Structure sheaves

The four-lobe pattern may arise from the collection of rod-like liquid crystalline entities probably having fairly uniform dimension, whereas the inner + type pattern may be a consequence of inter-rod interference. The four-lobe pattern further moves to low scattering angles with continued drying and eventually merges with the + type pattern to form an incomplete spherulite or sheaf structure. At this stage, the concentration levels off around 85 wt7, and the sheaf structure persists until the film is completely dried. 

This behaviour is explicable by the sheaf structure of the core ... 

Lopez et al. [104] observed that the mold temperature produces signiflcant differences in the macroscopic morphology and properties of the injected SPS samples. In particular, samples molded at high temperatures had higher resistance to the organic solvent than samples molded at low temperatures. The core of the molded samples analyzed by TEM appeared spherulitic [104]. However, the spherulites were not fully developed but appeared sheaf structures type with an elliptical profile. The intermediate region presented lamellar crystals oriented perpendicular to the flow direction. 

Recent developments have allowed atomic force microscopic (AFM) studies to follow the course of spherulite development and the internal lamellar structures as the spherulite evolves [206-209]. The major steps in spherulite formation were followed by AFM for poly(bisphenol) A octane ether [210,211] and more recently, as seen in the example of Figure 12 for a propylene 1-hexene copolymer [212] with 20 mol% comonomer. Accommodation of significant content of 1-hexene in the lattice allows formation and propagation of sheaf-like lamellar structure in this copolymer. The onset of sheave formation is clearly discerned in the micrographs of Figure 12 after crystallization for 10 h. Branching and development of the sheave are shown at later times. The direct observation of sheave and spherulitic formation by AFM supports the major features that have been deduced from transmission electron and optical microscopy. The fibrous internal spherulite structure could be directly observed by AFM. 

As is shown in this section, the main idea to obtain the ADHM description is the Beilinson spectral sequence, and the reason why it is so efficient depends on the fact that it comes from a resolution of the structure sheaf C>a of the diagonal... 

The symplectic form on the moduli space M.r,c, c2 is described as follows. Let be a stable sheaf on X, then the tangent space of MriCuC2 at is given by TsM.r,cllC2 = Ext1 ( , ), and the symplectic structure is defined by... 

XVIH) Let f 35 — s be a morphism of finite type and let F be a coherent sheaf on X. There exists a partition of 5 into locally closed subsets S, such that, giving each Sj the reduced scheme structure, the restrictions of F to each XxsSt is flat over 5, for all i. 

As is seen from the micrographs, in the carbon structures there are multiple branched fibers, which consist of finer sub fibers (Fig. 2 d), sheaf-like bundles of particles, spherical, tubular and conical elements (Fig. 2 a,b,c)-... 

For comparison, think of a differentiable or complex-analytic manifold. There again one has a topological space together with some additional structure and again one can describe the structure by a sheaf of functions, prescribing for each open set which functions are C°° or analytic. Thus X = Spec A with our sheaf on it is a sort of geometric object, and obviously... 

Let N be an R-module with some algebraic structure. Show that Aut(iV) is a sheaf in the fpqc topology. 

Much effort has been devoted to investigating the detailed architectures and the construction of spherulites. Early investigations of the crystallization of polymers through optical microscopy (OM) [7,8] posited that polymer spherulites consisted of radiating fibrous crystals with dense branches to fill space. Later, when electron microscopy (EM) became available, spherulites were shown to be comprised of layer-like crystallites [9,10], which were named lamellae. The lamellae are separated by disordered materials. In the center of the spherulites, the lamellae are stacked almost in parallel [5,6,11-15]. Away from the center, the stacked lamellae splay apart and branch, forming a sheaf-like structure [11,13-15]. It was also found that the thicknesses of lamellae are different [5,6,11,12]. The thicker ones are believed to be dominant lamellae while the thinner ones are subsidiary lamellae. 

Fig. 9. Sheaf-like small globule. Rods outside the sheaf form the defects of the globular structure. Both the defects of the loop type (left) and of the tad type (right) are shown...

The sufficiently short macromolecules of this type can also form the sheaf-iike small globule, Eq. (5.18) being the criterion of the small globule. General features of the analysis of the globular structure are analogous to those described in the previous section. In particular, the values of n, n+. ip- and coincide with those found earlier. 

With continued solvent evaporation, the four-lobe (x type) and the + type patterns merge to form a sheaf-like structure. In... 

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