Local shape complexity

The global and local shape codes can be used for measuring global and local shape compexity, respectively. Let w(s(a,b,M)) and w(lli)(a,b,M)) denote the number of different entries of the n-dimensional global shape matrix s(a,b,M) and a k-dimensional local shape matrix lb(a,b,M), respectively. Simple global and local shape complexity measures of molecule M are defined as the following ratios ... 

These results concern the theoretical basis of chemistry. However, they are also of relevance to the application of local shape analysis and to the subsequent use of numerical local shape descriptors in correlations with various, not-well-understood experimental results, such as biochemical activities in complex biochemical systems, potency data in pharmaceutical drug design approaches, experimental toxicities in toxicological risk assessments, etc. [27-36]. 

Localization of complex 53 was also recently carried out by AFM-IR in hormone-dependent MCF-7 breast cancer cells [79]. In this case the cells, cultivated on Cap2 windows then deposited on the prism, also of Cap2, of the AFM-IR apparatus, have the expected shape for epithelial cells. The resolution of the AFM-IR images confirms unambiguously the perinuclear localization of the complex. 

The structural entropy introduced here as a localization quantity characteristic of the decay of the distribution function is related to the shape complexity as InCLMc-... 

Lopez-Ruiz R, Mandni HL, Calbet X (1995) Phys Lett A 209 321 Pipek J, Varga 1, Nagy T (1990) Int J Quantum Chem 37 529 Pipek J, Varga I (1992) Phys Rev A 46 3148 Pipek J, Varga I (2002) Phys Rev E 68 026202. The structural entropy Sstr introduced independently as a localization quantity characteristic of the decay of the distribution function is related to the shape complexity as In C... 

Wlrile tire Bms fonnula can be used to locate tire spectral position of tire excitonic state, tliere is no equivalent a priori description of the spectral widtli of tliis state. These bandwidtlis have been attributed to a combination of effects, including inlromogeneous broadening arising from size dispersion, optical dephasing from exciton-surface and exciton-phonon scattering, and fast lifetimes resulting from surface localization 1167, 168, 170, 1711. Due to tire complex nature of tliese line shapes, tliere have been few quantitative calculations of absorjDtion spectra. This situation is in contrast witli tliat of metal nanoparticles, where a more quantitative level of prediction is possible. 

A locally one-dlinensional scheme (LOS) for the heat conduction equation in an arbitrary domain. The method of summarized approximation can find a wide range of application in designing economical additive schemes for parabolic equations in the domains of rather complicated configurations and shapes. More a detailed exploration is devoted to a locally one-dimensional problem for the heat conduction equation in a complex domain G = G -f F of the dimension p. Let x — (sj, 2,..., a- p) be a point in the Euclidean space R. ... 

The finite-element method (FEM) is based on shape functions which are defined in each grid cell. The imknown fimction O is locally expanded in a basis of shape fimctions, which are usually polynomials. The expansion coefficients are determined by a Ritz-Galerkin variational principle [80], which means that the solution corresponds to the minimization of a functional form depending on the degrees of freedom of the system. Hence the FEM has certain optimality properties, but is not necessarily a conservative method. The FEM is ideally suited for complex grid geometries, and the approximation order can easily be increased, for example by extending the set of shape fimctions. 

The electrochemical machining (ECM) of metals rests on the selective local anodic dissolution of metal. It is used to give metal parts the required shape and size, to drill holes, create hollows, cut shaped slots, and fashion parts of a complex pattern (e.g., the blades of gas turbines). It is an advantage of this method that it can also be used for hard metals (high-alloy steels and other alloys, metals in the quenched state, etc.). 

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