Motivated by applications in nuclear magnetic resonance (NMR) spectroscopy, we introduce a novel blind source separation (BSS) approach to treat nonnegative and correlated data. We consider the (over)-determined case where n sources are to be separated from n linear mixtures (n). Among the n source signals, there are n partially overlapping (Po) sources and one positive everywhere (Pe) source. This condition is applicable for many real-world signals such as NMR spectra of urine and blood serum for metabolic fingerprinting and disease diagnosis. The geometric properties of the mixture matrix and the sparseness structure of the source signals (in a transformed domain) are crucial to the identification of the mixing matrix and the sources. The method first identifies the mixing coefficients of the Pe source by exploiting geometry in data clustering. Then subsequent elimination of variables leads to a sub

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We look for bounded periodic solutions for a parametric fractional problem involving a continuous nonlinearity with subcritical growth. By using a variant of Caffarelli and Silvestre extension method adapted to the periodic case and variational tools we prove the existence of at least three bounded periodic solutions when the parameter varies in an appropriate range.

We define the counting of holomorphic cylinders in log Calabi-Yau surfaces. Although we start with a complex log Calabi-Yau surface, the counting is achieved by applying methods from non-archimedean geometry. This gives rise to new geometric invariants. Moreover, we prove that the counting satisfies a property of symmetry. Explicit calculations are given for a del Pezzo surface in detail, which verify the conjectured wall-crossing formula for the focus-focus singularity. Our holomorphic cylinders are expected to give a geometric understanding of the combinatorial notion of broken line by Gross, Hacking, Keel and Siebert. Our tools include Berkovich spaces, tropical geometry, Gromov-Witten theory and the GAGA theorem for non-archimedean analytic stacks.

In this paper, we show that general homogeneous manifolds G/P satisfy Conjecture O of Galkin, Golyshev and Iritani which `underlies' Gamma conjectures I and II of them. Our main tools are the quantum Chevalley formula for G/P and a theory on nonnegative matrices including Perron-Frobenius theorem.