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Register a new userSwiss cheeses, rational approximation and universal plane curves
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In this paper we consider the compact plane sets known as Swiss cheese sets, which are a useful source of examples in the theory of uniform algebras and rational approximation. We introduce a notion of allocation map connected with Swiss cheeses, and we develop the theory of such maps. We use this theory to modify examples previously constructed in the literature to solve various problems, in order to obtain examples of Swiss cheese sets homeomorphic to the Sierpinski carpet which solve the same problems. In particular, this allows us to give examples of essential, regular uniform algebras on locally connected, compact plane sets. Our techniques also allow us to avoid certain technical difficulties in the literature.
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Swiss cheese sets are compact subsets of the complex plane obtained by deleting a sequence of open disks from a closed disk. Such sets have provided numerous counterexamples in the theory of uniform algebras. In this paper, we introduce a topological space whose elements are what we call abstract Swiss cheeses. Working within this topological space, we show how to prove the existence of classical Swiss cheese sets (as discussed in a paper of Feinstein and Heath from 2010) with various desired properties. We first give a new proof of the Feinstein-Heath classicalisation theorem. We then consider when it is possible to classicalise a Swiss cheese while leaving disks which lie outside a given region unchanged. We also consider sets obtained by deleting a sequence of open disks from a closed annulus, and we obtain an analogue of the Feinstein-Heath theorem for these sets. We then discuss regularity for certain uniform algebras. We conclude with an application of these techniques to obtain a classical Swiss cheese set which has the same properties as a non-classical example of OFarrell (1979).
Through reducing the problem to rational orthogonal system (Takenaka-Malmquist system), this note gives a proof for existence of n-best rational approximation to functions in the Hardy H2(D) space by using pseudohyperbolic distance.
We generalize the classical universal approximation theorem for neural networks to the case of complex-valued neural networks. Precisely, we consider feedforward networks with a complex activation function $sigma : mathbb{C} to mathbb{C}$ in which each neuron performs the operation $mathbb{C}^N to mathbb{C}, z mapsto sigma(b + w^T z)$ with weights $w in mathbb{C}^N$ and a bias $b in mathbb{C}$, and with $sigma$ applied componentwise. We completely characterize those activation functions $sigma$ for which the associated complex networks have the universal approximation property, meaning that they can uniformly approximate any continuous function on any compact subset of $mathbb{C}^d$ arbitrarily well. Unlike the classical case of real networks, the set of good activation functions which give rise to networks with the universal approximation property differs significantly depending on whether one considers deep networks or shallow networks: For deep networks with at least two hidden layers, the universal approximation property holds as long as $sigma$ is neither a polynomial, a holomorphic function, or an antiholomorphic function. Shallow networks, on the other hand, are universal if and only if the real part or the imaginary part of $sigma$ is not a polyharmonic function.
We propose a design of cylindrical elastic cloak for coupled in-plane shear waves consisting of concentric layers of sub-wavelength resonant stress-free inclusions shaped as swiss-rolls. The scaling factor between inclusions sizes is according to Pendrys transform. Unlike the hitherto known situations, the present geometric transform starts from a Willis medium and further assumes that displacement fields ${bf u}$ in original medium and ${bf u}$ in transformed medium remain unaffected (${bf u}={bf u}$), and this breaks the minor-symmetries of the rank-4 and rank-3 tensors in the Willis equation that describes the transformed effective medium. We achieve some cloaking for a shear polarized source at specific, resonant sub-wavelength, frequencies, when it is located near a clamped obstacle surrounded by the structured cloak. Such an effective medium allows for strong Willis coupling [Quan et al., Physical Review Letters {bf 120}(25), 254301 (2018)], notwithstanding potential chiral elastic effects [Frenzel et al., Science {bf 358}(6366), 1072 (2017)], and thus mitigates roles of Willis and Cosserat media in the achieved elastodynamic cloaking.
In this paper, we study unirational differential curves and the corresponding differential rational parametrizations. We first investigate basic properties of proper differential rational parametrizations for unirational differential curves. Then we show that the implicitization problem of proper linear differential rational parametric equations can be solved by means of differential resultants. Furthermore, for linear differential curves, we give an algorithm to determine whether an implicitly given linear differential curve is unirational and, in the affirmative case, to compute a proper differential rational parametrization for the differential curve.
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