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Commutants, Reducing Subspaces and von Neumann Algebras Associated with Multiplication Operators
Published in Kehe Zhu, Handbook of Analytic Operator Theory, 2019
After a careful study in [GSZZ] of the case when the order of B is equal to 3, Guo, Sun, Zheng and Zhong formulated a more delicate conjecture: the number of minimal reducing subspaces of MBonLa2(D)equals the number of connected components of the Riemann surfaceSBof B−1 ∘ B on the unit disk [GSZZ, DSZ]. Here by a Riemann surface we mean a complex manifold of complex dimension 1, not necessarily connected. The amazing part of this modified conjecture is that the operator-theoretic quantity (the number of minimal reducing subspaces of MB) is accurately linked to a geometric quantity.
Statistical Modeling and Monitoring of Geometrical Deviations in Complex Shapes With Application to Additive Manufacturing
Published in Technometrics, 2022
Riccardo Scimone, Tommaso Taormina, Bianca Maria Colosimo, Marco Grasso, Alessandra Menafoglio, Piercesare Secchi
Although, as discussed in Section 1, it is not immediate to identify “standard” competitors for the proposed approach when the target objects are in a complex manifold configuration as in the presented real case study, it is possible to refer to the common practice adopted in industry. In the framework of additively manufactured complex shapes, the most common approach consists of computing individual and global descriptors, like the overall volume of the manufactured shape (that can be derived by standard software for mesh analysis) or the density of the part (that can be measured via the Archimede’s method). By monitoring such global descriptors it is possible to determine the presence of internal porosity or anomalous lack or excess of material. In order to assess the benefits of the proposed approach against such industrial practice, the volume of each 3D-printed shape was estimated through a mesh reconstruction to obtain a “watertight” mesh. To this aim, the screened Poisson surface reconstruction approach was applied (Kazhdan and Hoppe 2013). The estimated volume of the mesh was then used to design a univariate control chart for the shapes included into the real case study. The result is shown in Figure 24.
Assembly-free design for additive manufacturing of articulated components based on layered precision assignment
Published in International Journal of Computer Integrated Manufacturing, 2022
Jinghua Xu, Hongsheng Sheng, Jiangtao Zhan, Shuyou Zhang, Jianrong Tan
In order to verify the effectiveness and robustness of the proposed method, the complex manifold spine structure, akin to vertebrae cantilever structure, is selected for numerical test example. The influence of the mechanical error on the printing machine for the complex model is considered on the basis of the uncertainty error. The human bones of rib and spine can be reconstructed from medical images, such as computed tomography (CT), magnetic resonance imaging (MRI) and even positron emission tomography (PET), etc. The Figure 3 displays homogeneous human spine bone in (a)(b). From Figure 3, the geometric information of human spine containing total five separated vertebrae in global coordinate system (GCS) is as following: Firstly, note that the metrical length unit is hereafter exceptional mm. The surface area is 70,978.3574, enclosed volume is 332,343.2113, specific surface area is 0.2136 (mm−1). It has approximate left-right symmetry. The minimum printing stroke space required for the model in x, y, z directions is (105.0874,88.8064,188.7030) by ratio 0.5569:0.4706:1.
Stochastic coordinate-exchange optimal designs with complex constraints
Published in Quality Engineering, 2019
Pratola et al. (2017) developed a systematic approach to the design and analysis of experiments on the non-convex regions, but they focused more on handling the complex manifold surface of the design region than the non-convexity of the constraints. Their motivating example is the design of the locations of a glacier stake network. The design points are supposed to be on the 3-dimensional manifold of the glacier surface. The author discussed the maximin distance and the (robust) IMSPE criteria. They used the Gaussian process regression as the response surface model. For either criterion, to construct the optimal design in the non-convex region (in the geodesic distance space or the transformed Euclidean distance space), a dense candidate set is generated and then the unfeasible candidate points are removed. The optimal design points are selected from the feasible ones. This approach is sufficient for the 2- or 3-dimensional space in the spatial design and modeling scenarios. Instead of this simple approach, the SCE algorithm can be used to construct the optimal design in the (transformed) non-convex region.