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Tong Yang

Tong Yang appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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math.APArtificial IntelligenceComputer VisionMachine LearningComputation and Languagemath-phmath.MPmath.NANumerical AnalysisRoboticscs.CYData Structures and Algorithmseess.IVphysics.comp-phPopulations and Evolutionquant-phcond-mat.dis-nncond-mat.mtrl-sciDatabasesmath.GRmath.OCNetworking and Internet Architecturephysics.chem-ph

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Published work

71 published item(s)

preprint2026arXiv

ARC: Active and Reflection-driven Context Management for Long-Horizon Information Seeking Agents

Large language models are increasingly deployed as research agents for deep search and long-horizon information seeking, yet their performance often degrades as interaction histories grow. This degradation, known as context rot, reflects a failure to maintain coherent and task-relevant internal states over extended reasoning horizons. Existing approaches primarily manage context through raw accumulation or passive summarization, treating it as a static artifact and allowing early errors or misplaced emphasis to persist. Motivated by this perspective, we propose ARC, which is the first framework to systematically formulate context management as an active, reflection-driven process that treats context as a dynamic internal reasoning state during execution. ARC operationalizes this view through reflection-driven monitoring and revision, allowing agents to actively reorganize their working context when misalignment or degradation is detected. Experiments on challenging long-horizon information-seeking benchmarks show that ARC consistently outperforms passive context compression methods, achieving up to an 11% absolute improvement in accuracy on BrowseComp-ZH with Qwen2.5-32B-Instruct.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

EntroCoT: Enhancing Chain-of-Thought via Adaptive Entropy-Guided Segmentation

Chain-of-Thought (CoT) prompting has significantly enhanced the mathematical reasoning capabilities of Large Language Models. We find existing fine-tuning datasets frequently suffer from the "answer right but reasoning wrong" probelm, where correct final answers are derived from hallucinated, redundant, or logically invalid intermediate steps. This paper proposes EntroCoT, a unified framework for automatically identifying and refining low-quality CoT supervision traces. EntroCoT first proposes an entropy-based mechanism to segment the reasoning trace into multiple steps at uncertain junctures, and then introduces a Monte Carlo rollout-based mechanism to evaluate the marginal contribution of each step. By accurately filtering deceptive reasoning samples, EntroCoT constructs a high-quality dataset where every intermediate step in each reasoning trace facilitates the final answer. Extensive experiments on mathematical benchmarks demonstrate that fine-tuning on the subset constructed by EntroCoT consistently outperforms the baseslines of full-dataset supervision.

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preprint2026arXiv

Formal Skill: Programmable Runtime Skills for Efficient and Accurate LLM Agents

Large Language Model (LLM) agents increasingly act inside real workspaces, where tools and skills determine whether model reasoning becomes reliable action. Existing skills remain largely informal: Markdown skills and instruction packs encode procedures as long natural-language documents, while function calling, Model Context Protocol (MCP) servers, and framework tools structure individual actions but usually leave workflow state, policy enforcement, and completion discipline outside the skill itself. We introduce Formal Skill, a runtime-native abstraction that represents reusable capability with JSON metadata and action schemas, reliable Python executors, hook-governed control logic, Formal Skill routing, and skill-local runtime state. By moving reusable procedure from repeated prompt text into executable state machines and hook policies, Formal Skill gives agents a token-efficient and enforceable control surface. We implement the abstraction in FairyClaw, an open-source event-driven runtime for executable, observable, and composable Formal Skills. On Harness-Bench, FairyClaw obtains highly competitive average scores while using substantially fewer tokens, with especially strong results on tasks that expose the role of Formal Skill.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

From Static Risk to Dynamic Trajectories: Toward World-Model-Inspired Clinical Prediction

Clinical decision-making is a feedback system where risk estimates influence treatment, which in turn changes disease trajectories, and both shape clinicians' measurement practices. Static prediction often fails clinically: models trained on observational care logs conflate disease biology with clinician behavior, particularly under treatment confounder feedback and irregular or informative observation. This Review focuses on intervention-aware disease trajectory modeling in clinical AI--methods estimating patient-specific longitudinal disease evolution and assessing trajectory changes under alternative treatments. We organize the field around six linked components: three decision tasks (factual forecasting, counterfactual estimation, policy evaluation) and three data-generating mechanisms (disease evolution, treatment assignment, observation process) that determine identifiability. We present the first unified framework bridging forecasting, counterfactual trajectories, and policy evaluation across discrete/continuous time, explicitly addressing treatment assignment, time-varying confounding, and observation bias. We synthesize key method families (multistate/joint models, temporal point-process, deep sequence architectures, longitudinal causal inference), map them to relevant components, and align evaluation with claim strength via overlap diagnostics, uncertainty quantification, off-policy robustness, and target-trial validation. This synthesis advances benchmark prediction to decision-grade clinical evidence, enabling treatment-sensitive individualized futures, pre-deployment policy stress-testing, and safer closed-loop learning health systems that adapt/abstain when evidence is insufficient.

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preprint2026arXiv

Leveraging Error Diversity in Group Rollouts for Reinforcement Learning

Reinforcement Learning from Verifiable Rewards (RLVR) typically samples multiple responses per prompt and assigns binary rewards based on individual correctness, yet the collective structure of the group output, specifically the distribution of errors, is largely discarded. We identify this as a missed opportunity: empirical analysis reveals that error diversity within a group is a strong predictor of training success, with problems eliciting diverse wrong answers benefiting substantially more from RLVR than those producing homogeneous failures. Motivated by this observation, we propose Error Diversity Advantage Shaping (EDAS), a lightweight, algorithm-agnostic technique that modulates the advantage signal for incorrect rollouts based on intra-group error diversity. EDAS amplifies penalties for dominant, repeated errors and attenuates penalties for rare, exploratory ones, thereby encouraging the model to maintain diverse reasoning paths and discouraging error perseveration. Crucially, EDAS operates as a simple post-hoc adjustment that can be seamlessly integrated into any RLVR algorithm. We validate EDAS on top of several mainstream RLVR methods across a series of models and seven challenging math benchmarks, demonstrating consistent improvements. Notably, EDAS yields an average improvement of 6.29 points over DAPO on Qwen3-8B across seven benchmarks, confirming that exploiting the latent information in group rollouts is a broadly effective strategy for strengthening RLVR.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

LPFQA: A Long-Tail Professional Forum-based Benchmark for LLM Evaluation

Large Language Models (LLMs) perform well on standard reasoning and question-answering benchmarks, yet such evaluations often fail to capture their ability to handle long-tail, expertise-intensive knowledge in real-world professional scenarios. We introduce LPFQA, a long-tail knowledge benchmark derived from authentic professional forum discussions, covering 7 academic and industrial domains with 430 curated tasks grounded in practical expertise. LPFQA evaluates specialized reasoning, domain-specific terminology understanding, and contextual interpretation, and adopts a hierarchical difficulty structure to ensure semantic clarity and uniquely identifiable answers. Experiments on over multiple mainstream LLMs reveal substantial performance gaps, particularly on tasks requiring deep domain reasoning, exposing limitations overlooked by existing benchmarks. Overall, LPFQA provides an authentic and discriminative evaluation framework that complements prior benchmarks and informs future LLM development.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

NL2Repo-Bench: Towards Long-Horizon Repository Generation Evaluation of Coding Agents

Recent advances in coding agents suggest rapid progress toward autonomous software development, yet existing benchmarks fail to rigorously evaluate the long-horizon capabilities required to build complete software systems. Most prior evaluations focus on localized code generation, scaffolded completion, or short-term repair tasks, leaving open the question of whether agents can sustain coherent reasoning, planning, and execution over the extended horizons demanded by real-world repository construction. To address this gap, we present NL2Repo Bench, a benchmark explicitly designed to evaluate the long-horizon repository generation ability of coding agents. Given only a single natural-language requirements document and an empty workspace, agents must autonomously design the architecture, manage dependencies, implement multi-module logic, and produce a fully installable Python library. Our experiments across state-of-the-art open- and closed-source models reveal that long-horizon repository generation remains largely unsolved: even the strongest agents achieve below 40% average test pass rates and rarely complete an entire repository correctly. Detailed analysis uncovers fundamental long-horizon failure modes, including premature termination, loss of global coherence, fragile cross-file dependencies, and inadequate planning over hundreds of interaction steps. NL2Repo Bench establishes a rigorous, verifiable testbed for measuring sustained agentic competence and highlights long-horizon reasoning as a central bottleneck for the next generation of autonomous coding agents.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

The Molecular Structure of Thought: Mapping the Topology of Long Chain-of-Thought Reasoning

Large language models (LLMs) often fail to learn effective long chain-of-thought (Long CoT) reasoning from human or non-Long-CoT LLMs imitation. To understand this, we propose that effective and learnable Long CoT trajectories feature stable molecular-like structures in unified view, which are formed by three interaction types: Deep-Reasoning (covalent-like), Self-Reflection (hydrogen-bond-like), and Self-Exploration (van der Waals-like). Analysis of distilled trajectories reveals these structures emerge from Long CoT fine-tuning, not keyword imitation. We introduce Effective Semantic Isomers and show that only bonds promoting fast entropy convergence support stable Long CoT learning, while structural competition impairs training. Drawing on these findings, we present Mole-Syn, a distribution-transfer-graph method that guides synthesis of effective Long CoT structures, boosting performance and RL stability across benchmarks.

0 citations0 reviewsNo code linkedOpen access
preprint2024arXiv

Slot-guided Volumetric Object Radiance Fields

We present a novel framework for 3D object-centric representation learning. Our approach effectively decomposes complex scenes into individual objects from a single image in an unsupervised fashion. This method, called slot-guided Volumetric Object Radiance Fields (sVORF), composes volumetric object radiance fields with object slots as a guidance to implement unsupervised 3D scene decomposition. Specifically, sVORF obtains object slots from a single image via a transformer module, maps these slots to volumetric object radiance fields with a hypernetwork and composes object radiance fields with the guidance of object slots at a 3D location. Moreover, sVORF significantly reduces memory requirement due to small-sized pixel rendering during training. We demonstrate the effectiveness of our approach by showing top results in scene decomposition and generation tasks of complex synthetic datasets (e.g., Room-Diverse). Furthermore, we also confirm the potential of sVORF to segment objects in real-world scenes (e.g., the LLFF dataset). We hope our approach can provide preliminary understanding of the physical world and help ease future research in 3D object-centric representation learning.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

3D hyperbolic Navier-Stokes equations in a thin strip: global well-posedness and hydrostatic limit in Gevrey space

We consider the hyperbolic version of three-dimensional anisotropic Naver-Stokes equations in a thin strip and its hydrostatic limit that is a hyperbolic Prandtl type equations. We prove the global-in-time existence and uniqueness for the two systems and the hydrostatic limit when the initial data belong to the Gevrey function space with index 2. The proof is based on a direct energy method by observing the damping effect in the systems.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Anchor DETR: Query Design for Transformer-Based Object Detection

In this paper, we propose a novel query design for the transformer-based object detection. In previous transformer-based detectors, the object queries are a set of learned embeddings. However, each learned embedding does not have an explicit physical meaning and we cannot explain where it will focus on. It is difficult to optimize as the prediction slot of each object query does not have a specific mode. In other words, each object query will not focus on a specific region. To solved these problems, in our query design, object queries are based on anchor points, which are widely used in CNN-based detectors. So each object query focuses on the objects near the anchor point. Moreover, our query design can predict multiple objects at one position to solve the difficulty: "one region, multiple objects". In addition, we design an attention variant, which can reduce the memory cost while achieving similar or better performance than the standard attention in DETR. Thanks to the query design and the attention variant, the proposed detector that we called Anchor DETR, can achieve better performance and run faster than the DETR with 10$\times$ fewer training epochs. For example, it achieves 44.2 AP with 19 FPS on the MSCOCO dataset when using the ResNet50-DC5 feature for training 50 epochs. Extensive experiments on the MSCOCO benchmark prove the effectiveness of the proposed methods. Code is available at \url{https://github.com/megvii-research/AnchorDETR}.

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preprint2022arXiv

Cannikin's Law in Tensor Modeling: A Rank Study for Entanglement and Separability in Tensor Complexity and Model Capacity

This study clarifies the proper criteria to assess the modeling capacity of a general tensor model. The work analyze the problem based on the study of tensor ranks, which is not a well-defined quantity for higher order tensors. To process, the author introduces the separability issue to discuss the Cannikin's law of tensor modeling. Interestingly, a connection between entanglement studied in information theory and tensor analysis is established, shedding new light on the theoretical understanding for modeling capacity problems.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Efficient Distance-Optimal Tethered Path Planning in Planar Environments: The Workspace Convexity

The main contribution of this paper is the proof of the convexity of the omni-directional tethered robot workspace (namely, the set of all tether-length-admissible robot configurations), as well as a set of distance-optimal tethered path planning algorithms that leverage the workspace convexity. The workspace is proven to be topologically a simply-connected subset and geometrically a convex subset of the set of all configurations. As a direct result, the tether-length-admissible optimal path between two configurations is proven exactly the untethered collision-free locally shortest path in the homotopy specified by the concatenation of the tether curve of the given configurations, which can be simply constructed by performing an untethered path shortening process in the 2D environment instead of a path searching process in the pre-calculated workspace. The convexity is an intrinsic property to the tethered robot kinematics, thus has universal impacts on all high-level distance-optimal tethered path planning tasks: The most time-consuming workspace pre-calculation (WP) process is replaced with a goal configuration pre-calculation (GCP) process, and the homotopy-aware path searching process is replaced with untethered path shortening processes. Motivated by the workspace convexity, efficient algorithms to solve the following problems are naturally proposed: (a) The optimal tethered reconfiguration (TR) planning problem is solved by a locally untethered path shortening (UPS) process, (b) The classic optimal tethered path (TP) planning problem (from a starting configuration to a goal location whereby the target tether state is not assigned) is solved by a GCP process and $n$ UPS processes, where $n$ is the number of tether-length-admissible configurations that visit the goal location, (c) The optimal tethered motion to visit a sequence of multiple goal locations, referred to as

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preprint2022arXiv

Efficient Search of the k Shortest Non-Homotopic Paths by Eliminating Non-k-Optimal Topologies

An efficient algorithm to solve the $k$ shortest non-homotopic path planning ($k$-SNPP) problem in a 2D environment is proposed in this paper. Motivated by accelerating the inefficient exploration of the homotopy-augmented space of the 2D environment, our fundamental idea is to identify the non-$k$-optimal path topologies as early as possible and terminate the pathfinding along them. This is a non-trivial practice because it has to be done at an intermediate state of the path planning process when locally shortest paths have not been fully constructed. In other words, the paths to be compared have not rendezvoused at the goal location, which makes the homotopy theory, modelling the spatial relationship among the paths having the same endpoint, not applicable. This paper is the first work that develops a systematic distance-based topology simplification mechanism to solve the $k$-SNPP task, whose core contribution is to assert the distance-based order of non-homotopic locally shortest paths before constructing them. If the order can be predicted, then those path topologies having more than $k$ better topologies are proven free of the desired $k$ paths and thus can be safely discarded during the path planning process. To this end, a hierarchical topological tree is proposed as an implementation of the mechanism, whose nodes are proven to expand in non-homotopic directions and edges (collision-free path segments) are proven locally shortest. With efficient criteria that observe the order relations between partly constructed locally shortest paths being imparted into the tree, the tree nodes that expand in non-$k$-optimal topologies will not be expanded. As a result, the computational time for solving the $k$-SNPP problem is reduced by near two orders of magnitude.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Global Well-posedness of a Prandtl Model from MHD in Gevrey Function Spaces

We consider a Prandtl model derived from MHD in the Prandtl-Hartmann regime that has a damping term due to the effect of the Hartmann boundary layer. A global-in-time well-posedness is obtained in the Gevrey function space with the optimal index $2$. The proof is based on a cancellation mechanism through some auxiliary functions from the study of the Prandtl equation and an observation about the structure of the loss of one order tangential derivatives through twice operations of the Prandtl operator

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preprint2022arXiv

Knowledgebra: An Algebraic Learning Framework for Knowledge Graph

Knowledge graph (KG) representation learning aims to encode entities and relations into dense continuous vector spaces such that knowledge contained in a dataset could be consistently represented. Dense embeddings trained from KG datasets benefit a variety of downstream tasks such as KG completion and link prediction. However, existing KG embedding methods fell short to provide a systematic solution for the global consistency of knowledge representation. We developed a mathematical language for KG based on an observation of their inherent algebraic structure, which we termed as Knowledgebra. By analyzing five distinct algebraic properties, we proved that the semigroup is the most reasonable algebraic structure for the relation embedding of a general knowledge graph. We implemented an instantiation model, SemE, using simple matrix semigroups, which exhibits state-of-the-art performance on standard datasets. Moreover, we proposed a regularization-based method to integrate chain-like logic rules derived from human knowledge into embedding training, which further demonstrates the power of the developed language. As far as we know, by applying abstract algebra in statistical learning, this work develops the first formal language for general knowledge graphs, and also sheds light on the problem of neural-symbolic integration from an algebraic perspective.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

LGD: Label-guided Self-distillation for Object Detection

In this paper, we propose the first self-distillation framework for general object detection, termed LGD (Label-Guided self-Distillation). Previous studies rely on a strong pretrained teacher to provide instructive knowledge that could be unavailable in real-world scenarios. Instead, we generate an instructive knowledge based only on student representations and regular labels. Our framework includes sparse label-appearance encoder, inter-object relation adapter and intra-object knowledge mapper that jointly form an implicit teacher at training phase, dynamically dependent on labels and evolving student representations. They are trained end-to-end with detector and discarded in inference. Experimentally, LGD obtains decent results on various detectors, datasets, and extensive tasks like instance segmentation. For example in MS-COCO dataset, LGD improves RetinaNet with ResNet-50 under 2x single-scale training from 36.2% to 39.0% mAP (+ 2.8%). It boosts much stronger detectors like FCOS with ResNeXt-101 DCN v2 under 2x multi-scale training from 46.1% to 47.9% (+ 1.8%). Compared with a classical teacher-based method FGFI, LGD not only performs better without requiring pretrained teacher but also reduces 51% training cost beyond inherent student learning. Codes are available at https://github.com/megvii-research/LGD.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Q-ViT: Fully Differentiable Quantization for Vision Transformer

In this paper, we propose a fully differentiable quantization method for vision transformer (ViT) named as Q-ViT, in which both of the quantization scales and bit-widths are learnable parameters. Specifically, based on our observation that heads in ViT display different quantization robustness, we leverage head-wise bit-width to squeeze the size of Q-ViT while preserving performance. In addition, we propose a novel technique named switchable scale to resolve the convergence problem in the joint training of quantization scales and bit-widths. In this way, Q-ViT pushes the limits of ViT quantization to 3-bit without heavy performance drop. Moreover, we analyze the quantization robustness of every architecture component of ViT and show that the Multi-head Self-Attention (MSA) and the Gaussian Error Linear Units (GELU) are the key aspects for ViT quantization. This study provides some insights for further research about ViT quantization. Extensive experiments on different ViT models, such as DeiT and Swin Transformer show the effectiveness of our quantization method. In particular, our method outperforms the state-of-the-art uniform quantization method by 1.5% on DeiT-Tiny.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

QCluster: Clustering Packets for Flow Scheduling

Flow scheduling is crucial in data centers, as it directly influences user experience of applications. According to different assumptions and design goals, there are four typical flow scheduling problems/solutions: SRPT, LAS, Fair Queueing, and Deadline-Aware scheduling. When implementing these solutions in commodity switches with limited number of queues, they need to set static parameters by measuring traffic in advance, while optimal parameters vary across time and space. This paper proposes a generic framework, namely QCluster, to adapt all scheduling problems for limited number of queues. The key idea of QCluster is to cluster packets with similar weights/properties into the same queue. QCluster is implemented in Tofino switches, and can cluster packets at a speed of 3.2 Tbps. To the best of our knowledge, QCluster is the fastest clustering algorithm. Experimental results in testbed with programmable switches and ns-2 show that QCluster reduces the average flow completion time (FCT) for short flows up to 56.6%, and reduces the overall average FCT up to 21.7% over state-of-the-art. All the source code in ns-2 is available in Github without.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Tensor-networks for High-order Polynomial Approximation: A Many-body Physics Perspective

We analyze the problem of high-order polynomial approximation from a many-body physics perspective, and demonstrate the descriptive power of entanglement entropy in capturing model capacity and task complexity. Instantiated with a high-order nonlinear dynamics modeling problem, tensor-network models are investigated and exhibit promising modeling advantages. This novel perspective establish a connection between quantum information and functional approximation, which worth further exploration in future research.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Unravelling Distance-Dependent Inter-Site Interactions and Magnetic Transition Effects of Heteronuclear Single Atom Catalysts on Electrochemical Oxygen Reduction

Inter-site interactions between single atom catalysts (SACs) in the high loading regime are critical to tuning the catalytic performance. However, the understanding on such interactions and their distance dependent effects remains elusive, especially for the heteronuclear SACs. In this study, we reveal the effects of the distance-dependent inter-site interaction on the catalytic performance of SACs. Using the density functional theory calculations, we systematically investigate the heteronuclear iron and cobalt single atoms co-supported on the nitrogen-doped graphene (FeN4-C and CoN4-C) for oxygen reduction reaction (ORR). We find that as the distance between Fe and Co SACs decreases, FeN4-C exhibits a reduced catalytic activity, which can be mitigated by the presence of an axial hydroxyl ligand, whereas the activity of CoN4-C shows a volcano-like evolution with the optimum reached at the intermediate distance. We further unravel that the transition towards the high-spin state upon adsorption of ORR intermediate adsorbates is responsible for the decreased activity of both FeN4-C and CoN4-C at short inter-site distance. Such high-spin state transition is also found to significantly shift the linear relation between hydroxyl (*OH) and hydroperoxyl (*OOH) adsorbates. These findings not only shed light on the SAC-specific effect of the distance-dependent inter-site interaction between heteronuclear SACs, but also pave a way towards shifting the long-standing linear relations observed in multiple-electron chemical reactions.

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preprint2022arXiv

Vector fields of Cancellation for the Prandtl Operators

It has been a fascinating topic in the study of boundary layer theory about the well-posedness of Prandtl equation that was derived in 1904. Recently, new ideas about cancellation to overcome the loss of tangential derivatives were obtained so that Prandtl equation can be shown to be well-posed in Sobolev spaces to avoid the use of Crocco transformation as in the classical work of Oleinik. This short note aims to show that the cancellation mechanism is in fact related to some intrinsic directional derivative that can be used to recover the tangential derivative under some structural assumption on the fluid near the boundary.

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preprint2021arXiv

Phase diagram and superlattice structures of monolayer phosphorus carbide (P$_x$C$_{1-x}$)

Phase stability and properties of two-dimensional phosphorus carbide, P$_x$C$_{1-x}$, are investigated using the first-principles method in combination with cluster expansion and Monte Carlo simulation. Monolayer P$_x$C$_{1-x}$ is found to be a phase separating system which indicates difficulty in fabricating monolayer P$_x$C$_{1-x}$ or crystalline P$_x$C$_{1-x}$ thin films. Nevertheless, a bottom-up design approach is used to determine the stable structures of P$_x$C$_{1-x}$ of various compositions which turn out to be superlattices consisting of alternating carbon and phosphorus nanoribbons along the armchair direction. Results of first-principles calculations indicate that once these structures are produced, they are mechanically and thermodynamically stable. All the ordered structures are predicted to be semiconductors, with band gap ranging from 0.2 to 1.2 eV. In addition, the monolayer P$_x$C$_{1-x}$ are predicted to have high carrier mobility, and high optical absorption in the ultraviolet region which shows a red-shift as the P:C ratio increases. These properties make 2D P$_x$C$_{1-x}$ promising materials for applications in electronics and optoelectronics.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

A Deep Learning Approach for COVID-19 Trend Prediction

In this work, we developed a deep learning model-based approach to forecast the spreading trend of SARS-CoV-2 in the United States. We implemented the designed model using the United States to confirm cases and state demographic data and achieved promising trend prediction results. The model incorporates demographic information and epidemic time-series data through a Gated Recurrent Unit structure. The identification of dominating demographic factors is delivered in the end.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

A new stability and convergence proof of the Fourier-Galerkin spectral method for the spatially homogeneous Boltzmann equation

Numerical approximation of the Boltzmann equation is a challenging problem due to its high-dimensional, nonlocal, and nonlinear collision integral. Over the past decade, the Fourier-Galerkin spectral method has become a popular deterministic method for solving the Boltzmann equation, manifested by its high accuracy and potential of being further accelerated by the fast Fourier transform. Albeit its practical success, the stability of the method is only recently proved by Filbet, F. & Mouhot, C. in [$ Trans. Amer. Math. Soc.$ 363, no. 4 (2011): 1947-1980.] by utilizing the "spreading" property of the collision operator. In this work, we provide a new proof based on a careful $L^2$ estimate of the negative part of the solution. We also discuss the applicability of the result to various initial data, including both continuous and discontinuous functions.

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preprint2020arXiv

Cellular Decomposition for Non-repetitive Coverage Task with Minimum Discontinuities

A mechanism to derive non-repetitive coverage path solutions with a proven minimal number of discontinuities is proposed in this work, with the aim to avoid unnecessary, costly end effector lift-offs for manipulators. The problem is motivated by the automatic polishing of an object. Due to the non-bijective mapping between the workspace and the joint-space, a continuous coverage path in the workspace may easily be truncated in the joint-space, incuring undesirable end effector lift-offs. Inversely, there may be multiple configuration choices to cover the same point of a coverage path through the solution of the Inverse Kinematics. The solution departs from the conventional local optimisation of the coverage path shape in task space, or choosing appropriate but possibly disconnected configurations, to instead explicitly explore the leaast number of discontinuous motions through the analysis of the structure of valid configurations in joint-space. The two novel contributions of this paper include proof that the least number of path discontinuities is predicated on the surrounding environment, independent from the choice of the actual coverage path; thus has a minimum. And an efficient finite cellular decomposition method to optimally divide the workspace into the minimum number of cells, each traversable without discontinuties by any arbitrary coverage path within. Extensive simulation examples and real-world results on a 5 DoF manipulator are presented to prove the validity of the proposed strategy in realistic settings.

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preprint2020arXiv

CovidNet: To Bring Data Transparency in the Era of COVID-19

Timely, creditable, and fine-granular case information is vital for local communities and individual citizens to make rational and data-driven responses to the COVID-19 pandemic. This paper presents CovidNet, a COVID-19 tracking project associated with a large scale epidemic dataset, which was initiated by 1Point3Acres. To the best of our knowledge, the project is the only platform providing real-time global case information of more than 4,124 sub-divisions from over 27 countries worldwide with multi-language supports. The platform also offers interactive visualization tools to analyze the full historical case curves in each region. Initially launched as a voluntary project to bridge the data transparency gap in North America in January 2020, this project by far has become one of the major independent sources worldwide and has been consumed by many other tracking platforms. The accuracy and freshness of the dataset is a result of the painstaking efforts from our voluntary teamwork, crowd-sourcing channels, and automated data pipelines. As of May 18, 2020, the project website has been visited more than 200 million times and the CovidNet dataset has empowered over 522 institutions and organizations worldwide in policy-making and academic researches. All datasets are openly accessible for non-commercial purposes at https://coronavirus.1point3acres.com via a formal request through our APIs.

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preprint2020arXiv

Learning Human-Object Interaction Detection using Interaction Points

Understanding interactions between humans and objects is one of the fundamental problems in visual classification and an essential step towards detailed scene understanding. Human-object interaction (HOI) detection strives to localize both the human and an object as well as the identification of complex interactions between them. Most existing HOI detection approaches are instance-centric where interactions between all possible human-object pairs are predicted based on appearance features and coarse spatial information. We argue that appearance features alone are insufficient to capture complex human-object interactions. In this paper, we therefore propose a novel fully-convolutional approach that directly detects the interactions between human-object pairs. Our network predicts interaction points, which directly localize and classify the inter-action. Paired with the densely predicted interaction vectors, the interactions are associated with human and object detections to obtain final predictions. To the best of our knowledge, we are the first to propose an approach where HOI detection is posed as a keypoint detection and grouping problem. Experiments are performed on two popular benchmarks: V-COCO and HICO-DET. Our approach sets a new state-of-the-art on both datasets. Code is available at https://github.com/vaesl/IP-Net.

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preprint2020arXiv

Magnetic effects on the solvability of 2D MHD boundary layer equations without resistivity in Sobolev spaces

In this paper, we are concerned with the magnetic effect on the Sobolev solvability of boundary layer equations for the 2D incompressible MHD system without resistivity. The MHD boundary layer is described by the Prandtl type equations derived from the incompressible viscous MHD system without resistivity under the no-slip boundary condition on the velocity. Assuming that the initial tangential magnetic field does not degenerate, a local-in-time well-posedness in Sobolev spaces is proved without the monotonicity condition on the velocity field. Moreover, we show that if the tangential magnetic field shear layer is degenerate at one point, then the linearized MHD boundary layer system around the shear layer profile is ill-posed in the Sobolev settings provided that the initial velocity shear flow is non-degenerately critical at the same point.

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preprint2020arXiv

NagE: Non-Abelian Group Embedding for Knowledge Graphs

We demonstrated the existence of a group algebraic structure hidden in relational knowledge embedding problems, which suggests that a group-based embedding framework is essential for designing embedding models. Our theoretical analysis explores merely the intrinsic property of the embedding problem itself hence is model-independent. Motivated by the theoretical analysis, we have proposed a group theory-based knowledge graph embedding framework, in which relations are embedded as group elements, and entities are represented by vectors in group action spaces. We provide a generic recipe to construct embedding models associated with two instantiating examples: SO3E and SU2E, both of which apply a continuous non-Abelian group as the relation embedding. Empirical experiments using these two exampling models have shown state-of-the-art results on benchmark datasets.

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preprint2020arXiv

NTIRE 2020 Challenge on Perceptual Extreme Super-Resolution: Methods and Results

This paper reviews the NTIRE 2020 challenge on perceptual extreme super-resolution with focus on proposed solutions and results. The challenge task was to super-resolve an input image with a magnification factor 16 based on a set of prior examples of low and corresponding high resolution images. The goal is to obtain a network design capable to produce high resolution results with the best perceptual quality and similar to the ground truth. The track had 280 registered participants, and 19 teams submitted the final results. They gauge the state-of-the-art in single image super-resolution.

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preprint2020arXiv

Perceptual Extreme Super Resolution Network with Receptive Field Block

Perceptual Extreme Super-Resolution for single image is extremely difficult, because the texture details of different images vary greatly. To tackle this difficulty, we develop a super resolution network with receptive field block based on Enhanced SRGAN. We call our network RFB-ESRGAN. The key contributions are listed as follows. First, for the purpose of extracting multi-scale information and enhance the feature discriminability, we applied receptive field block (RFB) to super resolution. RFB has achieved competitive results in object detection and classification. Second, instead of using large convolution kernels in multi-scale receptive field block, several small kernels are used in RFB, which makes us be able to extract detailed features and reduce the computation complexity. Third, we alternately use different upsampling methods in the upsampling stage to reduce the high computation complexity and still remain satisfactory performance. Fourth, we use the ensemble of 10 models of different iteration to improve the robustness of model and reduce the noise introduced by each individual model. Our experimental results show the superior performance of RFB-ESRGAN. According to the preliminary results of NTIRE 2020 Perceptual Extreme Super-Resolution Challenge, our solution ranks first among all the participants.

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preprint2020arXiv

Space- and Computationally-Efficient Set Reconciliation via Parity Bitmap Sketch (PBS)

Set reconciliation is a fundamental algorithmic problem that arises in many networking, system, and database applications. In this problem, two large sets A and B of objects (bitcoins, files, records, etc.) are stored respectively at two different network-connected hosts, which we name Alice and Bob respectively. Alice and Bob communicate with each other to learn $AΔB$, the difference between A and B, and as a result the reconciled set $A\bigcup B$. Current set reconciliation schemes are based on either Invertible Bloom Filters (IBF) or Error-Correction Codes (ECC). The former has a low computational complexity of O(d), where d is the cardinality of $AΔB$, but has a high communication overhead that is several times larger than the theoretical minimum. The latter has a low communication overhead close to the theoretical minimum, but has a much higher computational complexity of $O(d^2)$. In this work, we propose Parity Bitmap Sketch (PBS), an ECC- based set reconciliation scheme that gets the better of both worlds: PBS has both a low computational complexity of O(d) just like IBF-based solutions and a low communication overhead of roughly twice the theoretical minimum. A separate contribution of this work is a novel rigorous analytical framework that can be used for the precise calculation of various performance metrics and for the near-optimal parameter tuning of PBS.

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preprint2020arXiv

Spectral Clustering with Smooth Tiny Clusters

Spectral clustering is one of the most prominent clustering approaches. The distance-based similarity is the most widely used method for spectral clustering. However, people have already noticed that this is not suitable for multi-scale data, as the distance varies a lot for clusters with different densities. State of the art(ROSC and CAST ) addresses this limitation by taking the reachability similarity of objects into account. However, we observe that in real-world scenarios, data in the same cluster tend to present in a smooth manner, and previous algorithms never take this into account. Based on this observation, we propose a novel clustering algorithm, which con-siders the smoothness of data for the first time. We first divide objects into a great many tiny clusters. Our key idea is to cluster tiny clusters, whose centers constitute smooth graphs. Theoretical analysis and experimental results show that our clustering algorithm significantly outperforms state of the art. Although in this paper, we singly focus on multi-scale situations, the idea of data smoothness can certainly be extended to any clustering algorithms

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preprint2020arXiv

Variance Regularization for Accelerating Stochastic Optimization

While nowadays most gradient-based optimization methods focus on exploring the high-dimensional geometric features, the random error accumulated in a stochastic version of any algorithm implementation has not been stressed yet. In this work, we propose a universal principle which reduces the random error accumulation by exploiting statistic information hidden in mini-batch gradients. This is achieved by regularizing the learning-rate according to mini-batch variances. Due to the complementarity of our perspective, this regularization could provide a further improvement for stochastic implementation of generic 1st order approaches. With empirical results, we demonstrated the variance regularization could speed up the convergence as well as stabilize the stochastic optimization.

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preprint2020arXiv

Well-posedness in Gevrey function space for 3D Prandtl equations without Structural Assumption

We establish the well-posedness in Gevrey function space with optimal class of regularity 2 for the three dimensional Prandtl system without any structural assumption. The proof combines in a novel way a new cancellation in the system with some of the old ideas to overcome the difficulty of the loss of derivatives in the system.This shows that the three dimensional instabilities in the system leading to ill-posedness are not worse than the two dimensional ones.

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preprint2020arXiv

Well-posedness of the MHD boundary layer system in Gevrey function space without Structural Assumption

We establish the well-posedness of the MHD boundary layer system in Gevrey function space without any structural assumption. Compared to the classical Prandtl equation, the loss of tangential derivative comes from both the velocity and magnetic fields that are coupled with each other. By observing a new type of cancellation mechanism in the system for overcoming the loss derivative degeneracy, we show that the MHD boundary layer system is well-posed with Gevrey index up to $3/2$ in both two and three dimensional spaces.

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preprint2019arXiv

DetNAS: Backbone Search for Object Detection

Object detectors are usually equipped with backbone networks designed for image classification. It might be sub-optimal because of the gap between the tasks of image classification and object detection. In this work, we present DetNAS to use Neural Architecture Search (NAS) for the design of better backbones for object detection. It is non-trivial because detection training typically needs ImageNet pre-training while NAS systems require accuracies on the target detection task as supervisory signals. Based on the technique of one-shot supernet, which contains all possible networks in the search space, we propose a framework for backbone search on object detection. We train the supernet under the typical detector training schedule: ImageNet pre-training and detection fine-tuning. Then, the architecture search is performed on the trained supernet, using the detection task as the guidance. This framework makes NAS on backbones very efficient. In experiments, we show the effectiveness of DetNAS on various detectors, for instance, one-stage RetinaNet and the two-stage FPN. We empirically find that networks searched on object detection shows consistent superiority compared to those searched on ImageNet classification. The resulting architecture achieves superior performance than hand-crafted networks on COCO with much less FLOPs complexity.

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preprint2016arXiv

A Shifting Bloom Filter Framework for Set Queries

Set queries are fundamental operations in computer systems and applications.This paper addresses the fundamental problem of designing a probabilistic data structure that can quickly process set queries using a small amount of memory. We propose a Shifting Bloom Filter (ShBF) framework for representing and querying sets. We demonstrate the effectiveness of ShBF using three types of popular set queries: membership, association, and multiplicity queries. The key novelty of ShBF is on encoding the auxiliary information of a set element in a location offset. In contrast, prior BF based set data structures allocate additional memory to store auxiliary information. To evaluate ShBF in comparison with prior art, we conducted experiments using real-world network traces. Results show that ShBF significantly advances the state-of-the-art on all three types of set queries.

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preprint2016arXiv

Ill-posedness of the Prandtl equations in Sobolev spaces around a shear flow with general decay

Motivated by the paper by D. Gerard-Varet and E. Dormy [JAMS, 2010] about the linear ill-posedness for the Prandtl equations around a shear flow with exponential decay in normal variable, and the recent study of well-posedness on the Prandtl equations in Sobolev spaces, this paper aims to extend the result in \cite{GV-D} to the case when the shear flow has general decay. The key observation is to construct an approximate solution that captures the initial layer to the linearized problem motivated by the precise formulation of solutions to the inviscid Prandtl equations.

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preprint2016arXiv

Measure valued solutions to the spatially homogeneous Boltzmann equation without angular cutoff

A uniform approach is introduced to study the existence of measure valued solutions to the homogeneous Boltzmann equation for both hard potential with finite energy, and soft potential with finite or infinite energy, by using Toscani metric. Under the non-angular cutoff assumption on the cross-section, the solutions obtained are shown to be in the Schwartz space in the velocity variable as long as the initial data is not a single Dirac mass without any extra moment condition for hard potential, and with the boundedness on moments of any order for soft potential.

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preprint2016arXiv

Well-posedness of thermal layer equations for inviscid compressible flows

A semi-explicit formula of solution to the boundary layer system for thermal layer derived from the compressible Navier-Stokes equations with the non-slip boundary condition when the viscosity coefficients vanish is given, in particular in three space dimension. In contrast to the inviscid Prandtl system studied by [7] in two space dimension, the main difficulty comes from the coupling of the velocity field and the temperature field through a degenerate parabolic equation. The convergence of these boundary layer equations to the inviscid Prandtl system is justified when the initial temperature goes to a constant. Moreover, the time asymptotic stability of the linearized system around a shear flow is given, and in particular, it shows that in three space dimension, the asymptotic stability depends on whether the direction of tangential velocity field of the shear flow is invariant in the normal direction respective to the boundary.

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preprint2015arXiv

A characterization of probability measure with finite moment and an application to the Boltzmann equation

We characterize probability measure with finite moment of any order in terms of the symmetric difference operators of their Fourier transforms. By using our new characterization, we prove the continuity $f(t,v)\in C((0, \infty),L^1_{2k-2 +α})$, where $f(t, v)$ stands for the density of unique measure-valued solution $(F_t)_{t\ge0}$ of the Cauchy problem for the homogeneous non-cutoff Boltzmann equation, with Maxwellian molecules, corresponding to a probability measure initial datum $F_0$ satisfying \[ \int |v|^{2k-2+α} dF_0(v) < \infty, 0\leq α< 2,k= 2, 3, 4,\cdots \] provided that $F_0$ is not a single Dirac mass.

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preprint2015arXiv

Global existence of weak solutions to the three-dimensional Prandtl equations with A special structure

The global existence of weak solutions to the three space dimensional Prandtl equations is studied under some constraint on its structure. This is a continuation of our recent study on the local existence of classical solutions with the same structure condition. It reveals the sufficiency of the monotonicity condition on one component of the tangential velocity field and the favorable condition on pressure in the same direction that leads to global existence of weak solutions. This generalizes the result obtained by Xin-Zhang on the two-dimensional Prandtl equations to the three-dimensional setting.

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preprint2015arXiv

Justification of Diffusion limit for the Boltzmann Equation with a non-trivial Profile

Under the diffusion scaling and a scaling assumption on the microscopic component, a non-classical fluid dynamic system was derived in \cite{BGLY} that is related to the system of ghost effect derived in \cite{Sone-2} in different settings. This paper aims to justify this limit system for a non-trivial background profile with slab symmetry. The result reveals not only the diffusion phenomena in the temperature and density, but also the flow of higher order in Knudsen number due to the gradient of the temperature. Precisely, we show that the solution to the Boltzmann equation converges to a diffusion wave with decay rates in both Knudsen number and time.

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preprint2014arXiv

A new characterization and global regularity of infinite energy solutions to the homogeneous Boltzmann equation

The purpose of this paper is to introduce a new characterization of the characteristic functions for the study on the measure valued solution to the homogeneous Boltzmann equation so that it precisely captures the moment constraint in physics. This significantly improves the previous result by Cannone-Karch [CPAM 63(2010), 747-778] in the sense that the new characterization gives a complete description of infinite energy solutions for the Maxwellian cross section. In addition, the global in time smoothing effect of the infinite energy solution except for a single Dirac mass initial datum is justified as for the finite energy solution.

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preprint2014arXiv

A well-posedness theory for the Prandtl equations in three space variables

The well-posedness of the three space dimensional Prandtl equations is studied under some constraint on its flow structure. It reveals that the classical Burgers equation plays an important role in determining this type of flow with special structure, that avoids the appearance of the complicated secondary flow in the three-dimensional Prandtl boundary layers. And the sufficiency of the monotonicity condition on the tangential velocity field for the existence of solutions to the Prandtl boundary layer equations is illustrated in the three dimensional setting. Moreover, it is shown that this structured flow is linearly stable for any three-dimensional perturbation.

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preprint2014arXiv

Local well-posedness of Prandtl equations for compressible flow in two space variables

In this paper, we consider the local well-posedness of the Prandtl boundary layer equations that describe the behavior of boundary layer in the small viscosity limit of the compressible isentropic Navier-Stokes equations with non-slip boundary condition. Under the strictly monotonic assumption on the tangential velocity in the normal variable, we apply the Nash-Moser-Hörmander iteration scheme and further develop the energy method introduced in [1] to obtain the well-posedness of the equations locally in time.

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preprint2014arXiv

On the ill-posedness of the Prandtl equations in three space dimensions

In this paper, we give an instability criterion for the Prandtl equations in three space variables, which shows that the monotonicity condition of tangential velocity fields is not sufficient for the well-posedness of the three dimensional Prandtl equations, in contrast to the classical well-posedness theory of the Prandtl equations in two space variables under the Oleinik monotonicity assumption of the tangential velocity. Both of linear stability and nonlinear stability are considered. This criterion shows that the monotonic shear flow is linear stable for the three dimensional Prandtl equations if and only if the tangential velocity field direction is invariant with respect to the normal variable, and this result is an exact complement to our recent work \cite{LWY} on the well-posedness theory for the three dimensional Prandtl equations with special structure.

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preprint2014arXiv

Spectrum analysis and optimal decay rates of the bipolar Vlasov-Poisson-Boltzmann equations

In the present paper, we consider the initial value problem for the bipolar Vlasov-Poisson-Boltzmann (bVPB) system and its corresponding modified Vlasov-Poisson-Boltzmann (mVPB). We give the spectrum analysis on the linearized bVPB and mVPB systems around their equilibrium state and show the optimal convergence rate of global solutions. It was showed that the electric field decays exponentially and the distribution function tends to the absolute Maxwellian at the optimal convergence rate $(1+t)^{-3/4}$ for the bVPB system, yet both the electric field and the distribution function converge to equilibrium state at the optimal rate $(1+t)^{-3/4}$ for the mVPB system.

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preprint2014arXiv

Spectrum analysis for the Vlasov-Poisson-Boltzmann system

By identifying a norm capturing the effect of the forcing governed by the Poisson equation, we give a detailed spectrum analysis on the linearized Vlasov-Poisson-Boltzmann system around a global Maxwellian. It is shown that the electric field governed by the self-consistent Poisson equation plays a key role in the analysis so that the spectrum structure is genuinely different from the well-known one of the Boltzmann equation. Based on this, we give the optimal time decay rates of solutions to the equilibrium.

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preprint2014arXiv

Spectrum Structure and Behaviors of the Vlasov-Maxwell-Boltzmann Systems

The spectrum structures and behaviors of the Vlasov-Maxwell-Boltzmann (VMB) systems for both two species and one species are studied in this paper. The analysis shows the effect of the Lorentz force induced by the electro-magnetic field leads to some different structure of spectrum from the classical Boltzmann equation and the closely related Vlasov-Poisson-Boltzmann system. And the significant difference between the two-species VMB model and one-species VMB model are given. The structure in high frequency illustrates the hyperbolic structure of the Maxwell equation. Furthermore, the long time behaviors and the optimal convergence rates to the equilibrium of the Vlasov-Maxwell-Boltzmann systems for both two species and one species are established based on the spectrum analysis, and in particular the phenomena of the electric field dominating and magnetic field dominating are observed for the one-species Vlasov-Maxwell-Boltzmann system.

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preprint2014arXiv

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.

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preprint2013arXiv

Global Solvability and Vanishing Shear Viscosity Limit of Planar Magnetohydrodynamic Equations with Large Initial Data

By observing a new relation between the magnetic pressure and the hydrodynamic pressure, global existence of classical solution to the full perfect MHD equations with large data is established, in particular including the case when all the viscosity, heat conductivity and diffusivity coefficients are constant. This can be viewed as an analog of the classical work by Kazhikhov-Shelukhin for the Navier-Stokes equations to the MHD equations. In addition, the vanishing shear viscosity limit is proved.

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preprint2013arXiv

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.

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preprint2013arXiv

Villani conjecture on smoothing effect of the homogeneous Boltzmann equation with measure initial datum

We justify the Villani conjecture on the smoothing effect for measure value solutions to the space homogeneous Boltzmann equation of Maxwellian type cross sections. This is the first rigorous proof of the smoothing effect for any measure value initial data except the single Dirac mass, which gives the optimal description on the regularity of solutions for positive time, caused by the singularity in the cross section. The main new ingredient in the proof is the introduction of a time degenerate coercivity estimate by using the microlocal analysis.

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preprint2012arXiv

The Vlasov-Poisson-Boltzmann System for Soft Potentials

An important physical model describing the dynamics of dilute weakly ionized plasmas in the collisional kinetic theory is the Vlasov-Poisson-Boltzmann system for which the plasma responds strongly to the self-consistent electrostatic force. This paper is concerned with the electron dynamics of kinetic plasmas in the whole space when the positive charged ion flow provides a spatially uniform background. We establish the global existence and optimal convergence rates of solutions near a global Maxwellian to the Cauchy problem on the Vlasov-Poisson-Boltzmann system for angular cutoff soft potentials with $-2\leq γ<0$. The main idea is to introduce a time dependent weight function in the velocity variable to capture the singularity of the cross-section at zero relative velocity.

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preprint2012arXiv

Well-posedness of The Prandtl Equation in Sobolev Spaces

We develop a new approach to study the well-posedness theory of the Prandtl equation in Sobolev spaces by using a direct energy method under a monotonicity condition on the tangential velocity field instead of using the Crocco transformation. Precisely, we firstly investigate the linearized Prandtl equation in some weighted Sobolev spaces when the tangential velocity of the background state is monotonic in the normal variable. Then to cope with the loss of regularity of the perturbation with respect to the background state due to the degeneracy of the equation, we apply the Nash-Moser-Hormander iteration to obtain a well-posedness theory of classical solutions to the nonlinear Prandtl equation when the initial data is a small perturbation of a monotonic shear flow.

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preprint2011arXiv

The Limit of the Boltzmann Equation to the Euler Equations for Riemann Problems

The convergence of the Boltzmann equaiton to the compressible Euler equations when the Knudsen number tends to zero has been a long standing open problem in the kinetic theory. In the setting of Riemann solution that contains the generic superposition of shock, rarefaction wave and contact discontinuity to the Euler equations, we succeed in justifying this limit by introducing hyperbolic waves with different solution backgrounds to capture the extra masses carried by the hyperbolic approximation of the rarefaction wave and the diffusion approximation of contact discontinuity.

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preprint2010arXiv

Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential

It is known that the singularity in the non-cutoff cross-section of the Boltzmann equation leads to the gain of regularity and gain of weight in the velocity variable. By defining and analyzing a non-isotropy norm which precisely captures the dissipation in the linearized collision operator, we first give a new and precise coercivity estimate for the non-cutoff Boltzmann equation for general physical cross sections. Then the Cauchy problem for the Boltzmann equation is considered in the framework of small perturbation of an equilibrium state. In this part, for the soft potential case in the sense that there is no positive power gain of weight in the coercivity estimate on the linearized operator, we derive some new functional estimates on the nonlinear collision operator. Together with the coercivity estimates, we prove the global existence of classical solutions for the Boltzmann equation in weighted Sobolev spaces.

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preprint2010arXiv

Bounded Solutions of the Boltzmann Equation in the Whole Space

We construct bounded classical solutions of the Boltzmann equation in the whole space without specifying any limit behaviors at the spatial infinity and without assuming the smallness condition on initial data. More precisely, we show that if the initial data is non-negative and belongs to a uniformly local Sobolev space in the space variable with Maxwellian type decay property in the velocity variable, then the Cauchy problem of the Boltzmann equation possesses a unique non-negative local solution in the same function space, both for the cutoff and non-cutoff collision cross section with mild singularity. The known solutions such as solutions on the torus (space periodic solutions) and in the vacuum (solutions vanishing at the spatial infinity), and solutions in the whole space having a limit equilibrium state at the spatial infinity are included in our category.

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preprint2010arXiv

Diffusion in a continuum model of self-propelled particles with alignment interaction

In this paper, we provide the $O(ε)$ corrections to the hydrodynamic model derived by Degond and Motsch from a kinetic version of the model by Vicsek & coauthors describing flocking biological agents. The parameter $ε$ stands for the ratio of the microscopic to the macroscopic scales. The $O(ε)$ corrected model involves diffusion terms in both the mass and velocity equations as well as terms which are quadratic functions of the first order derivatives of the density and velocity. The derivation method is based on the standard Chapman-Enskog theory, but is significantly more complex than usual due to both the non-isotropy of the fluid and the lack of momentum conservation.

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preprint2010arXiv

Fluid Dynamic Limit to the Riemann Solutions of Euler Equations: I. Superposition of rarefaction waves and contact discontinuity

Fluid dynamic limit to compressible Euler equations from compressible Navier-Stokes equations and Boltzmann equation has been an active topic with limited success so far. In this paper, we consider the case when the solution of the Euler equations is a Riemann solution consisting two rarefaction waves and a contact discontinuity and prove this limit for both Navier-Stokes equations and the Boltzmann equation when the viscosity, heat conductivity coefficients and the Knudsen number tend to zero respectively. In addition, the uniform convergence rates in terms of the above physical parameters are also obtained. It is noted that this is the first rigorous proof of this limit for a Riemann solution with superposition of three waves even though the fluid dynamic limit for a single wave has been proved.

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preprint2010arXiv

Global existence and full regularity of the Boltzmann equation without angular cutoff

We prove the global existence and uniqueness of classical solutions around an equilibrium to the Boltzmann equation without angular cutoff in some Sobolev spaces. In addition, the solutions thus obtained are shown to be non-negative and $C^\infty$ in all variables for any positive time. In this paper, we study the Maxwellian molecule type collision operator with mild singularity. One of the key observations is the introduction of a new important norm related to the singular behavior of the cross section in the collision operator. This norm captures the essential properties of the singularity and yields precisely the dissipation of the linearized collision operator through the celebrated H-theorem.

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preprint2010arXiv

The Boltzmann equation without angular cutoff in the whole space: III, Qualitative properties of solutions

This is a continuation of our series of works for the inhomogeneous Boltzmann equation. We study qualitative properties of classical solutions, precisely, the full regularization in all variables, uniqueness, non-negativity and convergence rate to the equilibrium. Together with the results of Parts I and II about the well posedness of the Cauchy problem around Maxwellian, we conclude this series with a satisfactory mathematical theory for Boltzmann equation without angular cutoff.

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preprint2009arXiv

Hydrodynamic Limit of the Boltzmann Equation with Contact Discontinuities

The hydrodynamic limit for the Boltzmann equation is studied in the case when the limit system, that is, the system of Euler equations contains contact discontinuities. When suitable initial data is chosen to avoid the initial layer, we prove that there exists a unique solution to the Boltzmann equation globally in time for any given Knudsen number. And this family of solutions converge to the local Maxwellian defined by the contact discontinuity of the Euler equations uniformly away from the discontinuity as the Knudsen number $\varepsilon$ tends to zero. The proof is obtained by an appropriately chosen scaling and the energy method through the micro-macro decomposition.

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preprint2009arXiv

Regularizing effect and local existence for non-cutoff Boltzmann equation

The Boltzmann equation without Grad's angular cutoff assumption is believed to have regularizing effect on the solution because of the non-integrable angular singularity of the cross-section. However, even though so far this has been justified satisfactorily for the spatially homogeneous Boltzmann equation, it is still basically unsolved for the spatially inhomogeneous Boltzmann equation. In this paper, by sharpening the coercivity and upper bound estimates for the collision operator, establishing the hypo-ellipticity of the Boltzmann operator based on a generalized version of the uncertainty principle, and analyzing the commutators between the collision operator and some weighted pseudo differential operators, we prove the regularizing effect in all (time, space and velocity) variables on solutions when some mild regularity is imposed on these solutions. For completeness, we also show that when the initial data has this mild regularity and Maxwellian type decay in velocity variable, there exists a unique local solution with the same regularity, so that this solution enjoys the $C^\infty$ regularity for positive time.

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