Improved Cheeger's Inequality: Analysis of Spectral Partitioning Algorithms through Higher Order Spectral Gap

Let \phi(G) be the minimum conductance of an undirected graph G, and let 0=\lambda_1 <= \lambda_2 <=... <= \lambda_n <= 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2, \phi(G) = O(k) \lambda_2 / \sqrt{\lambda_k}, and this performance guarantee is achieved by the spectral partitioning algorithm. This improves Cheeger's inequality, and the bound is optimal up to a constant factor for any k. Our result shows that the spectral partitioning algorithm is a constant factor approximation algorithm for finding a sparse cut if \lambda_k$ is a constant for some constant k. This provides some theoretical justification to its empirical performance in image segmentation and clustering problems. We extend the analysis to other graph partitioning problems, including multi-way partition, balanced separator, and maximum cut

Microstructural Characteristics and Hardness Enhancement of Super Duplex Stainless Steel by Friction Stir Processing

In the present study, microstructural evolution and hardness of the friction stir processed (FSPed) SAF 2507 super duplex stainless steel fabricated at a rotational speed of 650 rpm and a traverse speed of 60 mm/min were investigated. A scanning electron microscope (SEM) equipped with an electron backscatter diffraction (EBSD) detector was used to study the microstructure of the stir zone. The grain sizes of austenite and ferrite in the FSPed 2507 were found to be smaller (0.75 and 0.96 &mu;m) than those of the substrate (6.6 and 5.6 &mu;m) attributed to the occurrence of continuous dynamic recrystallization (CDRX) in both phases. Higher degree of grain refinement and DRX were obtained at the advancing side of the FSPed specimens due to higher strain and temperature. A non-uniform hardness distribution was observed along the longitudinal direction of the SZ. The maximum hardness was obtained at the bottom (407 HV1)

The Technological Design of Geometrically Complex Ti-6Al-4V Parts by Metal Injection Molding

In this study, the metal injection molding (MIM) process is applied to produce Ti-6Al-4V parts using blended and prealloyed powders, respectively. The feedstocks are prepared from a polyformaldehyde-based binder system with a powder loading of 60 vol%, exhibiting a low viscosity. The decomposition behavior of the binders is investigated and the thermal debinding procedure is designed accordingly. The debound parts are subsequently sintered at 1200 and 1300 &#176;C. The results show the mechanical properties of the sintered samples prepared from blended powder are comparable to those prepared from prealloyed powder, with yield strength of 810 MPa, ultimate tensile strength (UTS) of 927 MPa, and elongation of 4.6%. The density of the as-sintered samples can reach 4.26 g/cm3 while oxygen content is ~0.3%. Based on the results, watch cases with complex shapes are successfully produced from Ti-6Al-4V blended powder. The case gives a good example of applying metal injection molding to mass production of precise Ti-6Al-4V parts with complex shapes in a cost-effective way