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

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

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.

STG2Seq: Spatial-temporal Graph to Sequence Model for Multi-step Passenger Demand Forecasting

Multi-step passenger demand forecasting is a crucial task in on-demand vehicle sharing services. However, predicting passenger demand over multiple time horizons is generally challenging due to the nonlinear and dynamic spatial-temporal dependencies. In this work, we propose to model multi-step citywide passenger demand prediction based on a graph and use a hierarchical graph convolutional structure to capture both spatial and temporal correlations simultaneously. Our model consists of three parts: 1) a long-term encoder to encode historical passenger demands; 2) a short-term encoder to derive the next-step prediction for generating multi-step prediction; 3) an attention-based output module to model the dynamic temporal and channel-wise information. Experiments on three real-world datasets show that our model consistently outperforms many baseline methods and state-of-the-art models.Comment: 7 page

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

Extreme Equilibria in a General Negotiation Model

We study a bargaining model with a disagreement game between o¤ers and countero¤ers. In order to characterize the set of its subgame perfect equilibrium payo¤s, we provide a recursive technique that relies on the Pareto frontier of equilibrium payo¤s. When players have di¤erent time preferences, reaching an immediate agreement may not be Pareto e ¢ cient. The recursive technique developed in this paper generalizes that of Shaked and Sutton (1984) by incorporating the possibility of making unacceptable proposals into the backward induction analysis. Results from this paper extend all the previous …ndings and resolve some open issues in the current literature