Double point

Before the pH electrode is used, it should be calibrated using two (or more) buffers of known pH. Many standard buffers are commercially available, with an accuracy of 0.01 pH unit. Calibration must be performed at the same temperature at which the measurement will be made care must be taken to match the temperature of samples and standards. The exact procedure depends on the model of pH meter used. Modem pH meters, such as the one shown in Figure 5-8, are microcomputer controlled, and allow double-point calibration, slope calculation, temperature adjustment, and accuracy to 0.001 pH unit, all with few basic steps. The electrode must... 

Curvale and Cesco suggest a double point equation... 

For single point methods fit with minor errors of 3.54%, but the method of Solomon-Ciuta is only valid for concentrations below 0.25% wt of gelatin. Double point error is low of 2.00%. 

While all methods of single and double point are used to characterize a polymer solution is always advisable to increase the statistical weight (to reduce errors) with at least four different concentrations of polymer in a given solvent. 

Curvale, R.A., Cesco, J.C. 2009. Intrinsic viscosity determination by "single-point" and "double-point" equations. Applied Rheology 19, 5, 53347. 

It is evident that the projections of the fixed points A and B, in the EJ map, always lie on relative equilibria of type I, but that the position of the overlapping projections of C and D depends on the sign of b — a. If > b the double point is isolated between the two type I equilibria, and quantum monodromy is expected, for a sufficiently dense quantum lattice. If, on the other hand, the critical point lies on the type II relative equilibrium line and... 

Then the above argument shows that an element of the component of the fixed point set corresponding to D consists of ai single-points, ao double points,. .., and multiple points of multiplicity n all sitting at the zero section E and vanishing in the fiber direction. A typical example is shown in Figure 7.5. We have shown the following proposition. 

VIII. Double Point-Group Symmetry and Selection Rules.26... 

Prepared State. Here the Hamiltonian H is the time-independent molecular Hamiltonian. Both H0 and T are time independent. The initial prepared state is an eigenket to H0 and thus is nonstationary with respect to H = H0 + T. One example is provided by considering H0 as the spin-free Hamiltonian 77sp and the perturbation T as a spin interaction. A second example is provided by considering H0 as the spin-free Born-Oppenheimer Hamiltonian and T as a spin-free nonadiabatic perturbation. In the first example spin-free symmetry is not conserved but double-point group symmetry may be. In the second example point-group symmetry is not conserved, but spin-free symmetry is. The initial prepared state arises from some other time-dependent process as, for example, radiative absorption which occurs at a rate very much faster than the rate at which our prepared state evolves. Mechanisms for radiationless transitions in excited benzene may involve such prepared states, as is discussed in Section XI. 

Vm. DOUBLE POINT-GROUP SYMMETRY AND SELECTION RULES... 

An example of a double point group is the double group of C2 ... 

In the derivation of these spin-interaction selection rules the harmonic approximation was made. In taking nuclear vibration into account2,77 these selection rules are often broken. In addition to coupling with the internal vibrational modes of a molecule, coupling with the phonon modes in the solid state may be important.124 Some use of double point group symmetry will be found in Sections IX, XI, and XII. 

The 3B1 state of methylene gives rise to three double-point group symmetries... 

Suppose that Y = X U X, where X and X are curves without common components, having t points in common which are nonsingular for both X and X and ordinary double points (nodes) for Y, i.e X and X have distinct tangents at those points. In this case we will say that X and X intersect quasi-transversal lv In this situation the Hilbert polynomials of the three curves are related by the identity ... 

Discrete valuation ring (DVR), 2-5, 3-9, 7-6 Double point (ordinary), 11-10, 11-11 Dual numbers, algebra of, A-6 Dual projective space, 1 -5... 

The excitation diagram was found to contain saddle-node, Hopf, period doubling, and homoclinic bifurcations for the stroboscopic map. In addition, many of these co-dimension one bifurcation curves were found to meet at the following co-dimension two bifurcation points Bogdanov points (double +1 multipliers), points with double -1 multipliers, points with multipliers at li and H, metacritical period-doubling points, and saddle-node cusp points. 

Exercise 8.1-1 Name the classes in the double point groups C4 and C2v. 

Listing conjectured that these two knot types cannot be transformed into one another by ambient isotopy. Figure 13 shows the two enantiomorphous isotopy types, arbitrarily designated as R and L. For each type, individual presentations are shown as projections in the plane, called diagrams. Crossings in these diagrams represent transverse double points, with over- and under-characteristics clearly... 

In summary, failure to detect a rigidly achiral presentation does not mean that such a presentation cannot be found among the infinitely many presentations of a knot failure to interconvert enantiomorphous presentations by ambient isotopy does not exclude the possibility that an interconversion pathway can be found among the infinitely many pathways that are available and a palindromic knot polynomial does not necessarily mean that the knot is amphicheiral. Consequently, it may be impossible in certain cases to determine with complete certainty whether a knot is topologically chiral or not. The fundamental task of the theory of knots was stated over a hundred years ago by its foremost pioneer Given the number of its double points, to find all the essentially different forms which a closed curve can assume. 15 Yet to find invariants that will definitively determine whether or not a knot is chiral remains an unsolved problem to this day.63a Vassiliev invariants have been conjectured to be such perfect invariants.63b... 

External hair of animals, generally called wool, was spun into yam and woven into fabrics. Like silk, wool is essentially protein it is composed of various amino acids, a majority of which are keratin. (Unfortunately, the keratin contains sulfur, which attracts certain insects that thrive on wool and contribute to the scarcity of historic woolen fabrics.) The outstanding morphological characteristic of wool fiber is its external scales that overlap in one direction toward the tip of the fiber. The scales can be chemically, mechanically, and temporally damaged and can disappear as the wool deteriorates. Outside of the scales is a membranous layer, the epicuticle inside them is the bulk of the wool fiber, the cortex, which consists of millions of double-pointed, needle-like cells neatly laid... 

As is to be expected from non-dipolar 2D NLO-phores, the ir system of simple benzene or naphthalene derivatives is too small to show practically useful NLO responses (Wortmann et al., 1993 Wolff et al., 1997). Especially the perpendicular transition is of quite low intensity as indicated by the smaller double-pointed arrow in Scheme 25. Therefore, the tri- and tetrasubstituted blueprint structures [98]-[100] (Scheme 20) were systematically elongated in one, two or all three directions by the interpolation of phenyl-ethynyl bridges. The use of triple bonds eliminates possible conformational isomerism. The trisubstituted type [124]-[127] is shown. By a combination of EOAM and polarization-dependent EFISHG and HRS, all four independent tensor components were evaluated (Wolff et al., 1997). Results are given in Table 5. The dipole moment ju, lies parallel to the z axis, the y axis within the molecular plane and the x axis perpendicular to the molecular plane. The /3 values (at 1064 nm) are in units of 10 ° C m° V . ... 

This procedure is repeated, using successive completed points to pull out the glass, and leaving 2.5 cm of undisturbed tube each time, until the whole length has been used up (see Fig. 6.19). All double points which are straight, strong and of usable length should be set aside for a later exercise. 

From a tube whose diameter is approximately four-fifths that of the required bulb, draw strong double points with the distance between the shoulders equal to the bulb diameter (see Fig. 6.40a). 

FIG. 6.19. Double points, showing the shoulders. Botli points should be concentric witli tlie tube. 

FIG. 6.35. Side tube near the end of a test-tube, (a) Double points are made from suitable tubing, one point is sealed off and the side tube attached as shown in (b). (c) Tlie sealed off end is blown out and rimmed as in (d). (e) When tlie newly made rim has cooled, the open end of tlie test tube is fitted witli a bored cork caiiying a handle tlie other end is sealed off and blown up to a neat hemisphere. 

FIG. 6.41. Riffled side tubes. These tubes are prepared in pairs from double points shown in (a), (b) Tire first constriction and (c) the second consUiction must be neat and of reasonable wall-thickness, (d) Shows the bulb blown up. (e) Shows tlie bulb drawn into shape after heating on one side. The procedure is repeated at tlie other end of tlie double point. 

FIG, 6.46. A constant level device, (a) Double points from 25 mm diameter tubing, (b) One end blown to a hemisphere, (c) Shows a 8 mm diameter hole blown in tlie hemispherical end. (d) Tlie prepared overflow tube, of 6 mm tube and witli a 10 mm diameter bulb, inserted into tlie body of tlie constant level device, (e) Tlie ring seal completed, and the inlet tube ready to join on. (f) The outlet tube joined on. All the heated glass, at this end of the device, should be annealed, (g) Shows tlie completed constant level device witli tlie top blown out and rimmed. 

---