arXiv subjects

Wei Dai

Publications and source records attributed to Wei Dai.

At least 253 records · Page 14Linked to original sources

arXiv2019

Uniform a priori estimates for positive solutions of higher order Lane-Emden equations in $\mathbb{R}^n$

In this paper, we study the existence of uniform a priori estimates for positive solutions to Navier problems of higher order Lane-Emden equations \begin{equation*} (-Δ)^{m}u(x)=u^{p}(x), \qquad \,\, x\inΩ\end{equation*} for all large exponents $p$, where $Ω\subset\mathbb{R}^{n}$ is a star-shaped or strictly convex bounded domain with $C^{2m-2}$ boundary, $n\geq4$ and $2\leq m\leq\frac{n}{2}$. Our results extend those of previous authors for second order $m=1$ to general higher order cases $m\geq2$.

math.AP↗

arXiv2019

Liouville type theorems, a priori estimates and existence of solutions for critical order Hardy-Hénon equations in $\mathbb{R}^{n}$

In this paper, we consider the critical order Hardy-Hénon equations \begin{equation*} (-Δ)^{\frac{n}{2}}u(x)=\frac{u^{p}(x)}{|x|^{a}}, \,\,\,\,\,\,\,\,\,\,\, x \, \in \,\, \mathbb{R}^{n}, \end{equation*} where $n\geq4$ is even, $-\infty<a<n$, and $1<p<+\infty$. We first prove a Liouville theorem (Theorem \ref{Thm0}), that is, the unique nonnegative solution to this equation is $u\equiv0$. Then as an immediate application, we derive a priori estimates and hence existence of positive solutions to critical order Lane-Emden equations in bounded domains (Theorem \ref{Thm1} and \ref{Thm2}). Our results seem to be the first Liouville theorem, a priori estimates, and existence on the critical order equations in higher dimensions ($n\geq3$). Extensions to super-critical order Hardy-Hénon equations and inequalities will also be included (Theorem \ref{Thm0-sc} and \ref{Thm1-sc}).

math.AP↗

arXiv2019

Liouville type theorem for critical order Hénon-Lane-Emden type equations on a half space and its applications

In this paper, we are concerned with the critical order Hénon-Lane-Emden type equations with Navier boundary condition on a half space $\mathbb{R}^n_+$: \begin{equation}\label{NPDE0}\\\begin{cases} (-Δ)^{\frac{n}{2}} u(x)=f(x,u(x)),\ u(x)\geq0,\ x\in\mathbb{R}^{n}_+, \\ u=(-Δ)u = \cdots = (-Δ)^{\frac{n}{2}-1}u = 0,\ x\in\partial\mathbb{R}^{n}_+, \end{cases}\end{equation} where $u\in C^{n}(\mathbb{R}^{n}_+)\cap C^{n-2}(\overline{\mathbb{R}^{n}_+})$ and $n\geq2$ is even. We first consider the typical case $f(x,u)=|x|^{a}u^{p}$ with $0\leq a<\infty$ and $1<p<\infty$. We prove the super poly-harmonic properties and establish the equivalence between (0.1) and the corresponding integral equations \begin{equation}\label{IE0} u(x)=\int_{\mathbb{R}^{n}_+}G(x,y)f(y,u(y))dy, \end{equation} where $G(x,y)$ denotes the Green's function for $(-Δ)^{\frac{n}{2}}$ on $\mathbb{R}^n_+$ with Navier boundary conditions. Then, we establish Liouville theorem for (0.2) via ``the method of scaling spheres" developed initially in \cite{DQ0} by Dai and Qin, and hence we obtain the Liouville theorem for (0.1) on $\mathbb{R}^n_+$. As an application of the Liouville theorem on $\mathbb{R}^n_+$ (Theorem 1.6) and Liouville theorems in $\mathbb{R}^{n}$ established in Chen, Dai and Qin [4] for $n\geq4$ and Bidaut-Véron and Giacomini [1] for $n=2$, we derive a priori estimates and existence of positive solutions to critical order Lane-Emden equations in bounded domains for all $n\geq2$ and $1<p<\infty$. Extensions to IEs and PDEs with general nonlinearities $f(x,u)$ are also included.

math.AP↗

arXiv2019

Global existence and lifespan for semilinear wave equations with mixed nonlinear terms

Firstly, we study the equation $\square u = |u|^{q_c}+ |\partial u|^p$ with small data, where $q_c$ is the critical power of Strauss conjecture and $p\geq q_c.$ We obtain the optimal lifespan $\ln({T_\varepsilon})\approx\varepsilon^{-q_c(q_c-1)}$ in $n=3$, and improve the lower-bound of $T_\varepsilon$ from $\exp({c\varepsilon^{-(q_c-1)}})$ to $\exp({c\varepsilon^{-(q_c-1)^2/2}})$ in $n=2$. Then, we study the Cauchy problem with small initial data for a system of semilinear wave equations $\square u = |v|^q,$ $ \square v = |\partial_t u|^p$ in 3-dimensional space with $q<2$. We obtain that this system admits a global solution above a $p-q$ curve for spherically symmetric data. On the contrary, we get a new region where the solution will blow up.

math.AP↗

arXiv2019

Optimal Number of Measurements in a Linear System with Quadratically Decreasing SNR

We consider the design of a linear sensing system with a fixed energy budget assuming that the sampling noise is the dominant noise source. The energy constraint implies that the signal energy per measurement decreases linearly with the number of measurements. When the maximum sampling rate of the sampling circuit is chosen to match the designed sampling rate, the assumption on the noise implies that its variance increases approximately linearly with the sampling rate (number of measurements). Therefore, the overall SNR per measurement decreases quadratically in the number of measurements. Our analysis shows that, in this setting there is an optimal number of measurements. This is in contrast to the standard case, where noise variance remains unchanged with sampling rate, in which case more measurements imply better performance. Our results are based on a state evolution technique of the well-known approximate message passing algorithm. We consider both the sparse (e.g. Bernoulli-Gaussian and least-favorable distributions) and the non-sparse (e.g. Gaussian distribution) settings in both the real and complex domains. Numerical results corroborate our analysis.

eess.SP↗

arXiv2019

Probing cold nuclear matter effects with the productions of isolated-$γ$ and $γ$+jet in p$+$Pb collisions at $\sqrt{s_{NN}}=$ 8.16 TeV

We investigate the cold nuclear matter (CNM) effects on the productions of the isolated prompt photon and $γ+$jet in proton-lead collisions at $\rm 8.16$ TeV under the next-to-leading order (NLO) perturbative quantum chromodynamics calculations with four parametrizations for nuclear parton distribution functions (nPDFs), i.e. DSSZ, EPPS16, nCTEQ15, nIMParton. Our theoretical calculations provide good descriptions of pp baseline in the ATLAS collaboration and make predictions for future experimental results at $\rm p$+$\rm Pb$ collisions. We calculate the dependence of the nuclear modification factor of isolated prompt photon on transverse momentum $\rm p_T^γ$ and pseudo-rapidity $η^γ$ at very forward and backward rapidity regions, and demonstrate that the forward-to-backward yield asymmetries $\rm Y_{\rm pPb}^{\rm asym}$ as a function of $\rm p_T^γ$ with different nPDFs parametrizations have diverse behaviors. Furthermore, the nuclear modification factor of isolated-$γ+$jet $\rm R_{\rm pPb}^{γJet}$ as a function of $γ+$jet's pseudo-rapidity $η_{γ\rm Jet}=\frac{1}{2}(η_γ+η_{\rm Jet})$ at different average transverse momentum $\rm p_T^{\rm avg}=\frac{1}{2}(\rm p_T^γ+\rm p_T^{\rm Jet})$ has been discussed, which can facilitate a tomography study of CNM effects with precise locations in a rather wide kinematic region by varying the transverse momenta and rapidities of both isolated photon and jet in p+A collisions.

nucl-th↗

arXiv2018

The production of $b\bar{b}$ dijet in heavy-ion collisions at the LHC

We report our recent theoretical calculations for $b\bar{b}$ dijet production in high-energy nuclear collisions. The NLO+parton shower (PS) event generator SHERPA has been employed to provide the pp baseline of $b\bar{b}$ dijet production. A framework which combines Langevin transport equation to describe the evolution of heavy quarks and the higher-twist scheme to consider the inelastic energy loss of both light and heavy partons has been implemented. Within this framework, we present the theoretical result of transverse momentum imbalance $x_J=p_{T1}/p_{T2}$ both for inclusive dijet and $b\bar{b}$ dijet in Pb+Pb collision at 5.02 TeV. The energy loss of b jet would shift $x_J$ to smaller value relative to p+p reference which is consistent with the CMS data. In addition, we show the medium modification for angular correlation of $b\bar{b}$ dijet in A+A collision at $\sqrt{s_{NN}}=5.02$~TeV. We observe a stronger suppression in small $Δϕ=|ϕ_{b1}-ϕ_{b2}|$ region where the gluon splitting processes dominate relative to large $Δϕ$ region. The difference leads to a modest suppression on near side ($Δϕ\sim 0$) and enhancement on away side ($Δϕ\simπ$).

nucl-th↗

arXiv2018

Jet tomography in high-energy nuclear collisions

When an energetic parton traversing the QCD medium, it may suffer multiple scatterings and lose energy. This jet quenching phenomenon may lead to the suppression of leading hadron productions as well as medium modifications of full jet observables in heavy-ion collisions. In this talk we discuss the nuclear modification factors and yield ratios of identified meson such as $η$, $ρ^0$, $ϕ$, $ω$, and $K^0_{\rm S}$ as well as $π$ meson at large $p_T$ in A+A collisions at the next-to-leading order (NLO) with high-twist approach of parton energy loss. Then we discuss a newly developed formalism of combing NLO matrix elements and parton shower (PS) for initial hard production with parton energy loss in the QGP, and its application in investigating massive gauge boson ($Z^0$/$W^{\pm}$) tagged jet productions and $b\bar{b}$ dijet correlations in Pb+Pb at the LHC.

nucl-th↗

arXiv2018

Improving Data Quality through Deep Learning and Statistical Models

Traditional data quality control methods are based on users experience or previously established business rules, and this limits performance in addition to being a very time consuming process with lower than desirable accuracy. Utilizing deep learning, we can leverage computing resources and advanced techniques to overcome these challenges and provide greater value to users. In this paper, we, the authors, first review relevant works and discuss machine learning techniques, tools, and statistical quality models. Second, we offer a creative data quality framework based on deep learning and statistical model algorithm for identifying data quality. Third, we use data involving salary levels from an open dataset published by the state of Arkansas to demonstrate how to identify outlier data and how to improve data quality via deep learning. Finally, we discuss future work.

cs.AI↗

arXiv2018

Dimension Reduction for Origin-Destination Flow Estimation: Blind Estimation Made Possible

This paper studies the problem of estimating origin-destination (OD) flows from link flows. As the number of link flows is typically much less than that of OD flows, the inverse problem is severely ill-posed and hence prior information is required to recover the ground truth. The basic approach in the literature relies on a forward model where the so called traffic assignment matrix maps OD flows to link flows. Due to the ill-posedness of the problem, prior information on the assignment matrix and OD flows are typically needed. The main contributions of this paper include a dimension reduction of the inquired flows from $O(n^2)$ to $O(n)$, and a demonstration that for the first time the ground truth OD flows can be uniquely identified with no or little prior information. To cope with the ill-posedness due to the large number of unknowns, a new forward model is developed which does not involve OD flows directly but is built upon the flows characterized only by their origins, henceforth referred as O-flows. The new model preserves all the OD information and more importantly reduces the dimension of the inverse problem substantially. A Gauss-Seidel method is deployed to solve the inverse problem, and a necessary condition for the uniqueness of the solution is proved. Simulations demonstrate that blind estimation where no prior information is available is possible for some network settings. Some challenging network settings are identified and discussed, where a remedy based on temporal patterns of the O-flows is developed and numerically shown effective.

cs.CE↗

arXiv2018

Toward Understanding the Impact of Staleness in Distributed Machine Learning

Many distributed machine learning (ML) systems adopt the non-synchronous execution in order to alleviate the network communication bottleneck, resulting in stale parameters that do not reflect the latest updates. Despite much development in large-scale ML, the effects of staleness on learning are inconclusive as it is challenging to directly monitor or control staleness in complex distributed environments. In this work, we study the convergence behaviors of a wide array of ML models and algorithms under delayed updates. Our extensive experiments reveal the rich diversity of the effects of staleness on the convergence of ML algorithms and offer insights into seemingly contradictory reports in the literature. The empirical findings also inspire a new convergence analysis of stochastic gradient descent in non-convex optimization under staleness, matching the best-known convergence rate of O(1/\sqrt{T}).

cs.LG↗

arXiv2018

Liouville type theorems, a priori estimates and existence of solutions for non-critical higher order Lane-Emden-Hardy equations

In this paper, we are concerned with the non-critical higher order Lane-Emden-Hardy equations \begin{equation*} (-Δ)^{m}u(x)=\frac{u^{p}(x)}{|x|^{a}} \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\, \mathbb{R}^{n} \end{equation*} with $n\geq3$, $1\leq m<\frac{n}{2}$, $0\leq a<2m$, $1<p<\frac{n+2m-2a}{n-2m}$ if $0\leq a<2$, and $1<p<\infty$ if $2\leq a<2m$. We prove Liouville theorems for nonnegative classical solutions to the above Lane-Emden-Hardy equations (Theorem \ref{Thm0}), that is, the unique nonnegative solution is $u\equiv0$. As an application, we derive a priori estimates and existence of positive solutions to non-critical higher order Lane-Emden equations in bounded domains (Theorem \ref{Thm1} and \ref{Thm2}). The results for critical order Hardy-Hénon equations have been established by Chen, Dai and Qin \cite{CDQ} recently.

math.AP↗

arXiv2018

Liouville type theorem for critical order Lane-Emden-Hardy equations in $\mathbb{R}^n$

In this paper, we are concerned with the critical order Lane-Emden-Hardy equations \begin{equation*} (-Δ)^{\frac{n}{2}}u(x)=\frac{u^{p}(x)}{|x|^{a}} \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\, \mathbb{R}^{n} \end{equation*} with $n\geq4$ is even, $0\leq a<n$ and $1<p<+\infty$. We prove Liouville theorem for nonnegative classical solutions to the above Lane-Emden-Hardy equations (Theorem \ref{Thm0}), that is, the unique nonnegative solution is $u\equiv0$. Our result seems to be the first Liouville theorem on the critical order equations in higher dimensions ($n\geq3$).

math.AP↗

arXiv2018

Production of $ρ^{0}$ meson with large $p_T$ at NLO in heavy-ion collisions

Production of large transverse momentum $ρ^{0}$ meson in high-energy nuclear collisions is investigated for the first time at the next-leading-order in the QCD improved parton model. The $ρ^0$ fragmentation functions (FFs) in vacuum at any scale $Q$ are obtained, by evolving a newly developed initial parametrization of $ρ^0$ FFs at a scale $\rm Q_{0}^2=1.5\ GeV^2$ from a broken SU(3) model through NLO DGLAP equations. The numerical simulations of $p_{\rm T}$ spectra of $ρ^{0}$ meson in the elementary $\rm p+p$ collisions at NLO give a decent description of STAR $\rm p+p$ data. In $\rm A+A$ reactions the jet quenching effect is taken into account with the higher-twist approach by the medium-modified parton FFs due to gluon radiation in the quark-gluon plasma, whose space-time evolution is described by a (3+1D) hydrodynamical model. The nuclear modification factors for $ρ^{0}$ meson and its double ratio with $π^\pm$ nuclear modification in central $\rm Au+Au$ collisions at the RHIC are calculated and found to be in good agreement with STAR measurement. Predictions of $ρ^{0}$ nuclear modification and the yield ratio $ρ^0/π^0$ in central Pb+Pb at the LHC are also presented. It is shown that the ratio $ρ^0/π^0$ in central Pb+Pb will approach to that in p+p reactions when $p_{\rm T}>12$~GeV.

nucl-th↗

arXiv2018

Nuclear suppression of $ϕ$ meson yields with large $p_T$ at the RHIC and the LHC

We calculate $ϕ$ meson transverse momentum spectra in p+p collisions as well as their nuclear suppressions in central A+A collisions both at the RHIC and the LHC in LO and NLO with the QCD-improved parton model. We have included the parton energy loss effect in hot/dense QCD medium with the effectively medium-modified $ϕ$ fragmentation functions in the higher-twist approach of jet quenching. The nuclear modification factors of $ϕ$ meson in central Au+Au collisions at the RHIC and central Pb+Pb collisions at the LHC are provided, and a nice agreement of our numerical results at NLO with the ALICE measurement is observed. Predictions of yield ratios of neutral mesons such as $ϕ/π^0$, $ϕ/η$ and $ϕ/ρ^0$ at large $p_T$ in relativistic heavy-ion collisions are also presented for the first time.

nucl-th↗

arXiv2018

Reinforced Auto-Zoom Net: Towards Accurate and Fast Breast Cancer Segmentation in Whole-slide Images

Convolutional neural networks have led to significant breakthroughs in the domain of medical image analysis. However, the task of breast cancer segmentation in whole-slide images (WSIs) is still underexplored. WSIs are large histopathological images with extremely high resolution. Constrained by the hardware and field of view, using high-magnification patches can slow down the inference process and using low-magnification patches can cause the loss of information. In this paper, we aim to achieve two seemingly conflicting goals for breast cancer segmentation: accurate and fast prediction. We propose a simple yet efficient framework Reinforced Auto-Zoom Net (RAZN) to tackle this task. Motivated by the zoom-in operation of a pathologist using a digital microscope, RAZN learns a policy network to decide whether zooming is required in a given region of interest. Because the zoom-in action is selective, RAZN is robust to unbalanced and noisy ground truth labels and can efficiently reduce overfitting. We evaluate our method on a public breast cancer dataset. RAZN outperforms both single-scale and multi-scale baseline approaches, achieving better accuracy at low inference cost.

cs.CV↗

arXiv2018

Unsupervised Domain Adaptation for Automatic Estimation of Cardiothoracic Ratio

The cardiothoracic ratio (CTR), a clinical metric of heart size in chest X-rays (CXRs), is a key indicator of cardiomegaly. Manual measurement of CTR is time-consuming and can be affected by human subjectivity, making it desirable to design computer-aided systems that assist clinicians in the diagnosis process. Automatic CTR estimation through chest organ segmentation, however, requires large amounts of pixel-level annotated data, which is often unavailable. To alleviate this problem, we propose an unsupervised domain adaptation framework based on adversarial networks. The framework learns domain invariant feature representations from openly available data sources to produce accurate chest organ segmentation for unlabeled datasets. Specifically, we propose a model that enforces our intuition that prediction masks should be domain independent. Hence, we introduce a discriminator that distinguishes segmentation predictions from ground truth masks. We evaluate our system's prediction based on the assessment of radiologists and demonstrate the clinical practicability for the diagnosis of cardiomegaly. We finally illustrate on the JSRT dataset that the semi-supervised performance of our model is also very promising.

cs.CV↗

arXiv2018

A Tight Converse to the Spectral Resolution Limit via Convex Programming

It is now well understood that convex programming can be used to estimate the frequency components of a spectrally sparse signal from $2m+1$ uniform temporal measurements. It is conjectured that a phase transition on the success of the total-variation regularization occurs when the distance between the spectral components of the signal to estimate crosses $1/m$. We prove the necessity part of this conjecture by demonstrating that this regularization can fail whenever the spectral distance of the signal of interest is asymptotically equal to $1/m$.

cs.IT↗