centrifuge Axially Symmetric Solution in the Timit of High Angular Speed Re.-. . EP-4422-198-62S, Research Laboratories for the Engineering Sciences, University of Virginia, ChadottesviUe, 1962. 

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 "... 

The orthogonal secular equation corresponding to the symmetric solutions, therefore, takes the form... 

The first attempt to calculate realistic wave functions for electrons in metals is that of Wigner and Seitz (1933). These authors pointed out that space in a body-or face-centred cubic crystal could be divided into polyhedra surrounding each atom, that these polyhedra could be replaced without large error by spheres of radius r0, so that for the lowest state one has to find spherically symmetrical solutions of the Schrodinger equation (6) subject to the boundary condition that... 

Note that the solution given by Eq. 5.106 holds for x > 0 because the positive root of y/a-i was used. The symmetric solution for x < 0 is easily obtained by changing the sign of x. All the heat stored in the specimen at the time t is represented by the integral... 

The quasi-steady-state theory has been applied particularly where a condensed phase exists whose volume changes slowly with time. This is true, for example, in the sublimation of ice or the condensation of water vapor from air on liquid droplets (M3, M4). In the condensation of water vapor onto a spherical drop of radius R(t), the concentration of water vapor in the surrounding atmosphere may be approximated by the well-known spherically symmetric solution of the Laplace equation ... 

For one-component plasmas (OCP) of ionic charge Ze, one obtains, A2 = -TTni +Z r, where n is the electron density of plasma far from the ionic site. The spherically symmetric solution of Equation (7) yields... 

From the symmetry of the problem it is clear that there are azymuthally symmetric solutions (y = 0), implying that the nontrivial. (y 0) solution is giveJiDy 10... 

It should be noted that for non-isothermal case (and also for isothermal case with autocatalytic kinetics) the local equation may have multiple solutions. When this occurs, the averaged model obtained by the L-S method captures the complete set of solutions of the full CDR equations only within the region of convergence of the local equation. For example, for the wall-catalyzed non-isothermal reaction case, we have shown that the averaged two-mode model can capture only the three azimuthally symmetric solutions of the full CDR equation. The latter has three symmetric solutions (of which two are stable) as... 

In this way we have already obtained two approximate solutions for our H2+ ion. The variation of energy as a function of the nuclear distance is given for both of them in Fig. 13. We see that the symmetrical solution S (with the same signs) yields attraction with the formation of a stable molecule, the antisymmetrical A (with opposite signs) only gives repulsion 2. 

Nevertheless only the classical Coulomb law appears as the interaction law the special feature of the atomic bond thus does not depend on a special non-classical interaction but on the particular wave-mechanical behaviour of the electrons, so that in the symmetrical solution a piling up of negative charge can be produced at a place where the potential energy for the electrons is low. In the individual configuration H>H the repulsion of the two nuclei predominates, as can be seen from Fig. 13 K, over the attraction of the one proton for the cloud of negative charge around the other proton, at all distances. 

It is now possible to give the energy for the symmetrical solution and for the antisymmetrical after some numerical work. The symmetrical solution here also gives a potential curve with a minimum and thus corresponds with the formation of a stable molecule, while the antisymmetrical solution gives repulsion for all distances (Fig. 14). 

Figure 6. Single-channel currents of synthetic PHB96. Left Representative current fluctuations obtained when the given voltage was applied at a planar bilayer made from 1-palmitoyl, 2-oleoyl, phosphatidylcholine (16 0,18 1, PC) containing0.1 to 1% of 96mer of PHB between symmetric solutions of 60 mM RbCI, 5 mM MgCh, 10 mM Hepes CsOH, pH 7.2. The solid horizontal bar in each record indicates the current level with the channel closed. pO is the probability that the channel is in the open state. Right Corresponding conductivity histograms. N indicates the total number of observations that have been analyzed.

Figure 13. Current-voltage relations for synthetic PHBus/polyP ( ) and E. coli PHB/polyP channel complexes (A). The conductance of the channel for Ca2+ in symmetric solutions, under the experimental conditions described in Figure 12, is 101 6 pS for the synthetic channels and 104 12 pS for the E. coli channels. The data points represent mean values of 10 observations.28...

Figure 16. A. Block of E. coli PHB/polyP channels by transition metal cation, La3+. Representative single-channel current steps at 120 mV for PHB/polyP complexes, extracted from E. coli, purified by size-exclusion chromatography, and incorporated into bilayers of 16 0,18 1, PC between symmetric solutions of 100 mM CaCh, 2 mM MgCh,10 mM Tris Hepes, pH 7.3, and with LaCl3 added to the cis side as stated.27 (a) No LaCb (b) 0.20 mM LaCb (c) 0.55 mM LaCb. B. Block of synthetic PHBi2s/polyP channels by transition metal cation, La3+. The bilayer composed of 16 0,18 1, PC and cholesterol (5 1 w/w) was formed between aqueous bathing solutions of 200 mM CaCb, 5 mM MgCb, 10 mM Tris Hepes, pH 7.4. After incorporation of the channel, activities were recorded for 5 min at a clamping potential of-80 mV. Then, LaCb was added to the trans compartment to achieve the indicated concentrations. The bath was stirred and activities were recorded after 1 min of addition of La+3. Data points represent mean values of the amplitude histograms error bars show the standard deviation from the mean.28...

Linear stability analysis indicates that the symmetric solution in Eq. (13.60) becomes unstable when A is greater than the following critical value Ac... 

Problems associated with the quantum-mechanical definition of molecular shape do not diminish the importance of molecular conformation as a chemically meaningful concept. To find the balanced perspective it is necessary to know that the same wave function that describes an isolated molecule, also describes the chemically equivalent molecule, closely confined. The distinction arises from different sets of boundary conditions. The spherically symmetrical solutions of the free molecule are no longer physically acceptable solutions for the confined molecule. 

The above approximation method can also be applied to the case of a sphere in a 2-1 symmetrical solution, yielding [4]... 

As V reduces to this form if is a spherically symmetrical function, it follows that spherically symmetrical solutions must satisfy the equation... 

The SEC partition coefficient [6] (.K sec) was measured on a Superose 6 column for three sets of well-characterized symmetrical solutes the compact, densely branched nonionic polysaccharide, Ficoll the flexible chain nonionic polysaccharide, pullulan and compact, anionic synthetic polymers, carboxylated starburst dendrimers. All three solutes display a congruent dependence of K ec on solute radius, R. In accord with a simple geometric model for SEC, all of these data conform to the same linear plot of i sEc versus R. This plot reveals the behavior of noninteracting spheres on this column. The mobile phase for the first two solutes was 0.2M NaH2P04-Na2HP04, pH 7.0. In order to ensure the suppression of electrostatic repulsive interactions between the dendrimer and the packing, the ionic strength was increased to 0.30M for that solute. 

In the absence of an external electric field, = 0, Goodson and Hershbach [40] have found that these equations have a symmetric solution with the two electrons equidistant from the nucleus, with r = r2 = r. 

This symmetric solution represents a minimum in the region where all the eigenvalues of the Hessian matrix are positive, Z>ZC = y/2. For values of Z smaller than Zc, the solutions become unsymmetrical with one electron much closer to the nucleus than the other (ri / r2). In order to describe this symmetry breaking, it is convenient to introduce new variables (r, q) of the form... 

The symmetric solution to the chain condition is obtained by setting the constant in the exponential to be independent of a, b(a) = b. The most general solution is obtained using... 

We now turn to some simple examples where Aab can be calculated. We still stay within the first-order deviation from ideal symmetrical solution. 

For purely imaginary solutions of re (Re re = 0, re = ik, fc-real), the function un becomes proportional to cos(kn) for symmetrical solutions, and proportional to sin (kn) for antisymmetrical solutions, respectively. In these cases we obtain bulk symmetrical and antisymmetrical solutions which are similar to the solutions in chains with Frenkel excitons only. 

Figure 1. The two symmetric solutions (first panel) and their coincidence to a single orbit (second panel) if the periodicity conditions apply...

Figure 8. Loci of the stationary corotation solutions of the 2/1 resonance for several mass ratios m2/mi. Top figures correspond to the symmetric solutions of the two left-hand side plots in Figure 6. The points corresponding to two early determinations of the elements of Gliese 876 are shown in one of these plots. The bottom figure corresponds to the asymmetric solutions of the two right-hand side plots in Figure 6. The line across these curves shows the values of the eccentricities for which 0.63(1 + ei) = (1 — e2). In all panels, the thick line shows the boundary between the domains of symmetric and asymmetric solutions.

The above equations were used to find Apsidal corotation solutions in the case of planets in 2/1 and 3/1 mean-motion resonances. The relationship between eccentricities and mass ratios in some of these solutions is shown in Figure 8. The top panels correspond to symmetric solutions. In the left-hand side panel, the periapses are anti-aligned. This is the... 

The functions /,(, /) and fi(x,t) allow solutions to be derived for special cases of inlet and initial conditions. Note that 5 is a dummy variable of integration. In the case where the column has been washed with eluent and a sample has not been introduced, initial condition 3 from Table I applies, and f 1 (x,t) = 0. After a sample has been introduced and washed into the column by eluent, boundary condition 1 from Table I with c = 0 applies and f x,t) = 0. The eluent volume is Q — Avta. The two limiting cases mentioned above may be superimposed, offset from one another by A v a t = Q = VF where Q is the feed volume. This superimposition is applicable only in the case of linear equilibrium, which yields symmetric solutions. Figure 4 shows the results of these equations graphically for selected values. These values may be particularly applicable to proteins in size exclusion supports. 

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