Determining a Rotation of a Tetrahedron from a Projection

The following problem, arising from medical imaging, is addressed: Suppose that T is a known tetrahedron in ?³ with centroid at the origin. Also known is the orthogonal projection U of the vertices of the image ?T of T under an unknown rotation ? about the origin. Under what circumstances can ? be determined from T and U?     

EBSD-based continuum dislocation microscopy

Recent advances in high-resolution electron backscatter diffraction (EBSD)-based microscopy are applied to the characterization of elastic fields and incompatibility structures near the grain boundaries (GBs) in polycrystals. Two main recoveries are reported here: surface geometrically necessary dislocation (density) tensors, as described by Kröner, and the elastic fields near cracks (unconsolidated portions of interface) in loaded samples. Context for the application of these recoveries is described, using Green’s function solutions for combined heterogeneity and dislocation. Featured recoveries required the cross-correlation based determination of the elastic distortion tensor, aided by application of the simulated pattern method, and determination of the absolute pattern center utilizing the expected pattern properties in a spherical Kikuchi reference frame. High-resolution data obtained along an ultrasonically consolidated nickel boundary of varying amalgamation indicates that the imposed traction free boundary condition at free surfaces is well observed in the data structure. Further, high-resolution data acquired near a single grain boundary in well-annealed, low content steel suggests that it may be possible to measure the intrinsic elastic properties of GBs.     

Material Design in Poly(Lactic Acid) Systems: Block Copolymers,Star Homo- and Copolymers,and Stereocomplexes

Abstract

This paper gives a general overview of several approaches we have investigated for designing new PLA-based polymers with a broad range of properties and improved processability. These approaches include: copolymerization (block and stereoblock copolymers), microstructure and architecture control, and stereocomplexation. Multiblock copolymers with alternating “soft” and “hard” segments, synthesized over a broad range of chemical compositions, show properties ranging from hard plastics to elastomers. Stereoblock copolymers with alternating amorphous and semicrystalline PLA blocks combine the advantages of PLA homopolymers (crystallinity) and random copolymers (processability). Independent control of polymer architecture and microstructure allows for the synthesis of star polymers with various arm morphologies. A new method for stereocomplex formation between L-PLA and D-PLA, which combines in-situ polymerization with stereocomplexation, is also described. For the synthesis of these new materials we took advantage of: 1) chirality of lactide monomer, 2) retention of configuration during polymerization, 3) living nature of the ring-opening polymerization (ROP) of lactide in the presence of active hydrogen groups such as OH and NH₂, and 4) control of the level of transesterification reactions.     

Synthesis of mixed cyclotriveratrylenes

Cyclotriveratrylenes containing one benzene ring and two indole rings linked through N1 and C2 can be prepared by acid-catalysed reactions of formaldehyde and aryl aldehydes with 1,2-di-(1-indolylmethyl)benzene compounds.     

Nuclear charge radii and electromagnetic moments of scandium isotopes and isomers in the f7/2 shell

Collinear laser spectroscopy experiments on the ScII transition 3d4s ³D₂→3d4p ³F₃ at λ ≈ 363.1 nm were performed on the ^42–46Sc isotopic chain using an ion guide isotope separator with a cooler–buncher. Isotope and isomer shifts and hyperfine structures of five ground states and two isomers were measured. Preliminary results on the nuclear moments and charge radii changes deduced from these measurements are reported.     

Reconstruction of Convex Bodies from Brightness Functions

Abstract. Algorithms are given for reconstructing an approximation to an unknown convex body from finitely many values of its brightness function, the function giving the volumes of its projections onto hyperplanes. One of these algorithms constructs a convex polytope with less than a prescribed number of facets, while the others do not restrict the number of facets. Convergence of the polytopes to the body is proved under certain essential assumptions including origin symmetry of the body. Also described is an oracle-polynomial-time algorithm for reconstructing an approximation to an origin-symmetric rational convex polytope of fixed and full dimension that is only accessible via its brightness function. Some of the algorithms have been implemented, and sample reconstructions are provided.     

Bijections of #x211D; n onto itself

We characterize affine and continuous maps within the class of bijections of #x211D; ⁿ onto itself by the preservation of various geometric or topological figures. A characterization of similarity maps of Hilbert space is given.Supported in part by a grant from the National Science Foundation.     

A solution to Hammer's X-ray reconstruction problem

We propose algorithms for reconstructing a planar convex body K from possibly noisy measurements of either its parallel X-rays taken in a fixed finite set of directions or its point X-rays taken at a fixed finite set of points, in known situations that guarantee a unique solution when the data is exact. The algorithms construct a convex polygon Pk whose X-rays approximate (in the least squares sense) k equally spaced noisy X-ray measurements in each of the directions or at each of the points.It is shown that these procedures are strongly consistent, meaning that, almost surely, Pk tends to K in the Hausdorff metric as k→∞. This solves, for the first time in the strongest sense, Hammer's X-ray problem published in 1963.