Formal deformation

Definition 6.17. Two generalized simplicial complexes are said to have the same simple homotopy type if there exists a sequence of elementary collapses and expansions leading from one to the other. Such a sequence is called a formal deformation. 

Proposition 6.18. LetX be the geometric realization of an arbitrary abstract simplicial complex. Then there exists a formal deformation from X to BdX. 

Proof. To start with, since the barycentric subdivision can be represented as a sequence of stellar subdivisions, see Subsection 2.1.5, it is enough to find a formal deformation leading from X to sd(X, [Pg.95]

The discussion in Section 6.5 implies now that the polyhedral complex of all bipartite subgraphs of G, Bip(G), and the neighborhood complex Af G), have the same simple homotopy type, and yields an explicit formal deformation between these two complexes. ... 

Via an ad hoc extension of the viscoelastic Hertzian contact problem, Falsafi et al. [38] incorporated linear viscoelastic effects into the JKR formalism by replacing the elastic modulus with a viscoelastic memory function accounting for time and deformation, K t) ... 

Implicit in all these solutions is the fact that, when two spherical indentors are made to approach one another, the resulting deformed surface is also spherical and is intermediate in curvature between the shape of the two surfaces. Hertz [27] recognized this concept and used it in the development of his theory, yet the concept is a natural consequence of the superposition method based on Boussinesq and Cerutti s formalisms for integration of points loads. A corollary to this concept is that the displacements are additive so that the compliances can be added for materials of differing elastic properties producing the following expressions common to many solutions... 

The existence of yield stress Y at shear strains seems to be the most typical feature of rheological properties of highly filled polymers. A formal meaing of this term is quite obvious. It means that at stresses lower than Y the material behaves like a solid, i.e. it deforms only elastically, while at stresses higher than Y, like a liquid, i.e. it can flow. At a first approximation it may be assumed that the material is not deformed at all, if stresses are lower than Y. In this sense, filled polymers behave as visco-plastic media with a low-molecular and low-viscosity dispersion medium. This analogy is not random as will be stressed below when the values of the yield stress are compared for the systems with different dispersion media. The existence of yield stress in its physical meaning must be correlated with the strength of a structure formed by the interaction between the particles of a filler. 

One is purely formal, it concerns the departure from symmetry of an approximate solution of the Schrodinger equation for the electrons (ie within the Bom-Oppenheimer approximation). The most famous case is the symmetry-breaking of the solutions of the Hartree-Fock equations [1-4]. The other symmetry-breaking concerns the appearance of non symmetrical conformations of minimum potential energy. This phenomenon of deviation of the molecular structure from symmetry is so familiar, confirmed by a huge amount of physical evidences, of which chirality (i.e. the existence of optical isomers) was the oldest one, that it is well accepted. However, there are many problems where the Hartree-Fock symmetry breaking of the wave function for a symmetrical nuclear conformation and the deformation of the nuclear skeleton are internally related, obeying the same laws. And it is one purpose of the present review to stress on that internal link. 

A structure must be designed to resist gross plastic deformation and collapse under all the conditions of loading. The loads to which a process vessel will be subject in service are listed below. They can be classified as major loads, that must always be considered in vessel design, and subsidiary loads. Formal stress analysis to determine the effect of the subsidiary loads is only required in the codes and standards where it is not possible to demonstrate the adequacy of the proposed design by other means such as by comparison with the known behaviour of existing vessels. 

Conventional implementations of MaxEnt method for charge density studies do not allow easy access to deformation maps a possible approach involves running a MaxEnt calculation on a set of data computed from a superposition of spherical atoms, and subtracting this map from qME [44], Recourse to a two-channel formalism, that redistributes positive- and negative-density scatterers, fitting a set of difference Fourier coefficients, has also been made [18], but there is no consensus on what the definition of entropy should be in a two-channel situation [18, 36,41] moreover, the shapes and number of positive and negative scatterers may need to differ in a way which is difficult to specify. 

According to the aspherical-atom formalism proposed by Stewart [12], the one-electron density function is represented by an expansion in terms of rigid pseudoatoms, each formed by a core-invariant part and a deformable valence part. Spherical surface harmonics (multipoles) are employed to describe the directional properties of the deformable part. Our model consisted of two monopole (three for the sulfur atom), three dipole, five quadrupole, and seven octopole functions for each non-H atom. The generalised scattering factors (GSF) for the monopoles of these species were computed from the Hartree-Fockatomic functions tabulated by Clementi [14]. 

Note that the Zeeman interaction for a cubic system results in an isotropic g-value, but the combination with strain lowers the symmetry at least to axial (at least one of the 7 -, 0), and generally to rhombic. In other words, application of a general strain to a cubic system produces a symmetry identical to the one underlying a Zeeman interaction with three different g-values. In yet other words, a simple S = 1/2 system subject to a rhombic electronic Zeeman interaction only, can formally be described as a cubic system deformed by strain. 

The photoinduced absorbance anisotropy in a TPD experiment relaxes according to the same correlation function as in Eq. (4.16).(29) Effects of spatial variations in the excitation and probe beams, and chromophore concentration, have been treated and shown not to alter the final result.(29) NMR dipolar relaxation rates are expressed in terms of Fourier transforms of the correlation functions, 4ji< T2m[fi(0)] T2m[i2(f)]>> where fl(f) denotes the orientation of a particular internuclear vector. In view of Eq. (4.7), these correlation functions are independent of the index m, hence formally the same as in Eq. (4.16). For the analysis of NMR relaxation data, it is necessary also to evaluate Fourier transforms of the correlation functions. Methods to accomplish this in the case of deformable DNAs have been developed and applied to analyze a variety of data.(81 83)... 

A number of different atom-centered multipole models are available. We distinguish between valence-density models, in which the density functions represent all electrons in the valence shell, and deformation-density models, in which the aspherical functions describe the deviation from the IAM atomic density. In the former, the aspherical density is added to the unperturbed core density, as in the K-formalism, while in the latter, the aspherical density is superimposed on the isolated atom density, but the expansion and contraction of the valence density is not treated explicitly. 

The deformation functions, however, must also describe density accumulation in the bond regions, which in the one-center formalism is represented by the atom-centered terms. They must be more diffuse, with a different radial dependence. Since the electron density is a sum over the products of atomic orbitals, an argument can be made for using a radial dependence derived from the atomic orbital functions. The radial dependence is based on that of hydrogenic orbitals, which are valid for the one-electron atom. They have Slater-type radial functions, equal to exponentials multiplied by r1 times a polynomial of degree n — l — 1 in the radial coordinate r. As an example, the 2s and 2p hydrogenic orbitals are given by... 

The coefficients n, have to obey the condition n, f, imposed by Poisson s electrostatic equation, as pointed out by Stewart (1977). The radial dependence of the multipole density deformation functions may be related to the products of atomic orbitals in the quantum-mechanical electron density formalism of Eq. (3.7). The ss, sp, and pp type orbital products lead, according to the rules of multiplication of spherical harmonic functions (appendix E), to monopolar, dipolar, and quadrupolar functions, as illustrated in Fig. 3.6. The 2s and 2p hydrogenic orbitals contain, as highest power of r, an exponential multiplied by the first power of r, as in Eq. (3.33). This suggests n, = 2 for all three types of product functions of first-row atoms (Hansen and Coppens 1978). 

The multipole formalism described by Stewart (1976) deviates from Eq. (3.35) in several respects. It is a deformation density formalism in which the deformation from the IAM density is described by multipole functions with Slater-type radial dependence, without the K-type expansion and contraction of the valence shell. While Eq. (3.35) is commonly applied using local atomic coordinate systems to facilitate the introduction of chemical constraints (chapter 4), Stewart s formalism has been encoded using a single crystal-coordinate system. 

The aspherical density formalism of Hirshfeld is a deformation model with angular functions which are a sum over spherical harmonics. It will be described in more detail in section 3.2.6. All three models have been applied extensively in charge density studies (for a comparison, see Lecomte 1991). 

Hirshfeld (1971) was among the first to introduce atom-centered deformation density functions into the least squares procedure. Hirshfeld s formalism is a deformation model, in which the leading term is the unperturbed IAM density, and the deformation functions are of the form cos" 0jk, where 9jk is the angle between the radius vector r7 and axis k of a set of (n + l)(n + 2)/2 polar axes on each atom /, as defined in Table 3.8 (Hirshfeld 1977). The atomic deformation on atom j is described as... 

When observed structure factors are used, the thermally averaged deformation density, often labeled the dynamic deformation density, is obtained. An attractive alternative is to replace the observed structure factors in Eq. (5.8) by those calculated with the multipole model. The resulting dynamic model deformation map is model dependent, but any noise not fitted by the muitipole functions will be eliminated. It is also possible to plot the model density directly using the model functions and the experimental charge density parameters. In that case, thermal motion can be eliminated (subject to the approximations of the thermal motion formalism ), and an image of the static model deformation density is obtained, as discussed further in section 5.2.4. 

Following Pethica and Sutton (1988), the mechanical loop of STM responses to the force and exhibits an elastic deformation. By formally introduce an elastic constant a, the deformation is... 

The third chapter is about finite fiat group schemes. A good understanding of finite flat group schemes will give insight in the local structure of moduli spaces of abelian varieties. Indeed, one can show that the natural map Def(A, A) — Def(Ker(A),e ) from the deformation space of a polarized abelian variety to the deformation space of the kernel of A with its canonical pairing is formally smooth (at least if the characteristic of the base field is not equal to 2). A similar... 

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