Irrational

A superlattice is temied commensurate when all matrix elements uij j are integers. If at least one matrix element uij j is an irrational number (not a ratio of integers), then the superlattice is temied incommensurate. A superlattice can be inconnnensiirate in one surface dimension, while commensurate in the other surface dimension, or it could be mconmiensurate in both surface dimensions. 

The quasiperiodic route to chaos is historically important. It arises from a succession of Hopf birfurcations. As already noted, a single Hopf bifurcation results in a limit cycle. The next Hopf bifurcation produces a phase flow tliat can be represented on tire surface of a toms (douglmut). This flow is associated witli two frequencies if tire ratio of tliese frequencies is irrational tlien tire toms surface is densely covered by tire phase trajectory, whereas if... 

The scientific method, as mentioned, involves observation and experimentation (research) to discover or establish facts. These are followed by deduction or hypothesis, establishing theories or principles. This sequence, however, may be reversed. The noted twentieth-century philosopher Karl Popper, who also dealt with science, expressed the view that the scientist s work starts not with collection of data (observation) but with selection of a suitable problem (theory). In fact, both of these paths can be involved. vSignificant and sometimes accidental observations can be made without any preconceived idea of a problem or theory and vice versa. The scientist, however, must have a well-prepared, open mind to be able to recognize the significance of such observations and must be able to follow them through. Science always demands rigorous standards of procedure, reproducibility, and open discussion that set reason over irrational belief. 

Science develops ever more rigorous standards of procedure and evaluation for setting reason aside from irrational belief. Flowever,... 

Round-off errors. These are the consequence of using a number specified by m correct digits to approximate a number which requires more than m digits for its exacd specification. For example, approximate the irrational number V2 by I.4I4. Such errors are often... 

The somewhat irrational reason which propelled us into this endeavor was the questioning of Charlie Morgan who often asked if one or another obscure compound had been synthesized. As often as not, we were uncertain and now at least have a reference which, if not absolutely complete, will at least save face. 

R. Baumann, K. Kassner, C. Misbah, D. E. Temkin. Spatial subharmonics, irrational patterns, and disorder in eutectic growth. Phys Rev Lett 74 1597, 1995. 

J. M. Rawson and J. J. Longridge, Sulfur-Nitrogen Chains Rational and Irrational Behaviour, Chem. Soc. Rev., 53 (1997). 

These outrage factors arc not distortions in the public s perception of risk. They are inborn parts of what is interpreted as risk. They are explanations of why the public fears pollutants in the air and water more tlum tliey do geological radon. The problem is that many risk experts resist tlie use of the public s "irrational fear" in their risk management. 

Unsinn, m. nonsense, absurdity, tin-sinnig <2. nonsensical irrational insane, -spaltbar, a. uncleavable. -stabU, a. im-stable. -stat, -statig, etc., see unstet, un-stetig, etc. -statthaft, a. inadmissible illicit, -stet, -stetig, a. unsteady, unstable, variable, discontinuous. 

Tt is not difficult to show that such binary expansions - in fact expansions to an arbitrary base 6 > 1 - are complete in the unit interval see I. Niven, Irrational Numbers , The Cams Mathematical Monographs 11 (1956). 

What of the properties of th actual orbit of xqI Since / effectively reads off the digits in the binary expansion of xq, the properties of the orbit depend on whether Xq is itself rational or irrational. 

Xo is Irrational Using the same argument as given above, orbits for irrational Xq must be nonperiodic, with the attractor in this case being the entire interval. Because any finite sequence of digits appears infinitely many times within the binary decimal representation of almost all irrational numbers in [0,1] (all except for a set of measure zero), the orbit of almost all irrationals approaches any x G [0,1] to within a distance e << 1 an infinite number of times i.e., the Bernoulli shift is ergodic. 

Nesting of KAM-tori For two-degree-of-freedom system, the KAM theorem states that for sufficiently weak perturbations, all sufficiently irrational KAM-tori are pre-... 

Sufficiently irrational tori here means those tori whose frequency ratio uj /u>2 is such that... 

At the risk of oversimplifying, there are essentially three different dynamical regimes of the one-dimensional circle map (we have not yet formed our CML) (I) j A < 1 - for which we find mode-locking within the so-called AmoW Tongues (see section 4.1.5) and the w is irrational (11) k = 1 - for which the non mode-locked w intervals form a self-similar Cantor set of measure zero (111) k > 1 - for which the map becomes noninvertible and the system is, in principle, ripened for chaotic behavior (the real behavior is a bit more complicated since, in this regime, chaotic and nonchaotic behavior is actually densely interwoven in A - w space). 

Every attempt to employ mathematical methods in the study of chemical questions must he considered profoundly irrational and contrary to the spirit of chemistry. .. If mathematical analysis should ever hold a prominent place in chemistry — an aberration which is happily almost impossible — it would occasion a rapid and widespread degeneration of that science. 

This will demystify the nature of mentoring - a vital step in avoiding the build-up of irrational fears and jealousy. 

The system is regarded as commensurate if there is a rational value for the ratio a=al2iT=L/N, where a denotes the mean space of the particles, 27t is the potential period, and L indicates the number of potential periods within the calculation window. By using different combinations of L and N, one may be able to alter the commensurability of the system, which will become incommensurate when a approaches an irrational value. The golden mean, a=( /5- l)/2 144/233, for example, is usually chosen to characterize a typical incommensurate case. 

Solid contacts are incommensurate in most cases, except for two crystals with the same lattice constant in perfect alignment. That is to say, a commensurate contact will become incommensurate if one of the objects is turned by a certain angle. This is illustrated in Fig. 30, where open and solid circles represent the top-layer atoms at the upper and lower solids, respectively. The left sector shows two surfaces in commensurate contact while the right one shows the same solids in contact but with the upper surface turned by 90 degrees. Since the lattice period on the two surfaces, when measured in the x direction, are 5 3 A and 5 A, respectively, which gives a ratio of irrational value, the contact becomes incommensurate. 

The division of two numbers of s significant digits each can yield results that require nearly any number of digits (only a few if one number is a simple multiple of the other, many if one or both numbers are irrational) usually the number of bytes the computer displays or uses for calculation... 

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