Theory predicts that a two-pump fiber optical parametric amplifier or wavelength converter operated near the fiber zero-dispersion wavelength can exhibit a gain spectrum approximated by a Chebyshev polynomial of order 8. Under realistic conditions of pump spacing and fiber dispersion, very low-gain ripple can be obtained over a large bandwidth. For example, a dispersion-shifted fiber can provide a signal amplifier with a gain of 20 dB with 0.2-dB uniformity over a 45-nm bandwidth. Potential limitations are discussed.
In [13], Hillar and Lim famously demonstrated that "multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard". Despite many recent advancements, the state-of-the-art methods for computing such 'tensor analogues' still suffer severely from the curse of dimensionality. In this paper we show that the Tucker core of a tensor however, retains many properties of the original tensor, including the CP rank, the border rank, the tensor Schatten quasi norms, and the Z-eigenvalues. When the core tensor is smaller than the original tensor, this property leads to considerable computational advantages as confirmed by our numerical experiments. In our analysis, we in fact work with a generalized Tucker-like decomposition that can accommodate any full column-rank factor matrices.
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