The local structure of

Let k be a (not necessarily algebraically closed) field and X a smooth quasipro-jective variety of dimension d over k. In this section we study the structure of the stratum (Xj j)re(j which parametrizes subschemes of X which are concentrated in a (variable) point in X. 

Definition 2.1.1. Let X be a smooth projective variety over a field k. Let A C XxXbe the diagonal and 2 /xxx its ideal. Let A C X x X be the closed subscheme which is defined by I /XxX4 Let 

Proof Let k be an algebraic closure of k and X = X x k. Let Z C X be a subscheme of length n of X concentrated in a point, Iz its ideal in the local ring Ox,x and mx,z the maximal ideal of Ox,x- Then we have Iz D x (cf. 1.3.2). So we see that Hilbn(An/X)red and (Jf )re i axe closed subschemes of -Xlnl with the reduced induced structure, which have the same geometric points. Thus they are equal. The assertion on 7r follows directly from the definitions.  

Lemma 2.1.4. n (X j)red — X is a locally trivial fibre bundle in the Zariski topology with fibre Hilbn(J )red. 

Z f ( t1) s a l°ca y trivial fibre bundle with fibre Ar over 

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Figure A2.4.3. The localized structure of a hydrated metal cation in aqueous solution (the metal ion being assumed to have a primary hydration number of six). From [5].

A multitude of different variants of this model has been investigated using Monte Carlo simulations (see, for example [M])- The studies aim at correlating the phase behaviour with the molecular architecture and revealing the local structure of the aggregates. This type of model has also proven useful for studying rather complex structures (e.g., vesicles or pores in bilayers). 

In complexes with Cro, the overall bend and twist of the DNA are similar to those in the repressor complexes, but there is a significant difference in the local structure of two of the nucleotides in each half-site. Binding of 434 Cro or repressor fragment thus imposes a distinct local structure (Figure 8.13), as a result of differences in both the identity and conformations of various amino acid residues that interact with the DNA. The DNA conformational details are significant for the relative affinities of Cro and repressor for various sites, as we describe in a later section. 

EXAFS is a nondestructive, element-specific spectroscopic technique with application to all elements from lithium to uranium. It is employed as a direct probe of the atomic environment of an X-ray absorbing element and provides chemical bonding information. Although EXAFS is primarily used to determine the local structure of bulk solids (e.g., crystalline and amorphous materials), solid surfaces, and interfaces, its use is not limited to the solid state. As a structural tool, EXAFS complements the familiar X-ray diffraction technique, which is applicable only to crystalline solids. EXAFS provides an atomic-scale perspective about the X-ray absorbing element in terms of the numbers, types, and interatomic distances of neighboring atoms. 

The order parameter field (r) characterizes the local structure of the phases we have investigated. The most interesting is the topology of the phases,... 

The X-ray absorption fine structure (XAFS) methods (EXAFS and X-ray absorption near-edge structure (XANES)) are suitable techniques for determination of the local structure of metal complexes. Of these methods, the former provides structural information relating to the radial distribution of atom pairs in systems studied the number of neighboring atoms (coordination number) around a central atom in the first, second, and sometimes third coordination spheres the... 

The local structure of these complexes and adducts of calf-thymus DNA with different organotin(IV) moieties were determined by means of EXAFS. The... 

Comparison of the Mossbauer I Agxp I values with those calculated on the basis of the pqs formalism revealed that the organotin(IV) moiety has Tbp geometry, and in certain cases geometry too. Some of the complexes contain the organotin(IV) cation in two different environments. Finally, the local structure of the maleic acid complex formed with [Bu2Sn(IV)l was determined by an EXAFS method.The structural data are collected in Table 6. Similar studies... 

A second simplihcation results from introducing the Born-Oppenheimer separation of electronic and nuclear motions for convenience, the latter is most often considered to be classical. Each excited electronic state of the molecule can then be considered as a distinct molecular species, and the laser-excited system can be viewed as a mixture of them. The local structure of such a system is generally described in terms of atom-atom distribution functions t) [22, 24, 25]. These functions are proportional to the probability of Ending the nuclei p and v at the distance r at time t. Building this information into Eq. (4) and considering the isotropy of a liquid system simplifies the theory considerably. 

Herein we present a fully theoretical work based on density functional theory (DFT) enabling the investigation of the local structure of lanthanide-doped compounds, the calculation of... 

The atomic positions were allowed to relax until all forces were smaller than 0.005 eV/A The local structure of the Eu2+ impurity in CsMgBr3 is tackled via geometry optimization based on the cluster approach.40 The optimized lattice structure of CsMgBr3 is used. An appropriate cluster is obtained as a selective cut along the 3-fold c axis of the unit cell of CsMgBr3 (Figure la). A moiety containing five units... 

The EXAFS, which occurs at higher energies above the edge, is due to the interference between the outgoing and the backscattered photoelectron waves (10-14). EXAFS provides information about the local structure of the x-ray absorbing atom. Typically, nearest neighbor bond lengths and coordination numbers can be determined to 0.02 A (1%) and one atom in four (25%) (4 ). The accuracy of these determinations is somewhat worse for outer-shell atoms, for disordered systems, or for systems with asymmetric distributions of atoms within a shell (15,16). 

Conventional HRTEM operates at ambient temperature in high vacuum and directly images the local structure of a catalyst at the atomic level, in real space. In HRTEM, as-prepared catalyst powders can be used without additional sample preparation. The method does not normally require special treatment of thin catalyst samples. In HRTEM, very thin samples can be treated as WPOs, whereby the image intensity can be correlated with the projected electrostatic potential of the crystal, leading to the atomic structural information characterizing the sample. Furthermore, the detection of electron-stimulated XRE in the EM permits simultaneous determination of the chemical composition of the catalyst. Both the surface and sub-surface regions of catalysts can be investigated. 

A refinement of previous hypotheses concerning the local structure of turbulence in viscous incompressible fluid for high Reynolds numbers. Journal of Fluid Mechanics 13, 82-85. 

There are four parameters which must be fixed to use the MLE algorithm Embedding dimension, de, maximum scale, Sm, minimum scale, Sm and evolution time, O. Basically, de is the attractor dimension where the orbits were embedded, Sm is the estimate value of the length scale on which the local structure of the attractor is not longer being proved. Sm is the length scale in which noise is expected to appear. O is fixed for compute of divergence measurements which is the necessary time to renormalize the distances between trajectories (for more details see [50]). 

Due to its unique ability to directly image the local structure of a thin object with atomic resolution, HRTEM is an extremely powerful tool for materials research. Metals, ceramics, and semiconductors are some examples of prominent materials of interest. HRTEM imaging used to be a high-end research tool mostly used in academia, but has now become standard for a wide variety of applications from materials science research to defect analysis in industrial semiconductor fabrication lines. 

Neto and co-workers examined the ex situ Pt L3 EXAFS for a series of PtRu catalyst powders in air of varying nominal composition from 90 10 through to 60 40 atom %. The catalysts were prepared using a formic acid reduction method developed by the authors which resulted in very poorly alloyed particles, even after heat treatment to 300 °C under a hydrogen atmosphere. Unfortunately, the authors were not able to obtain Ru K edge data to identify the local structure of the Ru in their catalysts. 

Shimizu, K Takamatsu, M Nishi, K Yoshida, H Satsuma, A Tanaka, T Yoshida, S Hattori, T. Alumina-supported gallium oxide catalysts for NO selective reduction Influence of the local structure of surface gallium oxide species on the catalytic activity, J. Phys. Chem., B, 1999, Volume 103, Issue 9, 1542-1549. 

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