Undulants numbers

To understand the monk s passionate response, we must digress to some simple mathematics. Undulating numbers are of the form ababababab. For example, 171,717 and 28,282 are undulating numbers. A square number is of the formy = x. For example, 25 is a square number. So is 16. An undulating square is simply a square number that undulates. 

Schwlngungs-gieichung, /. vibration equation, -kreis, m. (Elec.) oscillatory circuit, -me-thode, /. swing method. -quanten2ahl, /. vibrational quantum number, -schreiber, m. oscillograph, -weite, /. amplitude of vibra tion. -Welle,/, undulation, vibrational wave, -zahl, /. vibration nmnber, vibration fre quency. -zeit,/. time of vibration, -zustand, m. state of vibration or oscillation. 

Under low-frequency excitation, the flame front is wrinkled by velocity modulations (Fig. 5.2.5). The number of undulations is directly linked to frequency. This is true as far as the frequency remains low (in this experiment, between 30 and 400 Hz). The flame deformation is created by hydrodynamic perturbations initiated at the base of the flame and convected along the front. When the velocity modulation amplitude is low, the undulations are sinusoidal and weakly damped as they proceed to the top of the flame. When the modulation amplitude is augmented, a toroidal vortex is generated at the burner outlet and the flame front rolls over the vortex near the burner base. Consumption is fast enough to suppress further winding by the structure as it is convected away from the outlet. This yields a cusp formed toward burnt gases. This process requires some duration and it is obtained when the flame extends over a sufficient axial distance. If the acoustic modulation level remain low (typically v /v < 20%),... 

Figure 2. Synchrotron X-ray source (schematic). The electrons execute circular motions in the storage ring and emit intense X-rays along the tangent of the orbit. This radiation is enhanced by undulator magnets that are often placed inside the vacuum vessel for enhanced performance. The storage ring has a number of straight sections for undulators and wigglers (not shown).

For the Buck, we see that the input capacitor RMS actually maxes out at D = 50%, whereas the output capacitor RMS current (curve number 12) increases dramatically at low D (high input). Does that really mean that we have to worry about the dissipation in the output capacitor Think about it. The output capacitor in a Buck is barely responsible for any of its losses, since it sees only the smoothened (undulating) inductor current. So yes, as a... 

Axial flow impellers, 16 672-673, 684 Axial-flow propeller-type pumps, 22 67, 68 Axially split pumps, 22 67, 68 Axial Peclet number, 10 763 Axial undulations, in hemodialysis, 26 830-831... 

Let us examine the instability oi strained thin films. In Fig. 3, thin films of30 ML are coherently bonded to the hard substrates. The film phase has a misfit strain, e = 0.01, relative to the substrate phase, and the periodic length is equal to 200 a. The three interface energies are identical to each other = yiv = y = Y Both phases are elastically isotropic, but the shear modulus of the substrate is twice that of the film (p = 2p). On the left-hand side, an infinite-torque condition is imposed to the substrate-vapor and film-substrate interfaces, whereas torque terms are equal to zero on the right. In the absence of the coherency strain, these films are stable as their thickness is well over 16 ML. With a coherency strain, surface undulations induced by thermal fluctuations become growing waves. By the time of 2M, six waves are definitely seen to have established, and these numbers are in agreement with the continuum linear elasticity prediction [16]. 

When this gratuitously titled machinery is set into rotary motion, the striped and displaced circles appear to undulate. This movement produces a hypnotic illusion of pulsating depth, or what Robert Lebel poetically called a screen for suggestive metamorphosis. Fascinated by this quirky optical experiment, Duchamp (especially in 1926) created a number of substitute disks decorated with either spiral-linear and colored patterns, many of which included similarly inscribed onomatopoetically, pseudo-emblematic patterns (see MD-125, MD-126, MD-135, MD-139). These motorized illusionistic doodles continued in production, culminating eventually in 1935, with the editions of some 1500, flat cardboard disks, collectively called the Rotoreliefs (MD-144). Another approach in this direction of rotary symbolism was represented in the 1926 film Anemic Cinema (MD-140), where spirals alternate with puns inscribed within slowly turning disks. ... 

In examining wool it is sometimes of importance to note the uniformity and number of the undulations of the fibre. This number, which normally increases with the fineness of the wool, is referred to 1 cm. 

Throughout this chapter we focus on the extended hydrodynamic description for smectic A-type systems presented in [42,43], We discuss the possibility of an undulation instability of the layers under shear flow keeping the layer thickness and the total number of layers constant. In contrast to previous approaches, Auernhammer et al. derived the set of macroscopic dynamic equations within the framework of irreversible thermodynamics (which allows the inclusion of dissipative as well as reversible effects) and performed a linear stability analysis of these equations. The key point in this model is to take into account both the layer displacement u and the director field ft. The director ft is coupled elastically to the layer normal p = in such a way that ft and p are parallel in equilibrium z is the coordinate perpendicular to the plates. 

Fig. 10.16. Final rotational state distributions of NO following the dissociation of C1NO through the T state. The quantum numbers n and k specify the initial vibrational and bending excitation of the ClNO(Ti) complex. The undulations for the excited bending states reflect the nodal structures of the dissociation wavefunction at the transition state. The open and the filled circles indicate different P and Q branches. The corresponding absorption spectrum is depicted in Figure 7.14. Adapted from Qian, Ogai, Iwata, and Reisler (1990).

Regarding the spatial aspects of the enzymatic degradation of CA-g-PLLA, a surface characterization [30] was carried out for melt-molded films by atomic force microscopy (AFM) and attenuated total-reflection Fourier-transform infrared spectroscopy (ATR-FTIR) before and after the hydrolysis test with proteinase K. As exemplified in Fig. 3 for a copolymer of MS = 22, the AFM study showed that hydrolysis for a few weeks caused a transformation of the original smooth surface of the test specimen (Fig. 3a) into a more undulated surface with a number of protuberances of 50-300 nm in height and less than a few micrometers in width (Fig. 3b). The ATR-FTIR measurements proved a selective release of lactyl units in the surface region of the hydrolyzed films, and the absorption intensity data monitored as a function of time was explicable in accordance with the AFM result. 

Out of the mouth of the six-legged triangle spill a number of tiny gems colored translucent emerald or sapphire. The gems undulate, sprout legs, and scurry away into the distance. 

These deviations were accounted by Strey et al.,8 who carried out experiments with the binary water—C12E5 system, by noting that the amplitude of the thermal undulations increased with the repeat distance d. If one considers the total area of the interface a constants which depends only on the number of surfactant molecules, the projection So of the total area on a plane perpendicular to d will decrease with increasing d. Hence, the apparent area per surfactant molecule, which is defined as the ratio between So and the total number of surfactant molecules, decreases with increasing d, while the ideal dilution law implies that the apparent area per surfactant molecule is a constant. The excess area, defined as AS = S — So, was related to the bending modulus of the interface,8 and the experimental results for the deviations from the ideal dilution law were used to determine. Kc-3,11 However, it should be noted that there are binary systems for which the deviations from the ideal dilution law occur in the opposite directions. For instance, in the binary systems of fatty acid alkali soaps/water, the apparent area per headgroup increases with water dilution, because of the incorporation of water in the interface.1... 

Having established that bilayer flexibility and bilayer interaction are the mesoscopic determinants, the next question is whether these determinants can be coupled to molecular parameters. In fact, this has been done to quite some extent. In general, bilayer flexibility can be shown (both experimentally as well as theoretically by simulation methods) to be directly related to bilayer thickness, lateral interaction between heads and tails of the surfactants, type of head group (ethoxylate, sugar, etc.), type of tail (saturated, unsaturated) and specific molecular mixes (e.g. SDS with or without pen-tanol). The bilayer interaction is known to be related to characteristics such as classical electrostatics. Van der Waals, Helfrich undulation forces (stemming from shape fluctuations), steric hindrance, number, density of bilayers, ionic strength, and type of salt. Two examples will be dicussed. 

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