Characteristic multiplier

Owing to the different relationship between t and y for a non-Newtonian fluid, the shear rate at the wall is not given by equation 3.13 but can be expressed as the flow characteristic multiplied by a correction factor as shown in Section 3.2. 

As 9 continues to increase (1 +< 9), the characteristic multipliers 2C an(l 2d Pa-ss through = — 1, and so these 2-period solutions become unstable. At this stage, we look at the fourth iterate and we find, as might now be expected, that a 4-cycle periodic solution appears (Figure 3.3 C, fixed point of period 4 for 9 = 3.5). The period doubles repeatedly and goes to infinity as one approaches a critical point 9C at which instability sets in for all periodic solutions, e.g., for the model (3.1), 9C 3.5699456. Above 9C all fixed points are unstable and the system is chaotic. 

Stability is determined by the eigenvalue analysis at an equilibrium point for flows and by the characteristic multiplier analysis of a periodic solution at a fixed point for maps [3]. 

Characteristic multiplier, Chapter 3 Correlation length, Chapter 1 ... 

In general, the characteristic multipliers can only be found by numerical integration (see Exercise 8.7.10). The following examples are two of the rare exceptions. 

Find the characteristic multiplier for the limit cycle of Example 8.7.1. 

Remark 11.2 The eigenvalues Pj of M are known as Floquet multipliers or characteristic multipliers and the eigenvalues v, of B are known as the Floquet exponents or characteristic exponents. They are related by pi = exp(n,T) [435]. 

Recall that a fixed point 0 x = xq) is called structurally stable if none of its characteristic multipliers, i.e. the roots of the characteristic equation (7.5.2), lies on the unit circle, A topological type (m,p) is assigned to it, where m is the number of roots inside the unit circle and p is that outside of the unit circle. If m = n (m = 0), the fixed point is stable (completely unstable). The fixed point is of saddle type when m 0,n. The set of all points whose trajectories converge to xq when iterated positively (negatively) is called the stable (unstable) manifold of the fixed point and denoted by Wq Wq). In the case where m = n, the attraction basin of O is Wq. If the fixed point is a saddle, the manifolds Wq and Wq are C -smooth embeddings of and MP in respectively. 

A square matrix has the eigenvalue A if there is a vector x fulfilling the equation Ax = Ax. The result of this equation is that indefinite numbers of vectors could be multiplied with any constants. Anyway, to calculate the eigenvalues and the eigenvectors of a matrix, the characteristic polynomial can be used. Therefore (A - AE)x = 0 characterizes the determinant (A - AE) with the identity matrix E (i.e., the X matrix). Solutions can be obtained when this determinant is set to zero. 

FIG. 10-28 Characteristic curve of a centrifugal pump operating at a constant speed of. 3450 r/min. To convert gallons per minute to cubic meters per hour, multiply hy 0.2271 to convert feet to meters, multiply hy 0..3048 to convert horsepower to kilowatts, multiply hy 0.746 and to convert inches to centimeters, multiply hy 2.54. 

The distinguishing performance characteristic of the torque converter, in contrast to the fluid coupling, is that It IS capable of multiplying torque. Torque multiplication is made possible by vane curvature and the presence of the reactor. When the converter is stalled—that is, the turbine and the reactor are stationary—the torque delivered to the gearbox is typically 2... 

If foaming characteristics of the system are less than air-water, results will be conservative. For systems tending to greater foam and bubbles than the air-water system, approximate a value of h i by multiplying calculated value by 2, or 3 or known relative relationship. 

In discussing the proton transfer (66), we saw that one of the neutral species could be a solvent molecule. We shall discuss that case below. Here we may notice that, when all four species are solute particles, the number of solute particles is unchanged by the reaction, or Aq = 0. In such a case AF° happens to be equal to the characteristic unit U multiplied by Avogadro s constant. 

Phototubes, multiplier, 56-59, 222 Placement error, 285-287 Planck s constant, 7, 8 Plastics, characterization by absorptiometry, 78, 79 Plateau, characteristic, 60 Platinum, determination by x-ray emission spectrography, 161, 328 L peaks, measured by Bragg, 25, 26, 35 L spectra, 35... 

In the so-called "wrinkled flame regime," the "turbulent flame speed" was expected to be controlled by a characteristic value of the turbulent fluctuations of velocity u rather than by chemistry and molecular diffusivities. Shchelkin [2] was the first to propose the law St/Sl= (1 + A u /Si) ), where A is a universal constant and Sl the laminar flame velocity of propagation. For the other limiting regime, called "distributed combustion," Summerfield [4] inferred that if the turbulent diffusivity simply replaces the molecular one, then the turbulent flame speed is proportional to the laminar flame speed but multiplied by the square root of the turbulence Reynolds number Re. ... 

The molar mass (MM) of any substance is the mass of one mole of that substance. As described in Section 2-1. each isotope of a particular element has a different mass. Therefore, the mass of one mole of any isotope has a unique value, its isotopic moiar mass. This characteristic molar mass can be found by multiplying the mass of one atom of that isotope by Avogadro s number. For example, mass spectrometry experiments reveal that one atom of carbon-13 has a mass of 2.15928 X 10- g, from which we can calculate the isotopic molar mass of... 

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