Undulants squares

The student thought for a few seconds, and replied, That is too easy. Master. It is the largest undulating square known to humanity. The teacher pondered this answer, and himself started to undulate in a mixture of excitement and perhaps even terror. 

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. 

When I conceived the idea of undulating squares a few years ago, it was not known whether any such numbers existed. It turns out that 69,696 = 264 is indeed the largest undulating square known to humanity, and most mathematicians believe we will never find a larger one. 

Dr. Helmut Richter from Germany is the world s most famous undulation hunter, and he has indicated to me that it is not necessary to restrict the mod searches to powers of ten, and that arbitrary primes work very well. He has searched for undulating squares with 1 million digits or fewer, using a Control Data Cyber 2000. No undulating squares greater than 69,696 have been found. 

Randy Tobias of the SAS Institute in North Carolina notes that there are larger undulating squares in other number bases. For example, 292 = 85,264 = 41,414 base 12. And 121 is an undulating square in any base. (121 base is ( + 1) ). 

Fig. 6.15 Dynamic structure factor from the junction-labelled triblock copolymer for different Q-values. T=433 K filled circles Q=0.20 A y filled squares Q=0.18 A open triangles down Q=0.14 A open triangles up Q=0.114 A open circles Q=0.08 A open squares Q=0.05 A T=473 K filled circles Q=0.20 A open triangles up Q=0.10 A open circles Q=0.08 A open squares Q=0.05 A"k The solid lines are result of the fit with the complete structure factor for surface undulations and Rouse motion. (Reprinted with permission from [284]. Copyright 2002 EDP Sciences)...

Clearly flow aligning behavior of the director is present and do increases linearly with the tilt angle, do. Above a threshold in the Spain rate, y 0.011, undulations in vorticity direction set in. In Fig. 14 the results of simulations for y 0.015 are shown. In Fig. 15 we have plotted the undulation amplitude obtained as a function of the shear rate. The dashed line indicates a square root behavior corresponding to a forward bifurcation near the onset of undulations. This is, indeed, what is expected, when a weakly nonlinear analysis based on the underlying macroscopic equations is performed [54], In Fig. 16 we have plotted an example for the dynamic behavior obtained from molecular dynamics simulations. It shows the time evolution after a step-type start for two shear rates below the onset of undulations. The two solid lines correspond to a fit to the data using the solutions of the averaged linearized form of (27). The shear approaches its stationary value for small tilt angle (implied by the use of the linearized equation) with a characteristic time scale t = fi/Bi. 

Fig. 15 Undulation amplitude due to shear. The amplitude of the undulations A is given as a function of the strain rate y. At a shear rate y 0.01 undulations set in. The amplitude of these undulations grows continuously with increasing shear rate. The dashed line shows a fit to data points starting at y > 0.02 assuming a square root dependence of the amplitude above the undulation onset. Fig. 5.6 of [54]...

The dependence of the interaction force between two undulating phospholipid bilayers and of the root-mean-square fluctuation of their separation distances on the average separation can be determined once the distribution of the intermembrane separation is known as a function of the applied pressure. However, most of the present theories for interacting membranes start by assuming that the distance distribution is symmetric, a hypothesis invalidated by Monte Carlo simulations. Here we present an approach to calculate the distribution of the intermembrane separation for any arbitrary interaction potential and applied pressure. The procedure is applied to a realistic interaction potential between neutral lipid bilayers in water, involving the hydration repulsion and van der Waals attraction. A comparison with existing experiments is provided. 

Square grid effect Before the electric field becomes great enough to cause the cholesteric-nematic phase transition, a periodic deformation may appear in cholesteric liquid crystals. The layer undulation occurs in two orthogonal directions so that a square pattern is observed. This effect is more likely to happen for cholesteric liquid crystals of large pitch (about microns). 

The crystal to IP distances for 0.5 A and 0.3 A are, of course, increased. This is advantageous because of the inverse square law effect in reducing the background under the diffraction spot. However, a crystal to film distance of, say, 0.5 m with a 1 mrad divergence beam would lead to a sizeable increase in the diffraction spot size (i.e. by 0.5mm). On an undulator, however, beam divergences are intrinsically —0.1 mrad and so the spot size over a 0.5 m distance would only increase by 0.05 mm due to this effect (table 6.4). Mosaic spreads of specimens also need to be narrow but —0.1 mrad is a quite reasonable expectation for these samples (Helliwell (1988) and Colapietro, Helliwell, Spagna and Thompson, unpublished, see figure 2.8(c)). 

Experiments have been carried out on p-(nitrobenzyloxy)-biphenyl [30] and typical patterns in the conductive range at onset are shown in Fig. 6. At low frequencies disordered rolls without point defects have been observed with a strong zig-zag (ZZ) modulation (see Fig. 6a) which can be interpreted as the isotropic version of oblique rolls. Above a critical frequency, a square pattern is observed which retains the ZZ character because the lines making up the squares are undulated. At onset the structure is disordered however, after a transient period defects are pushed out and the structure relaxes into a nearly defect-free, long-wave modulated, quasi-periodic square pattern (see Fig. 6b). 

Figure 6. Snapshots of EC patterns slightly above onset for case B. a ZZ modulated disordered rolls, b undulated (soft) squares.

The electrons enter the wiggler/undulator in unstructured packets (e.g. storage ring case) or in a quasi-continuous stream. As stated above, the electromagnetic waves, produced by oscillation of the various electrons traversing the device, add incoherently. There will be partial destruction since positive and negative amplitudes add algebraically. The emitted amplitude is proportional to the square root of the number of electrons in the packet, i.e. to so that the radiation intensity will be proportional to Ng. 

---