Solitary waves and excited states for Boson stars

We study the nonlinear, nonlocal, time-dependent partial differential equation [Formula: see text] which is known to describe the dynamics of quasi-relativistic boson stars in the mean-field limit. For positive mass parameter [Formula: see text] we establish existence of infinitely many (corresponding to distinct energies [Formula: see text]) traveling solitary waves, [Formula: see text], with speed [Formula: see text], where [Formula: see text] corresponds to the speed of light in our choice of units. These traveling solitary waves cannot be obtained by applying a Lorentz boost to a solitary wave at rest (with [Formula: see text]) because Lorentz covariance fails. Instead, we study a suitable variational problem for which the functions [Formula: see text] arise as solutions (called boosted excited states) to a Choquard-type equation in [Formula: see text], where the negative Laplacian is replaced by the pseudo-differential operator [Formula: see text] and an additional term [Formula: see text] enters. Moreover, we give a new proof for existence of boosted ground states. The results are based on perturbation methods in critical point theory.

Stability of the Enhanced Area Law of the Entanglement Entropy

We consider a multi-dimensional continuum Schrödinger operator which is given by a perturbation of the negative Laplacian by a compactly supported potential. We establish both an upper bound and a lower bound on the bipartite entanglement entropy of the ground state of the corresponding quasi-free Fermi gas. The bounds prove that the scaling behaviour of the entanglement entropy remains a logarithmically enhanced area law as in the unperturbed case of the free Fermi gas. The central idea for the upper bound is to use a limiting absorption principle for such kinds of Schrödinger operators.

Experimental and theoretical charge density, intermolecular interactions and electrostatic properties of metronidazole

Metronidazole is a radiosensitizer; it crystallizes in the monoclinic system with space group P21/c. The crystal structure of metronidazole has been determined from high-resolution X-ray diffraction measurements at 90 K with a resolution of (sin θ/λ)max = 1.12 Å−1. To understand the charge-density distribution and the electrostatic properties of metronidazole, a multipole model refinement was carried out using the Hansen–Coppens multipole formalism. The topological analysis of the electron density of metronidazole was performed using Bader's quantum theory of atoms in molecules to determine the electron density and the Laplacian of the electron density at the bond critical point of the molecule. The experimental results have been compared with the corresponding periodic theoretical calculation performed at the B3LYP/6-31G** level using CRYSTAL09. The topological analysis reveals that the N—O and C—NO2 exhibit less electron density as well as negative Laplacian of electron density. The molecular packing of crystal is stabilized by weak and strong inter- and intramolecular hydrogen bonding and H...H interactions. The topological analysis of O—H...N, C—H...O and H...H intra- and intermolecular interactions was also carried out. The electrostatic potential of metronidazole, calculated from the experiment, predicts the possible electrophilic and nucleophilic sites of the molecule; notably, the hydroxyl and the nitro groups exhibit large electronegative regions. The results have been compared with the corresponding theoretical results.

CNLLRR: A Novel Low-Rank Representation Method for Single-cell RNA-seq Data Analysis

The development of single-cell RNA-sequencing (scRNA-seq) technology has enabled the measurement of gene expression in individual cells. This provides an unprecedented opportunity to explore the biological mechanisms at the cellular level. However, existing scRNA-seq analysis methods are susceptible to noise and outliers or ignore the manifold structure inherent in the data. In this paper, a novel method called Cauchy non-negative Laplacian regularized low-rank representation (CNLLRR) is proposed to alleviate the above problem. Specifically, we employ the Cauchy loss function (CLF) instead of the conventional norm constraints in the noise matrix of CNLLRR, which will enhance the robustness of the method. In addition, graph regularization term is applied to the objective function, which can capture the paired geometric relationships between cells. Then, alternating direction method of multipliers (ADMM) is adopted to solve the optimization problem of CNLLRR. Finally, extensive experiments on scRNA-seq data reveal that the proposed CNLLRR method outperforms other state-of-the-art methods for cell clustering, cell visualization and prioritization of gene markers. CNLLRR contributes to understand the heterogeneity between cell populations in complex biological systems.Author summaryAnalysis of single-cell data can help to further study the heterogeneity and complexity of cell populations. The current analysis methods are mainly to learn the similarity between cells and cells. Then they use the clustering algorithm to perform cell clustering or downstream analysis on the obtained similarity matrix. Therefore, constructing accurate cell-to-cell similarity is crucial for single-cell data analysis. In this paper, we design a novel Cauchy non-negative Laplacian regularized low-rank representation (CNLLRR) method to get a better similarity matrix. Specifically, Cauchy loss function (CLF) constraint is applied to punish noise matrix, which will improve the robustness of CNLLRR to noise and outliers. Moreover, graph regularization term is applied to the objective function, which will effectively encode the local manifold information of the data. Further, these will guarantee the quality of the cell-to-cell similarity matrix learned. Finally, single-cell data analysis experiments show that our method is superior to other representative methods.

Elliptic equations with indefinite and unbounded potential and a nonlinear concave boundary condition

We consider an elliptic problem driven by the negative Laplacian plus an indefinite and unbounded potential and a superlinear reaction. The boundary condition is parametric, nonlinear and superlinear near zero. Thus, the problem is a new version of the classical “convex–concave” problem (problem with competing nonlinearities). First, we prove a bifurcation-type result describing the set of positive solutions as the parameter [Formula: see text] varies. We also show the existence of a smallest positive solution [Formula: see text] and investigate the properties of the map [Formula: see text]. Finally, by imposing bilateral conditions on the reaction we generate two more solutions, one of which is nodal.

Space-time fractional stochastic equations on regular bounded open domains

Fractional (in time and in space) evolution equations defined on Dirichlet regular bounded open domains, driven by fractional integrated in time Gaussian spatiotemporal white noise, are considered here. Sufficient conditions for the definition of a weak-sense Gaussian solution, in the mean-square sense, are derived. The temporal, spatial and spatiotemporal Hölder continuity, in the mean-square sense, of the formulated solution is obtained, under suitable conditions, from the asymptotic properties of the Mittag-Leffler function, and the asymptotic order of the eigenvalues of a fractional polynomial of the Dirichlet negative Laplacian operator on such bounded open domains.