Global curvature

As shown in the figure, the curvature of the entropy function always causes it to fall below its tangent planes. A mathematical object having such distinctive global curvature (such as an eggshell or an upside-down bowl) is called convex. Accordingly, we may restate the Gibbs criterion in terms of this intrinsic convexity property of the entropy function S = S(U,V,N) ... 

In principle, the integrand in (13.10) might be evaluated with Taylor series expansions such as (12.96), based on successively higher derivatives of the initial state. In practice, however, direct experimental evaluation of the functional dependence of each My on path variables would be needed to evaluate C along extended paths. Further discussion of global curvature or other descriptors of the Riemannian geometry of real substances therefore awaits acquisition of appropriate experimental data, well beyond that required to describe individual points on a reversible path. 

Heiden and Brickmann. The surface topography index s may be defined on the basis of two global curvatures, c, and C2, as follows ... 

The STI values vary within the interval 0 < 5 < 4. When calculated from the relation of both global curvatures [each of which can be either concave ( + ), flat (0), or convex (-)], where c, > C2, the STI gives an expression for the regional shape of every surface point. The shape varies continuously among five basic shape descriptors, namely, bag ( + / + ), cleft (+ /O), saddle ( + /-), ridge (0/ - ), knob (-/-), and as a special case, plateau (0/0). However, information about the absolute curvature is lost during the process of STI calculation. 

A theorem proved by O. Gonzalez and J. Maddocks (Global curvature, thickness, and the ideal shapes of knots. Proceedings ofthe National Academy of Sciences, USA, 96 [1999], 4769) in the context of knot topologies shows the remarkable connection between the three-point recipe and the tube thickness. 

Figure 8. Global curvature of CGCGAATTCGCG simulated with and without a CIS, syn dimer at T7-T8. (Reproduced with permission from ref. 29. Copyright 1996 American Chemical Society)...

Example 3.10 If a moment of Af, = 100 Nm/m is applied to the unidirectional composite described in the previous Example, calculate the curvatures which will occur. Determine also the stress and strain distributions in the global and local (1-2) directions. 

Example 3.12 For the laminate [0/352/ - determine the elastic constants in the global directions using the Plate Constitutive Equation. When stresses of = 10 MN/m, o-y = —14 MN/m and = —5 MN/m are applied, calculate the stresses and strains in each ply in the local and global directions. If a moment of 10(X) N m/m is added, determine the new stresses, strains and curvatures in the laminate. The plies are each 1 mm thick. 

Example 3.16 A unidirectional carbon hbre/PEEK laminate has the stacking sequence [O/SSa/—352]t- If it has an in-plane stress of = 100 MN/m applied, calculate the strains and curvatures in the global directions. The properties of the individual plies are... 

The surface dividing the components of the mixture formed by a layer of surfactant characterizes the structure of the mixture on a mesoscopic length scale. This interface is described by its global properties such as the surface area, the Euler characteristic or genus, distribution of normal vectors, or in more detail by its local properties such as the mean and Gaussian curvatures. 

In the standard approaches to the systems in which monolayers or bilayers are formed, one assumes that the width of the film is much smaller than the length characterizing the structure (oil or water domain size, for example). In such a case it is justified to represent the film by a mathematical surface and the structure can be described by the local invariants of the surface, i.e., the mean H and the Gaussian K curvatures and by the global (topological)... 

Hartree-Fock calculations of the three leading coefficients in the MacLaurin expansion, Eq. (5.40), have been made [187,232] for all atoms in the periodic table. The calculations [187] showed that 93% of rio(O) comes from the outermost s orbital, and that IIo(O) behaves as a measure of atomic size. Similarly, 95% of IIq(O) comes from the outermost s and p orbitals. The sign of IIq(O) depends on the relative number of electrons in the outermost s and p orbitals, which make negative and positive contributions, respectively. Clearly, the coefficients of the MacLaurin expansion are excellent probes of the valence orbitals. The curvature riQ(O) is a surprisingly powerful predictor of the global behavior of IIo(p). A positive IIq(O) indicates a type 11 momentum density, whereas a negative rio(O) indicates that IIo(O) is of either type 1 or 111 [187,230]. MacDougall has speculated on the connection between IIq(O) and superconductivity [233]. 

Copper ions have been reduced in colloidal assemblies differing in their structures (55,56). In all cases, copper metal particles are obtained. Figure 9.3.1 shows the freeze-fracture electron microscopy (FFEM) for the various parts of the phase diagram. Their structures have been determined by SAXS, conductivity, FFEM, and by predictions of microstructures that require only notions of local curvature and local and global packing constraints. 

In turn, in geometry A plays the role of connection (it defines the parallel transport around C) and F is the curvature of this connection. A global version of the Abelian Stokes theorem... 

Figure 15.7 is intended to show conceptually some important mathematical aspects of a scale-disparate, nonlinear problem. The lines are contours of a norm of the residual vector F in a two-dimensional space. The position on the figure (i.e., cartesian coordinates) represents the y vector. As illustrated, the residual norm is characterized by a long, narrow, valley. The solution, which is a global minimum in the residual norm, lies near one end of the valley. In a highly scale-disparate problem, the valleys are greatly elongated and the valley walls are very steep. Nonlinearity is represented by curvature of the valleys. 

Capillary forces in mixed fluid phase conditions are inversely proportional to the curvature of the interface. Therefore, menisci introduce elasticity to the mixed fluid, and mixtures of two Newtonian fluids exhibit global Maxwellian response. For more details see Alvarellos [1], his behavior is experimentally demonstrated with a capillary tube partially filled with a water droplet. The tube is tilted at an angle (3 smaller than the critical angle that causes unstable displacement. Then, a harmonic excitation is applied to the tube in the axial direction. For each frequency, the amplitude of the vibration is increased until the water droplet becomes unstable and flows in the capillary. Data in Figure 3 show a minimum required tube velocity between 40 and 50 Hz. This behavior indicates resonance of the visco-elastic system. The ratio of the relaxation time and characteristic time for pure viscous effect is larger than 11.64. 

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