Removable singularity

Abstract: We investigate the approximation ability of a multilayer perceptron (MLP) network when it is extended to the complex domain. The main challenge for processing complex data with neural networks has been the lack of bounded and analytic complex nonlinear activation functions in the complex domain, as stated by Liouville's theorem. To avoid the conflict between the boundedness and the analyticity of a nonlinear complex function in the complex domain, a number of ad hoc MLPs that include using two real-valued MLPs, one processing the real part and the other processing the imaginary part, have been traditionally employed. However, since nonanalytic functions do not meet the Cauchy-Riemann conditions, they render themselves into degenerative backpropagation algorithms that compromise the efficiency of nonlinear approximation and learning in the complex vector field. A number of elementary transcendental functions (ETFs) derivable from the entire exponential function ez that are analytic are defined as fully complex activation functions and are shown to provide a parsimonious structure for processing data in the complex domain and address most of the shortcomings of the traditional approach. The introduction of ETFs, however, raises a new question in the approximation capability of this fully complex MLP. In this letter, three proofs of the approximation capability of the fully complex MLP are provided based on the characteristics of singularity among ETFs. First, the fully complex MLPs with continuous ETFs over a compact set in the complex vector field are shown to be the universal approximator of any continuous complex mappings. The complex universal approximation theorem extends to bounded measurable ETFs possessing a removable singularity. Finally, it is shown that the output of complex MLPs using ETFs with isolated and essential singularities uniformly converges to any nonlinear mapping in the deleted annulus of singularity nearest to the origin.

Abstract: In this paper we obtain results on holomorphic continuation of proper holomorphic mappings between pseudoconvex domains with real-analytic boundaries in complex spaces of different dimensions. Equivalently, we obtain results concerning the analyticity of Cauchy-Riemann mappings between real-analytic pseudoconvex hypersurfaces in complex spaces of different dimensions. To begin with, we recall the corresponding results for mappings of equidimensional domains. Let D and D' be bounded pseudoconvex domains with smooth boundaries in C". If the boundaries of D and D' are strictly pseudoconvex or, more generally, of finite type in the sense of D'Angelo [15], then every proper holomorphic map of D onto D' extends smoothly t o / ) according to the results of Bell and Catlin [7, 8] and Diederich and Forn~ess [19]. For mappings between strictly pseudoconvex domains this was proved by Fefferman [26] and Nirenberg, Webster, and Yang [37]. If the boundaries of the pseudoconvex domains D, D ' c C" are real-analytic, then every proper holomorphic mapping of D onto D' extends holomorphically to a neighborhood of /3 according to Baouendi and Rothschild [2] and Diederich and Forn~ess [21]. This 'reflection principle' was first discovered by Lewy [34] and Pin6uk [38] for mappings between strictly pseudoconvex domains. In the case of biholomorphic mappings between weakly pseudoconvex domains with realanalytic boundaries the result follows from the work of Baouendi, Jacobowitz, and Treves [5]. Results in this direction were obtained in recent years by several authors; see the papers [4, 6, 18, 22, 33, 47, 49]. In this paper we are treating the case when the domains D and D' have different dimensions. To be specific, we assume that D e C " and D ' c C N are bounded pseudoconvex domains with real-analytic boundaries and N > n > l . In this situation a proper holomorphic map f : D , D ' need not be regular at the boundary. For instance, there exist proper holomorphic maps of balls of different dimensions

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Abstract: We introduce a functional that couples the nonlinear sigma model with a spinor field: $L=\int_M[|d\phi|^2+(\psi,\D\psi)]$. In two dimensions, it is conformally invariant. The critical points of this functional are called Dirac-harmonic maps. We study some geometric and analytic aspects of such maps, in particular a removable singularity theorem.