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2. Learning complex segments

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3. Designing Complex Fluids

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   Ewoldt, Randy H.; Saengow, Chaimongkol (January 2022, Annual Review of Fluid Mechanics)

   Taking a small step away from Newtonian fluid behavior creates an explosion in the range of possibilities. Non-Newtonian fluid properties can achieve diverse flow objectives, but the complexity introduces challenges. We survey useful rheological complexity along with organizing principles and design methods as we consider the following questions: How can non-Newtonian properties be useful? What properties are needed? How can we get those properties? 

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4. Complex harmonic capacitors

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   Iwaniec, Tadeusz; Onninen, Jani; Radice, Teresa (April 2025, Calculus of Variations and Partial Differential Equations)

   Abstract The concept of complex harmonic potential in a doubly connected condenser (capacitor) is introduced as an analogue of the real-valued potential of an electrostatic vector field. In this analogy the full differential of a complex potential plays the role of the gradient of the scalar potential in the theory of electrostatics. The main objective in the non-static fields is to rule out having the full differential vanish at some points. Nevertheless, there can be critical points where the Jacobian determinant of the differential turns into zero. The latter is in marked contrast to the case of real-valued potentials. Furthermore, the complex electric capacitor also admits an interpretation of the stored energy intensively studied in the theory of hyperelastic deformations. Engineers interested in electrical systems, such as energy storage devises, might also wish to envision complex capacitors aselectromagnetic condenserswhich, generally, store more energy that the electric capacitors. 

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5. Homotopy Reconstruction via the Cech Complex and the Vietoris-Rips Complex

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   Kim, Jisu; Shin, Jaehyeok; Chazal, Frédéric; Rinaldo, Alessandro; Wasserman, Larry (August 2020, Leibniz international proceedings in informatics)

   We derive conditions under which the reconstruction of a target space is topologically correct via the Čech complex or the Vietoris-Rips complex obtained from possibly noisy point cloud data. We provide two novel theoretical results. First, we describe sufficient conditions under which any non-empty intersection of finitely many Euclidean balls intersected with a positive reach set is contractible, so that the Nerve theorem applies for the restricted Čech complex. Second, we demonstrate the homotopy equivalence of a positive μ-reach set and its offsets. Applying these results to the restricted Čech complex and using the interleaving relations with the Čech complex (or the Vietoris-Rips complex), we formulate conditions guaranteeing that the target space is homotopy equivalent to the Čech complex (or the Vietoris-Rips complex), in terms of the μ-reach. Our results sharpen existing results. 

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