Search

Let μ M , D be a self-similar measure generated by an n × n expanding real matrix M = ρ - 1 ⁢ I and a finite digit set D ⊂ ℤ n , where 0 < | ρ | < 1 and I is an n × n unit matrix. In this paper, we study the existence of a Fourier basis for L 2 ⁢ (μ M , D) , i.e., we find a discrete set Λ such that E Λ = { e 2 ⁢ π ⁢ i ⁢ 〈 λ , x 〉 : λ ∈ Λ } is an orthonormal basis for L 2 ⁢ (μ M , D) . Under some suitable conditions for D, some necessary and sufficient conditions for L 2 ⁢ (μ M , D) to admit infinite orthogonal exponential functions are given. Then we set up a framework to obtain necessary and sufficient conditions for L 2 ⁢ (μ M , D) to have a Fourier basis. Finally, we demonstrate how these results can be applied to self-similar measures. [ABSTRACT FROM AUTHOR]

Let μ M , D be a self-similar measure generated by an n × n expanding real matrix M = ρ - 1 ⁢ I and a finite digit set D ⊂ ℤ n , where 0 < | ρ | < 1 and I is an n × n unit matrix. In this paper, we study the existence of a Fourier basis for L 2 ⁢ (μ M , D) , i.e., we find a discrete set Λ such that E Λ = { e 2 ⁢ π ⁢ i ⁢ 〈 λ , x 〉 : λ ∈ Λ } is an orthonormal basis for L 2 ⁢ (μ M , D) . Under some suitable conditions for D, some necessary and sufficient conditions for L 2 ⁢ (μ M , D) to admit infinite orthogonal exponential functions are given. Then we set up a framework to obtain necessary and sufficient conditions for L 2 ⁢ (μ M , D) to have a Fourier basis. Finally, we demonstrate how these results can be applied to self-similar measures. [ABSTRACT FROM AUTHOR]

Susceptibility to domain changes for image classification hinders the application and development of deep neural networks. Domain adaptation (DA) makes use of domain-invariant characteristics to improve the performance of a model trained on labeled data from one domain (source domain) on an unlabeled domain (target) with a different data distribution. But existing DA methods simply use pretrained models (e.g., AlexNet, ResNet) for feature extraction, which are convolutional models that are trapped in localized features and fail to acquire long-distance dependencies. Furthermore, many approaches depend too much on pseudo-labels, which can impair adaptation efficiency and lead to unstable and inconsistent results. In this research, we present S2AC, a novel approach for unsupervised deep domain adaptation, that makes use of a stacked attention architecture as a feature map extractor. Our method can fuse domain discrepancy with minimizing a linear transformation of the second statistics (covariances) extended by the p-norm, while simultaneously designing pretext tasks on heuristics to improve the generality of the learning representation. In addition, we have developed a new trainable relative position embedding that not only reduces the model parameters but also enhances model accuracy and expedites the training process. To illustrate our method's efficacy and controllability, we designed extensive experiments based on the Office31, Office_Caltech_10, and OfficeHome datasets. To the best of our knowledge, the proposed method is the first attempt at incorporating attention-based networks and self-supervised learning for image domain adaptation, and has shown promising results. [ABSTRACT FROM AUTHOR]

Two-dimensional ferromagnetic semiconductors with coupled valley physics and piezoelectric responses offer unprecedented opportunities to miniaturize low-power multifunctional integrated devices. Prompted by epitaxial fabrication of nonmagnetic 2H-TiTe2 monolayer on the Au(111) substrate, we predict through both density functional theory and Monte Carlo simulations that the semilithiated 2H-TiTe2 monolayer (Li@2H-TiTe2) is a stable near room-temperature semiconducting ferromagnet. Under an out-of-plane magnetization, Li@2H-TiTe2 exhibits a clean valley polarization up to 160 meV in its conduction band and a valley-contrasting Berry curvature due to the broken inversion and time-reversal symmetries, in favor of achievable anomalous valley Hall effect. Alternatively, the simultaneous charge, spin, valley Hall currents can be realized as well in the ferromagnetic system with circularly polarized light. Furthermore, the missing mirror symmetry generates a scarce vertical piezoelectricity as large as 0.89 pm V−1. These findings indicate that asymmetric surface functionalization by Li deposition on the 2H-TiTe2 monolayer opens up a vital avenue to predesign superior and tailored multifunctional materials. [ABSTRACT FROM AUTHOR]