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For the metric spaces, the homeomorphisms more general than conformal mappings are studied. It is proved that the families of ine indicated mappings are equicontinuous in their closure under definite conditions imposed on the boundaries of the corresponding domains.
Abstract Topological solitons are currently under investigation for their exotic properties, especially in nonlinear physics, optics, and material sciences. However, challenges of robust generation and limited stability over time have hindered their practical uses. To address this issue, an approach is developed to form structured arrays of solitons in films of polymerizable liquid crystals. Their complex molecular architecture is preserved by in situ photopolymerization forming a stable liquid crystal network. Most excitingly, their properties are advanced to include responsiveness functions. When thermally actuated, these topological solitons mediate the reconfiguration of surface topographies. Complex shape changes occur depending on the intrinsic complex spatial distribution of the director, which may even lead to full shape inversion and topographical changes as high as ≈40% of the initial thickness. Conversely, the shape changes provide information on the initial director profile, which is consistent with the mathematical model. The soliton‐containing polymer coatings are applicable in multiple domains, ranging from tunable optics to haptics, and from shape‐coupled sensing systems to temperature‐coupled heat management.
This paper presents a level-set topology optimization approach that uses conformal meshes for the analysis of the displacement field. The structure’s boundary is represented by the iso-contour of a level-set field discretized on a fixed background design mesh. The conformal mesh is updated for each design iteration via a PDE based mesh morphing process that identifies the set of facets in the background mesh that are homeomorphic to the boundary and relaxes the homeomorphic mesh to conform to the structure’s boundary and ensure high element quality. The conformal mesh allows for a more accurate computation of the response versus density and some level-set based methods which interpolate material properties using the volume fraction. Numerical examples illustrate the proposed approach by optimizing linear-elastic two- and three-dimensional structures, wherein insight into the performance of the mesh morphing process is provided. The examples also highlight the scalability of the approach.
Abstract We define and study the totally nonnegative part of the Chow quotient of the Grassmannian, or more simply the nonnegative configuration space . This space has a natural stratification by positive Chow cells , and we show that nonnegative configuration space is homeomorphic to a polytope as a stratified space. We establish bijections between positive Chow cells and the following sets: (a) regular subdivisions of the hypersimplex into positroid polytopes, (b) the set of cones in the positive tropical Grassmannian, and (c) the set of cones in the positive Dressian. Our work is motivated by connections to super Yang–Mills scattering amplitudes, which will be discussed in a sequel.
This slide presentation describes basic topological concepts, including topological spaces, homeomorphisms, homotopy, betti numbers. Scalar field topology explores finding topological features and scalar field visualization, and vector field topology explores finding topological features and vector field visualization.
Assume that Ω is a domain in the complex plane ℂ and A(z) is a symmetric 2×2 matrix function with measurable entries, detA = 1; and such that 1/K|ξ|{sup 2} ≤ 〈A(z)ξ, ξ〉 ≤ K|ξ|{sup 2}, ξ ∈ ℝ{sup 2}, 1 ≤ K < ∞ . In particular, for semilinear elliptic equations of the form div (A(z)∇u(z)) = f(u(z)) in Ω; we prove a factorization theorem that asserts that every weak solution u to the above equation can be expressed as the composition u = To𝜔; where 𝜔 : Ω → G stands for a K−quasiconformal homeomorphism generated by the matrix function A(z); and T(w) is a weak solution of the semilinear equation ∇T(w) = J(w)f(T(w)) in G: Here, the weight J(w) is the Jacobian of the inverse mapping 𝜔{sup −1}: Similar results hold for the corresponding nonlinear parabolic and hyperbolic equations. Some applications of these results to anisotropic media are given.
We study metric properties of ring Q-homeomorphisms with respect to the p-modulus, p > 2, in the complex plane and establish lower bounds for the areas of disks. An extremal problem concerning minimization of the area functional is also solved.
A quantization of the Sobolev space V=H{sub 0}{sup 1/2}(S{sup 1},R) of half- differentiable functions on the circle, which is closely connected with string theory, is constructed. The group QS(S{sup 1}) of quasisymmetric circle homeomorphisms acts on V by reparametrizations, but this action is not smooth. Nevertheless, a quantum infinitesimal action of QS(S{sup 1}) on V can be defined, which enables one to construct a quantum algebra of observables which is associated with the system (V,QS(S{sup 1})). Bibliography: 7 titles. (paper)