Nonlocal character

In this section we derive a nonpenetration condition between crack faces for inclined cracks in plates and discuss the equilibrium problem. As it turns out, the nonpenetration condition for inclined cracks is of nonlocal character. This means that by writing the condition at a fixed point we have to take into account the displacement values both at the point and at the other point chosen at the opposite crack face. As a corollary of this fact, the equilibrium equations hold only in a domain located outside the crack surface projection on the mid-surface of the plate. This section follows the papers (Khludnev, 1997b Kovtunenko et ah, 1998). 

Thus, the mutual nonpenetration condition between the crack faces is described by the inequalities (3.173), (3.176). The inequalities have a nonlocal character in particular, they contain values of the functions both at the point X and the point y = Px moreover the last values (i.e. at the point y = Px) are taken at the opposite crack faces. 

Another point of considerable importance is the nonlocal character of the exchange. The latter always favours non-uniform distributions of electrons (or holes) among the different d orbitals, e.g. eg and t2g orbitals, when the system is cubic. Direct Coulomb interactions as well as correlations favour uniform occupations of the different atomic orbitals and therefore counteract the effect of nonlocality of the exchange. Despite this, the anisotropies caused by exchange are important, in particular for bulk Ni (10) as well as for its surface (15). 

Tens of thousands of ceramic sherds have been excavated at the site, documenting the extensive use of pottery for cooking, storage, and transport of materials. Some of the ceramics also have a distinctively nonlocal character seen in the decoration of the pottery and in the temper used in the ceramic paste. Grit, bone, and lunestone are used as temper in the ceramics with exotic decoration, and not in the clearly local... 

The nonlocal character of the exchange operator in the HF approximation increases a lot the time needed for a self-consistent calculation relative to that of a local operator. For this purpose, a method was developed where one produces the one-particle orbitals by an Sldet satisfying an equation of the form... 

The account of the nonlocal character of the dielectric response modifies the Fresnel equations. In the framework of the jellium model for a metal, the reflection coefficients in s- and p-polarizations can be represented as °... 

However, the electronic states close to the Fermi energy contributing to the density of states in QCs seem to have a mixed nature. Electronic states in periodic crystals can be described by the Bloch theorem, that is, yielding delocahzed bulk states along with surface and surface resonance states. Angle-resolved photoemission has been used to probe the details of surface electronic structure of low-index surfaces of d-Al-Ni-Co and i-Al-Pd-Mn [187-190]. These experiments revealed a nearly-free-electron-hke dispersion in the region of s-p-derived states, indicating a nonlocalized character. However, the experiments could not determine whether localized states were present in addition to the nonlocalized ones. 

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