Isometric graph

Design Methods for Plastics using Deformation Data Isochronous and Isometric Graphs... 

As before, a similar result could have been achieved by taking a section across the creep curves at 1.5% strain, plotting an isometric graph (or a 1.5% modulus/time graph) and obtaining a value for modulus at 1 year (see Fig. 2.8)... 

Abstract. We present a method to construct pairs of isospectral quantum graphs which are not isometric. These graphs are the analogues of the family of isospectral domains in R2 which were first introduced by Gordon, Webb and Wolpert (C. Gordon et.al., 1992), recently enlarged by P. Buser el. al. (P. Buser et.al., 1994), and discussed further by Okada et. al. (Y. Okada et.al., 2001). 

In response to M. Kac s classical paper Can one hear the shape of a drum (M. Kac, 1966), much research effort was invested in two complementary problems - to identify classes of systems for which Kac s question is answered in the affirmative, or to find examples which are isospectral but not isometric. In the present paper we shall focus our attention to quantum graphs and in the following lines will review the subject of isospectrality from this intentionally narrowed point of view. The interested reader is referred to (T. Sunada, 1985 C. Gordon et.al., 1992 S. Chapman, 1995 P. Buser et.al., 1994 Y. Okada et.al., 2001 S. Zelditch, 2004) for a broader view of the field where spectral inversion and its uniqueness are discussed. 

DDS05] M. Deza, M. Dutour, and S. Shpectorov, Graphs 4D that are isometrically embeddable in hypercubes, Bulletin of South-East Asian Mathematical Society 29-3 (2005) 469-484. 

DGS04] M. Deza, V. P. Grishukhin, and M. I. Shtogrin, Scale-Isometric Polytopal Graphs in Hypercubes and Cubic Lattices, World Scientific and Imperial College Press, 2004. 

Figure 2 Dissociation potential vertically, and Z and Z2 from 2 to 36 horizontally, in almost isometric projection. Transition-metal and rare-earth molecules have been cut out, and the resulting pieces of the surface have been shd and joined to pieces at smaller Z. Homonuclear molecules are on the left-right diagonal, and the terrain is symmetrical with respect to a vertical plane through that diagonal. Any series of isoelectronic molecules has addresses going horizontally and normal to that plane. The scale is established by the value for N2 (9.79 eV), which lies between the peaks for CO and OC. The figure is constructed from stick graphs by drawing lines of least descent from each peak, by draping a surface from those lines down to the 0 eV valleys of rare-gas molecules. In addition, an attempt is made to indicate craters in the bottoms of which are found the alkaline-earth pairs.

A conventional creep curve as exhibited by most materials is illustrated in Fig. 2.25 although many engineers present the data using log axes to produce a graph of the form shown in Fig. 2.26. Data from families of strain-time curves at various values of constant stress are used to produce isochronous stress-strain curves (Fig. 2.27). These are obtained by cross-plotting stresses and strains at various times from the commencement of loading. The results of creep tests can also be used to derive constant strain, or isometric, curves of stress versus time, also as illustrated in Fig. 2.27. 

When a graph is truly three-dimensional (two independent variables and one dependent variable), one must be careful to use a perspective that allows the reader to determine easily the height (the dependent axis) at various points. The most useful such perspective is the isometric perspective, which involves rotating the straight-on coordinate system by 45° and then tilting it up in back by 30°. In this perspective, the distances perpendicular to the axes are not distorted. 

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