A compactness result for energy-minimizing harmonic maps with rough domain metric

In 1996, Shi generalized the epsilon-regularity theorem of Schoen and Uhlenbeck to energy-minimizing harmonic maps from a domain equipped with a bounded measurable Riemannian metric. In the present work we prove a compactness result for such energy-minimizing maps. As an application, we combine our result with Shi's theorem to give an improved bound on the Hausdorff dimension of the singular set, assuming that the map has bounded energy at all scales. This last assumption can be removed when the target manifold is simply-connected

Roton Instabilities and Wigner Crystallization of Rotating Dipolar Fermions in the Fractional Quantum Hall Regime

We point out the possibility of occurring instabilities in Laughlin liquids of rotating dipolar fermions with zero thickness. Previously such a system was predicted to be the Laughlin liquid for filling factors being greater and equal to 1/7. However, from intra-Landau-level excitations of the liquid in the single-mode approximation, the roton minima become negative and Laughlin liquids are unstable for filling factors being less and equal to 1/7. We then conclude that there are correlated Wigner crystals for filling factors being less and equal to 1/7.Comment: 4 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:0808.043

Conservation laws of some lattice equations

We derive infinitely many conservation laws for some multi-dimensionally consistent lattice equations from their Lax pairs. These lattice equations are the Nijhoff-Quispel-Capel equation, lattice Boussinesq equation, lattice nonlinear Schr\"{o}dinger equation, modified lattice Boussinesq equation, Hietarinta's Boussinesq-type equations, Schwarzian lattice Boussinesq equation and Toda-modified lattice Boussinesq equation

From Node-Line Semimetals to Large Gap QSH States in New Family of Pentagonal Group-IVA Chalcogenide

Two-dimensional (2D) topological insulators (TIs) have attracted tremendous research interest from both theoretical and experimental fields in recent years. However, it is much less investigated in realizing node line (NL) semimetals in 2D materials.Combining first-principles calculations and $k \cdot p$ model, we find that NL phases emerge in p-CS$_2$ and p-SiS$_2$, as well as other pentagonal IVX$_2$ films, i.e. p-IVX$_2$ (IV= C, Si, Ge, Sn, Pb; X=S, Se, Te) in the absence of spin-orbital coupling (SOC). The NLs in p-IVX$_2$ form symbolic Fermi loops centered around the $\Gamma$ point and are protected by mirror reflection symmetry. As the atomic number is downward shifted, the NL semimetals are driven into 2D TIs with the large bulk gap up to 0.715 eV induced by the remarkable SOC effect.The nontrivial bulk gap can be tunable under external biaxial and uniaxial strain. Moreover, we also propose a quantum well by sandwiching p-PbTe$_2$ crystal between two NaI sheets, in which p-PbTe$_2$ still keeps its nontrivial topology with a sizable band gap ($\sim$ 0.5 eV). These findings provide a new 2D materials family for future design and fabrication of NL semimetals and TIs.Comment: 6 pages, 5 figures,2 table

Squared eigenfunction symmetry of the DΔ\DeltamKP hierarchy and its constraint

In this paper squared eigenfunction symmetry of the differential-difference modified Kadomtsev-Petviashvili (D$\Delta$mKP) hierarchy and its constraint are considered. Under the constraint, the Lax triplets of the D$\Delta$mKP hierarchy, together with their adjoint forms, give rise to the positive relativistic Toda (R-Toda) hierarchy. An invertible transformation is given to connect the positive and negative R-Toda hierarchies. The positive R-Toda hierarchy is reduced to the differential-difference Burgers hierarchy. We also consider another D$\Delta$mKP hierarchy and show that its squared eigenfunction symmetry constraint gives rise to the Volterra hierarchy. In addition, we revisit the Ragnisco-Tu hierarchy which is a squared eigenfunction symmetry constraint of the differential-difference Kadomtsev-Petviashvili (D$\Delta$KP) system. It was thought the Ragnisco-Tu hierarchy does not exist one-field reduction, but here we find an one-field reduction to reduce the hierarchy to the Volterra hierarchy. Besides, the differential-difference Burgers hierarchy are also investigated in Appendix. A multi-dimensionally consistent 3-point discrete Burgers equation is given.Comment: 30 pages, with an Appendi

Discrete Crum's Theorems and Integrable Lattice Equations

In this paper, we develop discrete versions of Darboux transformations and Crum's theorems for two second order difference equations. The difference equations are discretised versions (using Darboux transformations) of the spectral problems of the KdV quation, and of the modified KdV equation or sine-Gordon equation. Considering the discrete dynamics created by Darboux transformations for the difference equations, one obtains the lattice potential KdV equation, the lattice potential modified KdV equation and the lattice Schwarzian KdV equation, that are prototypes of integrable lattice equations. It turns out that, along the discretisation processes using Darboux transformations, two families of integrable systems (the KdV family, and the modified KdV or sine-Gordon family), including their continuous, semi-discrete and lattice versions, are explicitly constructed. As direct applications of the discrete Crum's theorems, multi-soliton solutions of the lattice equations are obtained.Comment: Modified Introduction and Concluding remarks, add some relevant references, correct typo

The Dependent Random Measures with Independent Increments in Mixture Models

When observations are organized into groups where commonalties exist amongst them, the dependent random measures can be an ideal choice for modeling. One of the propositions of the dependent random measures is that the atoms of the posterior distribution are shared amongst groups, and hence groups can borrow information from each other. When normalized dependent random measures prior with independent increments are applied, we can derive appropriate exchangeable probability partition function (EPPF), and subsequently also deduce its inference algorithm given any mixture model likelihood. We provide all necessary derivation and solution to this framework. For demonstration, we used mixture of Gaussians likelihood in combination with a dependent structure constructed by linear combinations of CRMs. Our experiments show superior performance when using this framework, where the inferred values including the mixing weights and the number of clusters both respond appropriately to the number of completely random measure used

DPP-Net: Device-aware Progressive Search for Pareto-optimal Neural Architectures

Recent breakthroughs in Neural Architectural Search (NAS) have achieved state-of-the-art performances in applications such as image classification and language modeling. However, these techniques typically ignore device-related objectives such as inference time, memory usage, and power consumption. Optimizing neural architecture for device-related objectives is immensely crucial for deploying deep networks on portable devices with limited computing resources. We propose DPP-Net: Device-aware Progressive Search for Pareto-optimal Neural Architectures, optimizing for both device-related (e.g., inference time and memory usage) and device-agnostic (e.g., accuracy and model size) objectives. DPP-Net employs a compact search space inspired by current state-of-the-art mobile CNNs, and further improves search efficiency by adopting progressive search (Liu et al. 2017). Experimental results on CIFAR-10 are poised to demonstrate the effectiveness of Pareto-optimal networks found by DPP-Net, for three different devices: (1) a workstation with Titan X GPU, (2) NVIDIA Jetson TX1 embedded system, and (3) mobile phone with ARM Cortex-A53. Compared to CondenseNet and NASNet (Mobile), DPP-Net achieves better performances: higher accuracy and shorter inference time on various devices. Additional experimental results show that models found by DPP-Net also achieve considerably-good performance on ImageNet as well.Comment: 13 pages 9 figures, ECCV 2018 Camera Read

Effective Techniques for Message Reduction and Load Balancing in Distributed Graph Computation

Massive graphs, such as online social networks and communication networks, have become common today. To efficiently analyze such large graphs, many distributed graph computing systems have been developed. These systems employ the "think like a vertex" programming paradigm, where a program proceeds in iterations and at each iteration, vertices exchange messages with each other. However, using Pregel's simple message passing mechanism, some vertices may send/receive significantly more messages than others due to either the high degree of these vertices or the logic of the algorithm used. This forms the communication bottleneck and leads to imbalanced workload among machines in the cluster. In this paper, we propose two effective message reduction techniques: (1)vertex mirroring with message combining, and (2)an additional request-respond API. These techniques not only reduce the total number of messages exchanged through the network, but also bound the number of messages sent/received by any single vertex. We theoretically analyze the effectiveness of our techniques, and implement them on top of our open-source Pregel implementation called Pregel+. Our experiments on various large real graphs demonstrate that our message reduction techniques significantly improve the performance of distributed graph computation.Comment: This is a long version of the corresponding WWW 2015 paper, with all proofs include

Smoothed Hierarchical Dirichlet Process: A Non-Parametric Approach to Constraint Measures

Time-varying mixture densities occur in many scenarios, for example, the distributions of keywords that appear in publications may evolve from year to year, video frame features associated with multiple targets may evolve in a sequence. Any models that realistically cater to this phenomenon must exhibit two important properties: the underlying mixture densities must have an unknown number of mixtures, and there must be some "smoothness" constraints in place for the adjacent mixture densities. The traditional Hierarchical Dirichlet Process (HDP) may be suited to the first property, but certainly not the second. This is due to how each random measure in the lower hierarchies is sampled independent of each other and hence does not facilitate any temporal correlations. To overcome such shortcomings, we proposed a new Smoothed Hierarchical Dirichlet Process (sHDP). The key novelty of this model is that we place a temporal constraint amongst the nearby discrete measures $\{G_j\}$ in the form of symmetric Kullback-Leibler (KL) Divergence with a fixed bound $B$. Although the constraint we place only involves a single scalar value, it nonetheless allows for flexibility in the corresponding successive measures. Remarkably, it also led us to infer the model within the stick-breaking process where the traditional Beta distribution used in stick-breaking is now replaced by a new constraint calculated from $B$. We present the inference algorithm and elaborate on its solutions. Our experiment using NIPS keywords has shown the desirable effect of the model