We demonstrate a one-dimensional magnetic system can exhibit a Cantor-type spectrum using an example of a chain graph with coupling at the vertices exposed to a magnetic field perpendicular to the graph plane and varying along the chain. If the field grows linearly with an irrational slope, measured in terms of the flux through the loops of the chain, we demonstrate the character of the spectrum relating it to the almost Mathieu operator.

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Designing an unsupervised image denoising approach in practical applications is a challenging task due to the complicated data acquisition process. In the realworld case, the noise distribution is so complex that the simplified additive white Gaussian (AWGN) assumption rarely holds, which significantly deteriorates the Gaussian denoisers’ performance. To address this problem, we apply a deep neural network that maps the noisy image into a latent space in which the AWGN assumption holds, and thus any existing Gaussian denoiser is applicable. More specifically, the proposed neural network consists of the encoder-decoder structure and approximates the likelihood term in the Bayesian framework. Together with a Gaussian denoiser, the neural network can be trained with the input image itself and does not require any pre-training in other datasets. Extensive experiments on real-world noisy image datasets have shown that the combination of neural networks and Gaussian denoisers improves the performance of the original Gaussian denoisers by a large margin. In particular, the neural network+BM3D method significantly outperforms other unsupervised denoising approaches and is competitive with supervised networks such as DnCNN, FFDNet, and CBDNet.

We investigate approximations of the vertex coupling on a star-shaped graph by families of operators with singularly scaled rank-one interactions. We find a family of vertex couplings, generalizing the <i></i>-interaction on the line, and show that with a suitable choice of the parameters they can be approximated in this way in the norm-resolvent sense. We also analyze spectral properties of the involved operators and demonstrate the convergence of the corresponding on-shell scattering matrices.

The mutual intensity and its equivalent phase-space representations quantify an optical field's state of coherence and are important tools in the study of light propagation and dynamics, but they can only be estimated indirectly from measurements through a process called coherence retrieval, otherwise known as phase-space tomography. As practical considerations often rule out the availability of a complete set of measurements, coherence retrieval is usually a challenging high-dimensional ill-posed inverse problem. In this paper, we propose a trace-regularized optimization model for coherence retrieval and a provably convergent adaptive accelerated proximal gradient algorithm for solving the resulting problem. Applying our model and algorithm to both simulated and experimental data, we demonstrate an improvement in reconstruction quality over previous models as well as an increase in convergence speed compared to existing first-order methods.