Hyperbolic universe

A direct consequence of this feature of the hyperbolic heating schedule is that the relative desorption rates n = Nt/Nm are also universal functions of y, and their shape depends on the desorption order only. They read ... 

In estimating the value of Ed by means of the transcendental equations (28), the circumstance utilized is that the variation of em for a given change in Tm is much less than the variation of exp(em) (31). Until now, only particular solutions have been available for the hyperbolic and linear heating schedules and for the first-order and second-order desorptions. They can be found for example in the fundamental papers by Redhead (31) and Carter (32) or in the review by Contour and Proud homme (106), and therefore will not be repeated here. Recently, a universal procedure for the... 

The third independent possible estimation of Ed is given by an analysis of the peak shape. To this end, the hyperbolic heating schedule is most convenient. According to Eq. (38), the width w of the peak at a given value of the relative rate of desorption n and for a given desorption order is a universal constant in the e-scale w(n) = ej — h = f(n), which leads directly to... 

So far we have established an estimate for the rate of convergence in a very simple problem. It is possible to obtain a similar result for this problem by means of several other methods that might be even much more simpler. However, the indisputable merit of the well-developed method of energy inequalities is its universal applicability it can be translated without essential changes to the multidimensional case, the case of variable coefficients, difference schemes for parabolic and hyperbolic equations and other situations. 

It will be shown in another chapter that a possible type of distribution generated by certain processes is of the hyperbolic type (see Chapter 22). The number of particles increases with decreasing size until we reach the dimensions of molecules, atoms, and electrons. There appears to be no definite limit to which material may be subdivided. As the particles increase in size they become relatively fewer in number. The hyperbolic curve is a close approximation of the subdivision of matter in the universe. 

How do these unconventional ideas link with the standard view of a solid as a close packed array of atoms Evidently most of the frameworks discussed above cannot be so characterised. The two-dimensional hyperbolic picture does break down for very dense structures. Thus the densest four-coordinated silicate, coesite, violates this universality (see Fig. 2.12). (Its ring size is less than that of trid5m[ ite, cristobalite, keatite or quartz, in spite of its higher density.) This polymorph is too dense for a two-dimensional description to be useful and the Aree-dimensional description takes over. The notion of intrinsic curvature is less rigid for silicates than for the other frameworks, because the Si-O-Si angle usually differs from 180 . 

It is clear from the universal diagram (Fig. 4.11) that a variety of bilayer phases can form only if die surfactant parameter is between about 0.5 and 1.7. For higher values of the surfactant parameter, steric constraints e.g. head-group crowding) preclude the formation of curved hyperbolic bilayers or monolayers. The opportunity for bilayer pol)rmorphism exists for surfactant parameters l3dng between 1.0 and about 1.5. Iliese bilayer phases are expected to adopt cubic or rhombohedral symmetries, corresponding to the most homogeneous three-periodic minimal surfaces. 

The information contents must always be documented. Exaggerated statements are not admissible, and nor may universal or hyperbolic assertions or indemonstrable comparisons with no objective basis be made. 

Leveque, R. 2002 Finite Volume Methods for Hyperbolic Problems. Cambridge Cambridge University Press. 

The four-dimensional world, which we have for the sake of symmetry represented as spherical, is in reality hyperbolic, and consists of two sheets, which are only connected with each other at infinity. The formulae embrace both sheets, but only one of them represents the actual universe."... 

General relativity goes no further than this. The extent of curvature or the topology of space-time depends on additional observations and criteria. Assumptions to simplify the mathematics, but not supported by testable data, must inevitably produce spurios results. Examples of such simplifications include the assumption of universal time or Euclidean space-time. Einstein s thought experiment which established the non-Euclidean nature of space-time also predicts the nature of the curvature as positive, which excludes hyperbolic space. 

Movable hyperbolic tree of plant phylogeny, devised by Deep Green and hosted by the University of California, Berkeley... 

Figure 5.10 Bundle center temperature prediction for oven temperature profile with hyperbolic stages. Note The smooth curve is the hyperbola approximated by the given temperature ramps indicated by the dashed line. Source Reprinted from Heine M, PhD Dissertation, University of Karlsruhe, Germany, 1988.

FIGURE 18 (a) Practical thermal secondary concentrator referred to as the trumpet, which has been built and tested at the University of Chicago, (b) Profiles for a DCPC and two dielectric compound hyperbolic concentrators (DCHCs) used for photovoltaic secondaries. 

LeVeque RJ. Finite volume methods for hyperbolic problems. Cambridge Cambridge University Press 2002. 

A mass filter that separates ions according to their trajectories in a two-dimensional hyperbolic electrostatic field with DC and radiofrequency AC components. Ions with stable trajectories transit the filter, while other ions discharge on the electrodes. Unlike a quadrupole ion trap, the Chan Hee Chon, Dongqing Li Department of Mechanical Engineering, Vanderbilt University, Nashville, TN, USA Chanhee.chon vanderbilt.edu... 

The FD relation is characterized by a spectral density function and a temperature-dependent hyperbolic tangent factor that decreases with temperature. At 0 K, this factor merges into that for a harmonic bath, implying the universality of the nature of the bath as it approaches absolute zero. 

Bykov, V. V. [1978] On the structure of a neighborhood of a separatrix contour with a saddle-focus, in Methods of Qualitative Theory of Differential Equation (Gorky Gorky State University), 3-32 [1980] On bifurcations of dynamical systems with a separatrix contour containing a saddle-focus, ibid. 44-72 [1988] On the birth of a non-trivial hyperbolic set from a contour formed by separatrices of a saddle, ibid. 22-32. 

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