Steiner problem

These minimization problems were first investigated early in the nineteenth century by the mathematician Jacob Steiner They are often known collectively as the Steiner problem. The general problem of linking n points has not been solved analytically. It was the mathematician Richard Cour-anti4 i5 who first popularized the analogue solutions to these problems in the 1940 s. 

A Geometrical Proof of the Three Point Steiner Problem... 

In the Steiner problems, examined in Chapter 3, the assumption concerning continuous gradients is not justified. Most of the solutions contain intersections with gradients that change discontinuously. This will also be true of the majority of the problems to be discussed in section 4.5 and so it will not be possible to apply the Euler-Lagrange equations. 

A clue to the shape of the minimum surface can be obtained by recalling the solution to the two dimensional cube the solution to the Steiner problem for four points arranged at the corners of a square (Fig. 3.5). The configuration contains a line, not a point, at the centre. So the cubic framework might be expected to have a square, rather than a line, at the centre. 

When the height of the prism is sufficiently great a central horizontal cross-section of the film is similar to that produced by three pins separating two parallel perspex plates which was discussed in Chapter 3. For prisms with bases in the form of polygons one obtains minimum surfaces with a central horizontal section, at the half height of the prism, that are similar to those produced by the polygonal arrangement of pins in the solution of the Steiner problem. 

In the Steiner problem, Chapter 3, and the minimum surface area problems, discussed earlier in this chapter, the Laplace-Young equation had a simple form. This resulted from the zero excess pressure across any point on the surface of the soap film. In the case of a bubble, or clusters of bubbles, the excess pressure across any surface is not in general zero. However the Laplace-Young equation can be applied under these more general conditions. Plateau s rules concerning the angles at which surfaces and lines of soap films intersect apply also to the surfaces and lines of soap film produced by clusters of bubbles. 

The derivation of Eq. (6.8) assumes that the extremum curve is continuous and has a continuous derivative. Consequently it cannot be used to solve problems which have a discontinuous solution or a discontinuous derivative. Such problems as the Steiner problems and the minimum area problems, discussed in Chapters 3 and 4, can have discontinuous gradients and consequently cannot be solved using the Euler-Lagrange equation. 

M. Steiner, The Applicability of Mathematics as a Philosophical Problem, Harvard University Press, Cambridge, MA, 1998. 

To our knowledge, in the first approach described, Youden and Steiner introduced the term ruggedness test for a setup in which by means of an experimental design the influences of minor but deliberate and controlled changes in the method parameters or factors are evaluated in order to detect non-robust factors, i.e., with a large influence on the response. Controlling the latter factors avoids problems in the following interlaboratory study. 

A long-standing geochemical problem is the source of water in volcanic eruptions and geothermal systems how much is derived from the magma itself and how much is recycled meteoric water One of the principal and unequivocal conclusions drawn from stable isotope studies of fluids in volcanic hydrothermal systems is that most hot spring waters are meteoric waters derived from local precipitation (Craig et al. 1956 Clayton et al. 1968 Clayton and Steiner 1975 Truesdell and Hulston 1980, and others). 

While the underlying mathematical optimization problem, also referred to as Steiner-Weber-problem or minisum problem, is one of the classical models discussed in operations research literature on facility location (cf. Drezner et al. 2001), it is much too abstract to be of real value to actual industrial location decisions (cf. Gotze 1995, p. 56). A general criticism of Weber s theory can be found in Behrens (1971, pp. 15-19) and Meyer-Lindemann (1951, pp. 55-67). 

The majority of the materials with low flame spread (or low heat release) also exhibit low smoke release. However, it has been shown in several series of room-corner test projects (with the tested material lining either the walls or the walls and the ceiling), that -10% of the materials tested (8 out of 84) exhibited adequate heat-release (or fire growth) characteristics, but have very high smoke release (Table 21.17 and Figure 21.16).189190 These materials would cause severe obscuration problems if used in buildings. A combination of this work, and the concept that a visibility of 4 m is reasonable for people familiar with their environment,191 has led all the U.S. codes to include smoke pass/fail criteria when room-corner tests are used as alternatives to the ASTM E 84 Steiner tunnel test. 

The use of DMSO in recent studies has been largely upon the premise that the problem of base association would be avoided. Some doubt as to the validity of this assumption arises when the results of a conductometric study are considered (Exner and Steiner, 1974). Ion-pairing constants for lithium, sodium, potassium and cesium t-butoxide in DMSO have been evaluated as 108, 106, 270 and 200 M-1 respectively. Not only do these results suggest that there is base association in DMSO but they also imply that base-catalysed reactions involving alkali metal alkoxides in DMSO should be affected by the nature of the cation. If these conclusions are valid and if the possible involvement of the dimsyl anion in these reactions is also taken into consideration, then the choice of DMSO to remove the problem of base association can be a poor one, especially if the base is a lithium or sodium salt of a hindered alkoxide. It is far better to avoid association effects by the use of crown ethers (Bartsch et al., 1973, 1974, 1975). On the other hand, the use of lithium and potassium t-butoxide in DMSO solvent might aid in distinguishing reactivities of free ions and of ion pairs in certain processes. 

But something does seem to have struck one of our Holocausters as odd. Jean-Fran9ois Steiner vividly describes the problem resulting from the actual, enormous wood (fuel) requirements involved in cremation ... 

This test has been criticized because it does not simulate actual building fire conditions [5,6], An additional problem with foamed samples is that the specimens either retract out of the reach of the flame or drip on to the floor of the tunnel. In Canada this has been addressed by using a downward-facing burner and mounting the specimens on the floor of the tunnel. Despite its limitations, the Steiner Tunnel method continues to be used to test and rate thermoplastic foams. 

Perform Detailed Design Calculations, This subject has been dealt with in vatying degrees of complexity by several authors. The fitsl treatment of the problem was by Weller and Steiner in 1950, A... 

Steiner, M., and H. Hillemanns. 1990. Venostasin retard in the management of venous problems during pregnancy. Phlebdogy 5 41-44. 

Multi-reference Brillouin-Wigner theory overcomes the intruder state problem because the exact energy is contained in the denominator factors. Calculations are therefore state specific , that is they are performed for one state at a time. This is in contrast to multi-reference Rayleigh-Schrddinger perturbation theory which is applied to a manifold of states simultaneously. Multi-reference Brillouin-Wigner perturbation theory is applied to a single state. Wenzel and Steiner [105] write (see also [106]) ... 

Chapter 4 is about the comparison and choice of modern HPLC columns Stefan Lamotte, Stavros Kromidas, and Frank Steiner give an overview of different columns and come forward with proposals for pragmatic tests for columns as well as column portfolios, depending on the separation problem. 

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