On minimal additive complements of integers

Let $C,W\subseteq \mathbb{Z}$. If $C+W=\mathbb{Z}$, then the set $C$ is called an additive complement to $W$ in $\mathbb{Z}$. If no proper subset of $C$ is an additive complement to $W$, then $C$ is called a minimal additive complement. Let $X\subseteq \mathbb{N}$. If there exists a positive integer $T$ such that $x+T\in X$ for all sufficiently large integers $x\in X$, then we call $X$ eventually periodic. In this paper, we study the existence of a minimal complement to $W$ when $W$ is eventually periodic or not. This partially answers a problem of Nathanson.Comment: 13 page

Global analysis of quadrupole shape invariants based on covariant energy density functionals

Coexistence of different geometric shapes at low energies presents a universal structure phenomenon that occurs over the entire chart of nuclides. Studies of the shape coexistence are important for understanding the microscopic origin of collectivity and modifications of shell structure in exotic nuclei far from stability. The aim of this work is to provide a systematic analysis of characteristic signatures of coexisting nuclear shapes in different mass regions, using a global self-consistent theoretical method based on universal energy density functionals and the quadrupole collective model. The low-energy excitation spectrum and quadrupole shape invariants of the two lowest $0^{+}$ states of even-even nuclei are obtained as solutions of a five-dimensional collective Hamiltonian (5DCH) model, with parameters determined by constrained self-consistent mean-field calculations based on the relativistic energy density functional PC-PK1, and a finite-range pairing interaction. The theoretical excitation energies of the states: $2^+_1$, $4^+_1$, $0^+_2$, $2^+_2$, $2^+_3$, as well as the $B(E2; 0^+_1\to 2^+_1)$ values, are in very good agreement with the corresponding experimental values for 621 even-even nuclei. Quadrupole shape invariants have been implemented to investigate shape coexistence, and the distribution of possible shape-coexisting nuclei is consistent with results obtained in recent theoretical studies and available data. The present analysis has shown that, when based on a universal and consistent microscopic framework of nuclear density functionals, shape invariants provide distinct indicators and reliable predictions for the occurrence of low-energy coexisting shapes. This method is particularly useful for studies of shape coexistence in regions far from stability where few data are available.Comment: 13 pages, 3 figures, accepted for publication in Phys. Rev.

Multi-Person Brain Activity Recognition via Comprehensive EEG Signal Analysis

An electroencephalography (EEG) based brain activity recognition is a fundamental field of study for a number of significant applications such as intention prediction, appliance control, and neurological disease diagnosis in smart home and smart healthcare domains. Existing techniques mostly focus on binary brain activity recognition for a single person, which limits their deployment in wider and complex practical scenarios. Therefore, multi-person and multi-class brain activity recognition has obtained popularity recently. Another challenge faced by brain activity recognition is the low recognition accuracy due to the massive noises and the low signal-to-noise ratio in EEG signals. Moreover, the feature engineering in EEG processing is time-consuming and highly re- lies on the expert experience. In this paper, we attempt to solve the above challenges by proposing an approach which has better EEG interpretation ability via raw Electroencephalography (EEG) signal analysis for multi-person and multi-class brain activity recognition. Specifically, we analyze inter-class and inter-person EEG signal characteristics, based on which to capture the discrepancy of inter-class EEG data. Then, we adopt an Autoencoder layer to automatically refine the raw EEG signals by eliminating various artifacts. We evaluate our approach on both a public and a local EEG datasets and conduct extensive experiments to explore the effect of several factors (such as normalization methods, training data size, and Autoencoder hidden neuron size) on the recognition results. The experimental results show that our approach achieves a high accuracy comparing to competitive state-of-the-art methods, indicating its potential in promoting future research on multi-person EEG recognition.Comment: 10 page

Nuclear quantum shape-phase transitions in odd-mass systems

Microscopic signatures of nuclear ground-state shape phase transitions in odd-mass Eu isotopes are explored starting from excitation spectra and collective wave functions obtained by diagonalization of a core-quasiparticle coupling Hamiltonian based on energy density functionals. As functions of the physical control parameter -- the number of nucleons -- theoretical low-energy spectra, two-neutron separation energies, charge isotope shifts, spectroscopic quadrupole moments, and $E2$ reduced transition matrix elements accurately reproduce available data, and exhibit more pronounced discontinuities at neutron number $N=90$, compared to the adjacent even-even Sm and Gd isotopes. The enhancement of the first-order quantum phase transition in odd-mass systems can be attributed to a shape polarization effect of the unpaired proton which, at the critical neutron number, starts predominantly coupling to Gd core nuclei that are characterized by larger quadrupole deformation and weaker proton pairing correlations compared to the corresponding Sm isotopes.Comment: 6 pages, 4 figure