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Applied Differential Equations
Published in Harish Parthasarathy, Advanced Probability and Statistics: Applications to Physics and Engineering, 2023
Suppose we have a pseudo-elastic material like chalk. When we apply a twist (ie, torsion force) to it, then it develops fractures along certain curves on the boundary surface. We can thus partition the material into disjoint regions where the fractures occur along the boundaries between the disjoint region and within each disjoint region, the laws of elasticity are valid. Let therefore, D1, D2, .., DN denote the disjoint regions. The boundaries are therefore the sets Bjk= cl(Dj) ∩ (cl(Dk)), 1 ≤ j < k ≤ N where the D'js are open sets on the surface of the topological manifold on which the fractures appear. Let ua(t, r) denote the ath component of the displacement on the surface of the body. Then strain tensor is then given by in the union of the open sets Dk, k = 1, 2, ...„ sab(t,r)=(1/2)(ua,b+ub,a)(t,r)
Applications in String Theory and Quantum Field Theory
Published in Khodakhast Bibak, Restricted Congruences in Computing, 2020
A surface is a compact oriented two-dimensional topological manifold. Roughly speaking, a surface is a space that ‘locally’ looks like the Euclidean plane. Informally, a graph is said to be embedded into (or drawn on) a surface if it can be drawn on the surface in such a way that its edges meet only at their endpoints. A ribbon graph is a finite and connected graph together with a cyclic ordering on the set of half edges incident to each vertex. One can see that ribbon graphs and embedded graphs are essentially equivalent concepts; that is, a ribbon graph can be thought of as a set of disks (or vertices) attached to each other by thin stripes (or edges) glued to their boundaries. There are several other names for these graphs in the literature, for example, fat graphs, combinatorial maps, and unrooted maps. For a thorough introduction to the theory of embedded graphs, we refer the reader to the lovely book by Lando and Zvonkin [113].
Optimization of Neural Network Architectures
Published in Bogdan M. Wilamowski, J. David Irwin, Intelligent Systems, 2018
The above consideration suggests that the MLP can be used for the approximation of the canonical decomposition of any function specified on the compact topological manifold. The following question comes to mind: Why is the canonical decomposition needed? Usually, essential variables, which fully describe the input–output relation, are not precisely defined. Thus, the approximation of this relation can be difficult. The existence of the first layer allows us to transform real data to the form of the complete set of variables of an invertible mapping. If the input space agrees with the inverse image of the approximated mapping, the first hidden layer is unnecessary.
A generalized mountain pass lemma with a closed subset for locally Lipschitz functionals
Published in Applicable Analysis, 2022
Fengying Li, Bingyu Li, Shiqing Zhang
Let B be a nonempty closed subset of X and let be a class of nonempty compact sets in X. According to [9] Definition 1, they call that is a homotopy-stable family with extended boundaryB when for every such that on one has . Some meaningful situations are special cases of this notion. For instance, if Q denotes a compact set in X, is a non-empty closed subset of Q, , and , then enjoys the above-mentioned property with . In particular, it holds true when Q indicates a compact topological manifold in X having a nonempty boundary while .
Limit sets in global multiobjective optimization
Published in Optimization, 2022
Gabriele Eichfelder, Oliver Stein
This section studies if and how under Assumptions 2.1–2.4 enclosures and provisional nondominated sets approximate the nondominated set of Y for ε descreasing to zero. Observe that is known to be a subset of the boundary of the image set [10] so that, if Y is a topological manifold with boundary, the dimension of can be at most m−1. At the same time, for all boxes in (1) ‘become flat at least in one direction’ and thus at most -dimensional. This fits well to the expected dimension of , so that one may expect that the enclosures converge to in, for example, the Hausdorff metric.
A categorical view of Poincaré maps and suspension flows
Published in Dynamical Systems, 2022
Let be a homeomorphism on a topological manifold X. The mapping torus of f is the manifold defined by where ∼ is the smallest equivalence relation with for each There is a natural surjection which sends each point to the corresponding equivalence class. We denote a point in by where