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This paper introduces an algorithm for the nonnegative matrix factorization-and-completion problem, which aims to find nonnegative low-rank matrices <i>X</i> and <i>Y</i> so that the product <i>XY</i> approximates a nonnegative data matrix <i>M</i> whose elements are partially known (to a certain accuracy). This problem aggregates two existing problems: (i) nonnegative matrix factorization where all entries of <i>M</i> are given, and (ii) low-rank matrix completion where nonnegativity is not required. By taking the advantages of both nonnegativity and low-rankness, one can generally obtain superior results than those of just using one of the two properties. We propose to solve the non-convex constrained least-squares problem using an algorithm based on the classical alternating direction augmented Lagrangian method. Preliminary convergence properties of the algorithm and numerical simulation results are presented. Compared to

In this paper, we developed a feasible and efficient deep-learning-based framework to combine the United States (US) natality data for the last five decades, with changing variables and factors, into a consistent database. We constructed a graph based on the property and elements of databases, including variables, and conducted a graph convolutional network (GCN) to learn the embeddings of variables on the constructed graph, where the learned embeddings implied the similarity of variables. Specifically, we devised a loss function with a slack margin and a banlist mechanism (for a random walk) to learn the desired structure (two nodes sharing more information were more similar to each other.), and developed an active learning mechanism to conduct the harmonization. Toward a total of 9,321 variables from 49 databases (i.e., 783 stemmed variables, from 1970 to 2018), we applied our model iteratively together with human reviews for four rounds, then obtained 323 hyperchains of variables. During the harmonization, the first round of our model achieved recall and precision of 87.56%, 57.70%, respectively. Our harmonized graph neural network (HGNN) method provides a feasible and efficient way to connect relevant databases at a meta-level. Adapting to the database's property and characteristics, HGNN can learn patterns globally, which is powerful to discover the similarity between variables among databases. Our proposed method provides an effective way to reduce the manual effort in database harmonization and integration of fragmented data into useful databases for future research.

We discuss the discrete spectrum of N particles in a curved planar waveguide. If they are neutral fermions, the maximum number of particles that the waveguide can bind is given by a one-particle BirmanSchwinger bound in combination with the Pauli principle. On the other hand, if they are charged, e.g., electrons in a bent quantum wire, the Coulomb repulsion plays a crucial role. We prove a sufficient condition under which the discrete spectrum of such a system is empty.

We consider a pair of isoperimetric problems arising in physics. The first concerns a Schrdinger operator in L^2(\mathbb{R}^2) with an attractive interaction supported on a closed curve , formally given by (<i>x</i>); we ask which curve of a given length maximizes the ground state energy. In the second problem we have a loop-shaped thread in L^2(\mathbb{R}^2), homogeneously charged but not conducting, and we ask about the (renormalized) potential-energy minimizer. Both problems reduce to purely geometric questions about inequalities for mean values of chords of . We prove an isoperimetric theorem for <i>p</i>-means of chords of curves when <i>p</i> 2, which implies in particular that the global extrema for the physical problems are always attained when is a circle. The letter concludes with a discussion of the <i>p</i>-means of chords when <i>p</i> &gt; 2.