Hyperbolic pair

The interesting question is to enumerate, if possible, those elementary (R,q)gen-polycycles. Call an (R, q)gen-polycycle elliptic, parabolic, or hyperbolic if the number i + T — i (where r = max C/ i) is positive, zero, or negative, respectively. In Theorem 7.2.1, we will see that the number of elementary (r, [Pg.76]

Depending on the relative velocity in opposite pairs of the rollers, flow can be either purely rotational (X = — 1), simple shear (X = 0) or hyperbolic straining (X = 1, shown on this figure)... 

In terms of transient behaviors, the most important parameters are the fluid compressibility and the viscous losses. In most field problems the inertia term is small compared with other terms in Eq. (128), and it is sometimes omitted in the analysis of natural gas transient flows (G4). Equations (123) and (128) constitute a pair of partial differential equations with p and W as dependent variables and t and x as independent variables. The equations are hyperbolic as shown, but become parabolic if the inertia term is omitted from Eq. (128). As we shall see later, the hyperbolic form must be retained if the method of characteristics (Section V,B,1) is to be used in the solution. 

Figure 2.15. Schematic of a quadrupole analyzer, (a) A hyperbolic cross-section (b) cross-section of cylindrical rods (c) the operating principle of a quadrupole mass filter. The x-direction pair of rods acts like a high pass filter so ion C (with low m/z) is not allowed through, and the y-direction pair of rods acts like a low pass filter and takes care of ion A (with high m/z). Only ion B having an m/z in the stable range is allowed through the quadrupole mass filter for subsequent detection. Reprinted from A. Westman-Brinkmalm and G. Brinkmalm (2002). In Mass Spectrometry and Hyphenated Techniques in Neuropeptide Research, J. Silberring and R. Ekman (eds.) New York John Wiley Sons, 47-105. With permission of John Wiley Sons, Inc.

A linear quadrupole mass analyzer consists of four hyperbolically or cyclindrically shaped rod electrodes extending in the z-direction and mounted in a square configuration (xy-plane, Figs. 4.31, 4.32). The pairs of opposite rods are each held at the same potential which is composed of a DC and an AC component. 

A quadrupole mass analyzer is made of four hyperbolic or circular rods placed in parallel with identical diagonal distances from each other. The rods are electrically connected in diagonal. In addition to an alternating radiofrequency (RE) potential (V), a positive direct current (DC) potential (U) is applied on one pair of rods while a negative potential is applied to the other pair (Fig. 1.17). The ion trajectory is affected in x and y directions by the total electric field composed by a quadrupolar alternating field and a constant field. Because there is only a two-dimensional quadrupole field the ions, accelerated after ionization, maintain their velocity along the z axis. 

Several workers have concluded that under conditions used in their study ion-pairing in the mobile phase between amphiphilic hetaeron ions and oppositely charged sample components governed retention. Horvath et al. (34) examined the effect of alkyl sulfates and other alkyl anions on the retention of catecholamines in which both the concentration and the length of the alkyl chains of the hetaerons were varied. The hyperbolic concentration dependence of the retention factor shown in Fig. 48, was found to be similar to that reported by others. 

Quadrupoles are comprised of four metal rods, ideally of hyperbolic cross section, arranged as shown in Fig. 5.4. A combination of radiofrequency (RF) and direct current (DC) voltages are applied to each pair of rods, which creates an electric field within the region bounded by the rods. Depending on the RF/DC ratio, the electric field between the rods will allow ions in a narrow m/z range to pass, typically 0.8 m/z —just how narrow will depend on a number of factors which influence the resolution. Hence, by changing the RF/DC ratio in a controlled manner, the quadrupole can be... 

Figure 16.9—Representation of a quadrupole. Notice the pairing of oppositely charged electrodes. This experimental design requires high-precision machining of the hyperbolic electrodes. To the right a series of equipotential hyperbolic lines in the central part of the quadrupole is shown.

An ideal quadrupole field can be generated using four parallel electrodes (Z, = 5 to 20 cm) which have a hyperbolic cross-sectional field at their interior (Fig. 16.9). The electrodes are coupled in pairs and a potential difference U is applied across the pairs. If the distance between two opposite electrodes is 2 r0, then the potential d> within the xy plane of the quadrupole will be given by ... 

Early designs consisted of a cylindrical configuration. Later designs used a pair of truncated cones. Today s design consists of a hyperbolic shape. The advantages of this configuration include ... 

Figure 13. (The color version is available from the authors.) The previous pair of trajectories as seen in the normal-form coordinates in the hyperbolic direction. The complicated dynamics at the saddle has been smoothed out. The two trajectories approach from the top. Note how the axis acts as a separatrix between them. The reactive one intersects the TS (the diagonal) at the dot. Compare this with Figs. 5, 6, and 7. Primes on 1 3 and p (see the text) have been dropped.

Fig. 8 Schematic of a tandem quadrupole MS/MS instrument. A tandem quadrupole MS/MS instrument consists of two quad-rupole MS filters, MSI and MS2, separated by a collision cell. Each quadrupole MS filter consists of four cylindrical or hyperbolic shaped rods. A unique combination of direct current (dc) potential and radiofrequency (rf) potential is applied to each pair of rods (one pair 180° out of phase with the other). A mass spectrum results by varying the voltages at a constant rf/dc ratio. A variety of scan modes (e.g., full scan, product ion, precursor ion, neutral loss) provide unique capabilities for quantitative and qualitative structure analysis. (Courtesy of Micromass, Manchester, UK.)...

This domain consists of a bundle of a-helices packed in pairs against each other. The most common arrangement is illustrated in Fig. 6.1. The space between the four helices is occupied by hydrophobic side chains, whereas polar side chains are directed towards the surrounding solution. The a-helices are twisted with respect to each other, and their arrangement is similar to that of a fragment of a blue phase (c/. Figure 5.5). In both cases, the chirality of the structural units leads to hyperbolic curvature within the aggregate. 

If the resonant tori, which are the invariant tori whose rotational numbers are rational, are broken under perturbations, the pairs of elliptic and hyperbolic cycles are created in the resonance zone. This fact is known as a result of the Poincare-Birkhoff theorem [4], which holds only if the twist condition, Eq. (2), is satisfied. Around elliptic cycles thus created, new types of tori, which are... 

The operation of the QIT is based on the same physical principle as the quadrupole mass spectrometer described above. Both devices make use of the ability of RF fields to confine ions. However, the RF field of an ion trap is designed to trap ions in three dimensions rather than to allow the ions to pass through as in a QMF, which confines ions in only two dimensions. This difference has a significant unpact on the operation and limitations of the QIT, The physical arrangement of a QIT is different from that of a QMF. If an imaginary axis is drawn through the y-axis of the quadrupole rods, and the rods are rotated around the axis, a solid ring with a hyperbolic inner surface results from the x-axis pair of rods. 

Quadrupole mass analyzers consist of four parallel metal rods which are either round-shaped or ideally have a hyperbolic section. Ion separation is realized by applying an alternating electrical fleld to the four rods with each pair of adjacent rods having the opposite signs of the potential. The potential applied to two opposite rods is a superposition of a constant potential U and an alternating potential V cos mt (Eq. (5.1)). The other two opposite rods bear the potential . co describes the angular frequency and equals Inv when v is the frequency of the applied radio frequency field t is the time. 

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