Convergence to the Self-similar Solutions to the Homogeneous Boltzmann Equation

The Boltzmann H-theorem implies that the solution to the Boltzmann equation tends to an equilibrium, that is, a Maxwellian when time tends to infinity. This has been proved in varies settings when the initial energy is finite. However, when the initial energy is infinite, the time asymptotic state is no longer described by a Maxwellian, but a self-similar solution obtained by Bobylev-Cercignani. The purpose of this paper is to rigorously justify this for the spatially homogeneous problem with Maxwellian molecule type cross section without angular cutoff.Comment: 23 page

Entanglement Preserving in Quantum Copying of Three-qubit Entangled State

We study the degree to which quantum entanglement survives when a three-qubit entangled state is copied by using local and non-local processes, respectively, and investigate iterating quantum copying for the three-qubit system. There may exist inter-three-qubit entanglement and inter-two-qubit entanglement for the three-qubit system. We show that both local and non-local copying processes degrade quantum entanglement in the three-particle system due to a residual correlation between the copied output and the copying machine. We also show that the inter-two-qubit entanglement is preserved better than the inter-three-qubit entanglement in the local cloning process. We find that non-local cloning is much more efficient than the local copying for broadcasting entanglement, and output state via non-local cloning exhibits the fidelity better than local cloning.Comment: 6 pages, 3 figure

Global solutions to the Vlasov-Poisson-Landau System

Based on the recent study on the Vlasov-Poisson-Boltzmann system with general angular cutoff potentials [3, 4], we establish in this paper the global existence of classical solutions to the Cauchy problem of the Vlasov-Poisson-Landau system that includes the Coulomb potential. This then provides a different approach on this topic from the recent work [8].Comment: 10 page

Adaptive Stochastic Alternating Direction Method of Multipliers

The Alternating Direction Method of Multipliers (ADMM) has been studied for years. The traditional ADMM algorithm needs to compute, at each iteration, an (empirical) expected loss function on all training examples, resulting in a computational complexity proportional to the number of training examples. To reduce the time complexity, stochastic ADMM algorithms were proposed to replace the expected function with a random loss function associated with one uniformly drawn example plus a Bregman divergence. The Bregman divergence, however, is derived from a simple second order proximal function, the half squared norm, which could be a suboptimal choice. In this paper, we present a new family of stochastic ADMM algorithms with optimal second order proximal functions, which produce a new family of adaptive subgradient methods. We theoretically prove that their regret bounds are as good as the bounds which could be achieved by the best proximal function that can be chosen in hindsight. Encouraging empirical results on a variety of real-world datasets confirm the effectiveness and efficiency of the proposed algorithms.Comment: 13 page

Fast and Accurate Graph Stream Summarization

A graph stream is a continuous sequence of data items, in which each item indicates an edge, including its two endpoints and edge weight. It forms a dynamic graph that changes with every item in the stream. Graph streams play important roles in cyber security, social networks, cloud troubleshooting systems and other fields. Due to the vast volume and high update speed of graph streams, traditional data structures for graph storage such as the adjacency matrix and the adjacency list are no longer sufficient. However, prior art of graph stream summarization, like CM sketches, gSketches, TCM and gMatrix, either supports limited kinds of queries or suffers from poor accuracy of query results. In this paper, we propose a novel Graph Stream Sketch (GSS for short) to summarize the graph streams, which has the linear space cost (O(|E|), E is the edge set of the graph) and the constant update time complexity (O(1)) and supports all kinds of queries over graph streams with the controllable errors. Both theoretical analysis and experiment results confirm the superiority of our solution with regard to the time/space complexity and query results' precision compared with the state-of-the-art

The Vlasov-Maxwell-Boltzmann system near Maxwellians in the whole space with very soft potentials

Since the work [13] by Guo [Invent. Math. 153 (2003), no. 3, 593--630], how to establish the global existence of perturbative classical solutions around a global Maxwellian to the Vlasov-Maxwell-Boltzmann system with the whole range of soft potentials has been an open problem. This is mainly due to the complex structure of the system, in particular, the degenerate dissipation at large velocity, the velocity-growth of the nonlinear term induced by the Lorentz force, and the regularity-loss of the electromagnetic fields. This paper aims to resolve this problem in the whole space provided that initial perturbation has sufficient regularity and velocity-integrability.Comment: 41 page

One-dimensional Compressible Navier-Stokes Equations with Temperature Dependent Transport Coefficients and Large Data

This paper is concerned with the global smooth non-vacuum solutions with large data to the Cauchy problem of the one-dimensional compressible Navier-Stokes equations with degenerate temperature dependent transport coefficients which satisfy conditions from the consideration in kinetic theory. A Nishida-Smoller type result is obtained.Comment: 36 page

Deep Compressive Autoencoder for Action Potential Compression in Large-Scale Neural Recording

Understanding the coordinated activity underlying brain computations requires large-scale, simultaneous recordings from distributed neuronal structures at a cellular-level resolution. One major hurdle to design high-bandwidth, high-precision, large-scale neural interfaces lies in the formidable data streams that are generated by the recorder chip and need to be online transferred to a remote computer. The data rates can require hundreds to thousands of I/O pads on the recorder chip and power consumption on the order of Watts for data streaming alone. We developed a deep learning-based compression model to reduce the data rate of multichannel action potentials. The proposed model is built upon a deep compressive autoencoder (CAE) with discrete latent embeddings. The encoder is equipped with residual transformations to extract representative features from spikes, which are mapped into the latent embedding space and updated via vector quantization (VQ). The decoder network reconstructs spike waveforms from the quantized latent embeddings. Experimental results show that the proposed model consistently outperforms conventional methods by achieving much higher compression ratios (20-500x) and better or comparable reconstruction accuracies. Testing results also indicate that CAE is robust against a diverse range of imperfections, such as waveform variation and spike misalignment, and has minor influence on spike sorting accuracy. Furthermore, we have estimated the hardware cost and real-time performance of CAE and shown that it could support thousands of recording channels simultaneously without excessive power/heat dissipation. The proposed model can reduce the required data transmission bandwidth in large-scale recording experiments and maintain good signal qualities. The code of this work has been made available at https://github.com/tong-wu-umn/spike-compression-autoencoderComment: 19 pages, 13 figure

Excited states of holographic superconductors

In this paper we re-investigate the model of the anti-de Sitter gravity coupled to Maxwell and charged scalar fields, which has been studied as the gravitational dual to a superconductor for a long time since the famous work [Phys.\ Rev.\ Lett.\ {\bf 101}, 031601 (2008)]. By numerical method, we present a novel family of solutions of holographical superconductor with excited states, and find there exists a lower critical temperature in the corresponding excited state. Moreover, we study the condensate and conductivity in the excited states. It is very interesting that the conductivity $\sigma$ of each excited state has an additional pole in $\text{Im}[\sigma]$ and a delta function in $\text{Re}[\sigma]$ arising at the low temperature inside the gap, which is just the evidence of the existence of excited states.Comment: 13 pages, 3 figures; V2: discussions and references added, accepted for publication in JHE

Strong magnetization and Chern insulators in compressed graphene/CrI3_{3} van der Waals heterostructures

Graphene-based heterostructures are a promising material system for designing the topologically nontrivial Chern insulating devices. Recently, a two-dimensional (2D) monolayer ferromagnetic insulator CrI$_{3}$ was successfully synthesized in experiments [Huang et al., Nature 546, 270 (2017)]. Here, these two interesting materials are proposed to build a heterostructure (Gr/CrI$_{3}$). Our first-principles calculations show that the system forms a van der Waals (vdW) heterostructure, relatively facilely fabricated in experiments. A Chern insulating state is acquired in the Gr/CrI$_{3}$ heterostructure if the vdW gap is compressed to certain extents by applying an external pressure. Amazingly, very strong magnetization (about 150 meV) is found in graphene, induced by the substrate CrI$_{3}$, despite the vdW interactions between them. A low-energy effective model is employed to understand the mechanism. The work functions, contact types, and band alignments of the Gr/CrI$_{3}$ heterostructure system are also studied. Our work demonstrates that the Gr/CrI$_{3}$ heterostructure is a promising system to observe the quantum anomalous Hall effect at high temperatures (up to 45 K) in experiments.Comment: 9 pages, 5 figure