Shape regularity

Bloom, E. D. and R. C. Erdmann, 1980, The Observation of a Universal Shape Regularity in Earthquake Frequency - Magnitude Distributions, Bull. Seismol. Soc. Am. 70 p a49. 

Sliwka, J., Wasilewicz-Flis, I., Jakuczun, H., Gebhardt, C. (2008). Tagging quantitative trait loci for dormancy, tuber shape, regularity of tuber shape, eye depth and flesh colour in diploid potato originated from six Solanum species. Plant Breeding, 127, 49-55. 

Secondly, the linearized inverse problem is, as well as known, ill-posed because it involves the solution of a Fredholm integral equation of the first kind. The solution must be regularized to yield a stable and physically plausible solution. In this apphcation, the classical smoothness constraint on the solution [8], does not allow to recover the discontinuities of the original object function. In our case, we have considered notches at the smface of the half-space conductive media. So, notche shapes involve abrupt contours. This strong local correlation between pixels in each layer of the half conductive media suggests to represent the contrast function (the object function) by a piecewise continuous function. According to previous works that we have aheady presented [14], we 2584... 

The label liquid crystal seems to be a contradiction in tenns since a crystal cannot be liquid. However, tire tenn refers to a phase fonned between a crystal and a liquid, witli a degree of order intennediate between tire molecular disorder of a liquid and tire regular stmcture of a crystal. Wlrat we mean by order here needs to be defined carefully. The most important property of liquid crystal phases is tliat tire molecules have long-range orientational order. For tliis to be possible tire molecules must be anisotropic, whetlier tliis results from a rodlike or disclike shape. 

Methane, CH4, for example, has a central carbon atom bonded to four hydrogen atoms and the shape is a regular tetrahedron with a H—C—H bond angle of 109°28, exactly that calculated. Electrons in a lone pair , a pair of electrons not used in bonding, occupy a larger fraction of space adjacent to their parent atom since they are under the influence of one nucleus, unlike bonding pairs of electrons which are under the influence of two nuclei. Thus, whenever a lone pair is present some distortion of the essential shape occurs. 

When the ammonium ion NH is formed the lone pair becomes a bonding pair and the shape becomes a regular tetrahedron. 

Benzene is planar and its carbon skeleton has the shape of a regular hexagon There IS no evidence that it has alternating single and double bonds As shown m Figure 111 all the carbon-carbon bonds are the same length (140 pm) and the 120° bond angles... 

Another simple way of comparing the shape of the isotherm on a given solid with that on a reference sample is to read olT the ordinates of the two isotherms at regular intervals of relative pressure, calculate the value / of their ratio, and plot / against p/p°. Changes in the shape of the isotherm on the given solid from that on the reference sample will show up as deviations of the /-plot from the horizontal. 

The crack shape is defined by the function -ip. This function is assumed to be fixed. It is noteworthy that the problems of choice of the so-called extreme crack shapes were considered in (Khludnev, 1994 Khludnev, Sokolowski, 1997). We also address this problem in Sections 2.4 and 4.9. The solution regularity for biharmonic variational inequalities was analysed in (Frehse, 1973 Caffarelli et ah, 1979 Schild, 1984). The last paper also contains the results on the solution smoothness in the case of thin obstacles. As for general solution properties for the equilibrium problem of the plates having cracks, one may refer to (Morozov, 1984). Referring to this book, the boundary conditions imposed on crack faces have the equality type. In this case there is no interaction between the crack faces. 

The nonpenetration condition considered in this section leads to new effects such as the appearance of interaction forces between crack faces. It is of interest to establish the highest regularity of the solution up to the crack faces and thus to analyse the smoothness of the interaction forces. The regularity of the solution stated in this section entails the components of the strain and stress tensors to belong to in the vicinity of the crack and the interaction forces to belong to T. If the crack shape is not regular, i.e. 0 1), the interaction forces can be characterized by the nonnegative... 

Property Cuprammo-niu m Regular rayon Improved rayon Modal Polynosic Y-Shaped a rayon Solvent-spun b rayon... 

Because of the diversity of filler particle shapes, it is difficult to clearly express particle size values in terms of a particle dimension such as length or diameter. Therefore, the particle size of fillers is usually expressed as a theoretical dimension, the equivalent spherical diameter (esd), ie, the diameter of a sphere having the same volume as the particle. An estimate of regularity may be made by comparing the surface area of the equivalent sphere to the actual measured surface area of the particle. The greater the deviation, the more irregular the particle. 

CellgeometTy is governed predominantly by the final foam density and the external forces exerted on the cellular stmcture prior to its stabilization in the expanded state. In a foam prepared without such external forces, the cells tend to be spherical or ellipsoidal at gas volumes less than 70—80% of the total volume, and they tend toward the shape of packed regular dodecahedra at greater gas volumes. These shapes have been shown to be consistent with surface chemistry arguments (144,146,147). Photographs of actual foam cells (Fig. 2) show a broad range of variations in shape. 

These local stmctural rules make it impossible to constmct a regular, periodic, polyhedral foam from a single polyhedron. No known polyhedral shape that can be packed to fiU space simultaneously satisfies the intersection rules required of both the films and the borders. There is thus no ideal stmcture that can serve as a convenient mathematical idealization of polyhedral foam stmcture. Lord Kelvin considered this problem, and his minimal tetrakaidecahedron is considered the periodic stmcture of polyhedra that most nearly satisfies the mechanical constraints. 

A unit referred to as the Higgins Loop has been popular ia water treatmeat, as well as other appHcatioas. Resia is pulsed at regular iatervals arouad a rectangularly shaped loop. The diameter of the adsorption section is larger than that of the regeneration section. 

For skin packaging, ionomer films provide outstanding protection, especially for sharp objects. Films can be drawn down tightly around regular shapes without penetration. Characteristics of heat seals can be adjusted so that convenient peelable seals are obtained (31—35). 

The characteristics of a powder that determine its apparent density are rather complex, but some general statements with respect to powder variables and their effect on the density of the loose powder can be made. (/) The smaller the particles, the greater the specific surface area of the powder. This increases the friction between the particles and lowers the apparent density but enhances the rate of sintering. (2) Powders having very irregular-shaped particles are usually characterized by a lower apparent density than more regular or spherical ones. This is shown in Table 4 for three different types of copper powders having identical particle size distribution but different particle shape. These data illustrate the decisive influence of particle shape on apparent density. (J) In any mixture of coarse and fine powder particles, an optimum mixture results in maximum apparent density. This optimum mixture is reached when the fine particles fill the voids between the coarse particles. 

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