arXiv subjects

Dan Dai

Publications and source records attributed to Dan Dai.

At least 37 records · Page 2Linked to original sources

arXiv2020

On integrals of the tronquée solutions and the associated Hamiltonians for the Painlevé II equation

We consider a family of tronquée solutions of the Painelvé II equation \begin{equation*} q''(s)=2q(s)^3+sq(s)-(2α+\frac12), \qquad α> -\frac12, \end{equation*} which is characterized by the Stokes multipliers $$s_1=-e^{-2απi },\quad s_2=ω, \quad s_3=-e^{2 απi} $$ with $ω$ being a free parameter. These solutions include the well-known generalized Hastings-McLeod solution as a special case if $ω=0$. We derive asymptotics of integrals of the tronquée solutions and the associated Hamiltonians over the real axis for $α> -1/2$ and $ω\geq 0$, with the constant terms evaluated explicitly. Our results agree with those already known in the literature if the parameters $α$ and $ω$ are chosen to be special values. Some applications of our results in random matrix theory are also discussed.

math-ph↗

arXiv2019

Gaussian unitary ensembles with pole singularities near the soft edge and a system of coupled Painlevé XXXIV equations

In this paper, we study the singularly perturbed Gaussian unitary ensembles defined by the measure \begin{equation*} \frac{1}{C_n} e^{- n\textrm{tr}\, V(M;λ,\vec{t}\;)}dM, \end{equation*} over the space of $n \times n$ Hermitian matrices $M$, where $V(x;λ,\vec{t}\;):= 2x^2 + \sum_{k=1}^{2m}t_k(x-λ)^{-k}$ with $\vec{t}= (t_1, t_2, \ldots, t_{2m})\in \mathbb{R}^{2m-1} \times (0,\infty)$, in the multiple scaling limit where $λ\to 1$ together with $\vec{t} \to \vec{0}$ as $n\to \infty$ at appropriate related rates. We obtain the asymptotics of the partition function, which is described explicitly in terms of an integral involving a smooth solution to a new coupled Painlevé system generalizing the Painlevé XXXIV equation. The large $n$ limit of the correlation kernel is also derived, which leads to a new universal class built out of the $Ψ$-function associated with the coupled Painlevé system.

math-ph↗

arXiv2019

The distribution function for the maximal height of $N$ non-intersecting Bessel paths

In this paper, we consider $N$ non-intersecting Bessel paths starting at $x=a\geq 0$, and conditioned to end at the origin $x=0$. We derive the explicit formula of the distribution function for the maximum height. Depending on the starting point $a>0$ or $a=0$, the distribution functions are also given in terms of the Hankel determinants associated with the multiple discrete orthogonal polynomials or discrete orthogonal polynomials, respectively.

math-ph↗

arXiv2019

Automatic construction of Chinese herbal prescription from tongue image via CNNs and auxiliary latent therapy topics

The tongue image provides important physical information of humans. It is of great importance for diagnoses and treatments in clinical medicine. Herbal prescriptions are simple, noninvasive and have low side effects. Thus, they are widely applied in China. Studies on the automatic construction technology of herbal prescriptions based on tongue images have great significance for deep learning to explore the relevance of tongue images for herbal prescriptions, it can be applied to healthcare services in mobile medical systems. In order to adapt to the tongue image in a variety of photographic environments and construct herbal prescriptions, a neural network framework for prescription construction is designed. It includes single/double convolution channels and fully connected layers. Furthermore, it proposes the auxiliary therapy topic loss mechanism to model the therapy of Chinese doctors and alleviate the interference of sparse output labels on the diversity of results. The experiment use the real world tongue images and the corresponding prescriptions and the results can generate prescriptions that are close to the real samples, which verifies the feasibility of the proposed method for the automatic construction of herbal prescriptions from tongue images. Also, it provides a reference for automatic herbal prescription construction from more physical information.

cs.CV↗

arXiv2019

Stochastic Region Pooling: Make Attention More Expressive

Global Average Pooling (GAP) is used by default on the channel-wise attention mechanism to extract channel descriptors. However, the simple global aggregation method of GAP is easy to make the channel descriptors have homogeneity, which weakens the detail distinction between feature maps, thus affecting the performance of the attention mechanism. In this work, we propose a novel method for channel-wise attention network, called Stochastic Region Pooling (SRP), which makes the channel descriptors more representative and diversity by encouraging the feature map to have more or wider important feature responses. Also, SRP is the general method for the attention mechanisms without any additional parameters or computation. It can be widely applied to attention networks without modifying the network structure. Experimental results on image recognition datasets including CIAFR-10/100, ImageNet and three Fine-grained datasets (CUB-200-2011, Stanford Cars and Stanford Dogs) show that SRP brings the significant improvements of the performance over efficient CNNs and achieves the state-of-the-art results.

cs.CV↗

arXiv2018

Competitive Inner-Imaging Squeeze and Excitation for Residual Network

Residual networks, which use a residual unit to supplement the identity mappings, enable very deep convolutional architecture to operate well, however, the residual architecture has been proved to be diverse and redundant, which may leads to low-efficient modeling. In this work, we propose a competitive squeeze-excitation (SE) mechanism for the residual network. Re-scaling the value for each channel in this structure will be determined by the residual and identity mappings jointly, and this design enables us to expand the meaning of channel relationship modeling in residual blocks. Modeling of the competition between residual and identity mappings cause the identity flow to control the complement of the residual feature maps for itself. Furthermore, we design a novel inner-imaging competitive SE block to shrink the consumption and re-image the global features of intermediate network structure, by using the inner-imaging mechanism, we can model the channel-wise relations with convolution in spatial. We carry out experiments on the CIFAR, SVHN, and ImageNet datasets, and the proposed method can challenge state-of-the-art results.

cs.CV↗

arXiv2018

Chinese Herbal Recognition based on Competitive Attentional Fusion of Multi-hierarchies Pyramid Features

Convolution neural netwotks (CNNs) are successfully applied in image recognition task. In this study, we explore the approach of automatic herbal recognition with CNNs and build the standard Chinese herbs datasets firstly. According to the characteristics of herbal images, we proposed the competitive attentional fusion pyramid networks to model the features of herbal image, which mdoels the relationship of feature maps from different levels, and re-weights multi-level channels with channel-wise attention mechanism. In this way, we can dynamically adjust the weight of feature maps from various layers, according to the visual characteristics of each herbal image. Moreover, we also introduce the spatial attention to recalibrate the misaligned features caused by sampling in features amalgamation. Extensive experiments are conducted on our proposed datasets and validate the superior performance of our proposed models. The Chinese herbs datasets will be released upon acceptance to facilitate the research of Chinese herbal recognition.

cs.CV↗

arXiv2018

Metabolize Neural Network

The metabolism of cells is the most basic and important part of human function. Neural networks in deep learning stem from neuronal activity. It is self-evident that the significance of metabolize neuronal network(MetaNet) in model construction. In this study, we explore neuronal metabolism for shallow network from proliferation and autophagy two aspects. First, we propose different neuron proliferate methods that constructive the selfgrowing network in metabolism cycle. Proliferate neurons alleviate resources wasting and insufficient model learning problem when network initializes more or less parameters. Then combined with autophagy mechanism in the process of model self construction to ablate under-expressed neurons. The MetaNet can automatically determine the number of neurons during training, further, save more resource consumption. We verify the performance of the proposed methods on datasets: MNIST, Fashion-MNIST and CIFAR-10.

cs.NE↗

arXiv2018

Asymptotics of partition functions in a fermionic matrix model and of related $q$-polynomials

In this paper, we study asymptotics of the thermal partition function of a model of quantum mechanical fermions with matrix-like index structure and quartic interactions. This partition function is given explicitly by a Wronskian of the Stieltjes-Wigert polynomials. Our asymptotic results involve the theta function and its derivatives. We also develop a new asymptotic method for general $q$-polynomials.

math-ph↗

arXiv2018

Gap probability at the hard edge for random matrix ensembles with pole singularities in the potential

We study the Fredholm determinant of an integrable operator acting on the interval $(0,s)$ whose kernel is constructed out of a hierarchy of higher order analogues to the Painlevé III equation. This Fredholm determinant describes the critical behavior of the eigenvalue gap probability at the hard edge of unitary invariant random matrix ensembles perturbed by poles of order $k$ in the double scaling regime. Using the Riemann-Hilbert method, we obtain the large $s$ asymptotics of the Fredholm determinant. Moreover, we derive a Painlevé type formula of the Fredholm determinant, which is expressed in terms of an explicit integral involving a solution to the coupled Painlevé III system.

math-ph↗

arXiv2018

On the quasi-Ablowitz-Segur and quasi-Hastings-McLeod solutions of the inhomogeneous Painlevé II equation

We consider the quasi-Ablowitz-Segur and quasi-Hastings-McLeod solutions of the inhomogeneous Painlevé II equation $$ u"(x)=2u^3(x)+xu(x)-α\qquad \textrm{for } α\in \mathbb{R} \textrm{ and } |α| > \frac{1}{2}. $$ These solutions are obtained from the classical Ablowitz-Segur and Hastings-McLeod solutions via the Bäcklund transformation, and satisfy the same asymptotic behaviors when $x \to \pm \infty$. For $|α| > 1/2$, we show that the quasi-Ablowitz-Segur and quasi-Hastings-McLeod solutions possess $[ \, |α| + \frac{1}{2} \, ]$ simple poles on the real axis, which rigorously justifies the numerical results in Fornberg and Weideman (Found. Comput. Math., 14 (2014), no. 5, 985-1016).

math.CA↗

arXiv2018

Tracy-Widom distributions in critical unitary random matrix ensembles and the coupled Painlevé II system

We study Fredholm determinants of the Painlevé II and Painlevé XXXIV kernels. In certain critical unitary random matrix ensembles, these determinants describe special gap probabilities of eigenvalues. We obtain Tracy-Widom formulas for the Fredholm determinants, which are explicitly given in terms of integrals involving a family of distinguished solutions to the coupled Painlevé II system in dimension four. Moreover, the large gap asymptotics for these Fredholm determinants are derived, where the constant terms are given explicitly in terms of the Riemann zeta-function.

math-ph↗

arXiv2017

Connection formulas for the Ablowitz-Segur solutions of the inhomogeneous Painlevé II equation

We consider the second Painlevé equation $$ u"(x)=2u^3(x)+xu(x)-α, $$ where $α$ is a nonzero constant. Using the Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems, we rigorously prove the asymptotics as $x \to \pm \infty$ for both the real and purely imaginary Ablowitz-Segur solutions, as well as the corresponding connection formulas. We also show that the real Ablowitz-Segur solutions have no real poles when $α\in (-1/2, 1/2)$.

math.CA↗

arXiv2016

Asymptotics of orthogonal polynomials and the Painlevé transcendents

In this survey, we review asymptotic results of some orthogonal polynomials which are relate to the Painlevé transcendents. They are obtained by applying Deift-Zhou's nonlinear steepest descent method for Riemann-Hilbert problems. In the last part of this article, we list several open problems.

math.CA↗

arXiv2016

Hankel determinants for a singular complex weight and the first and third Painlevé transcendents

In this paper, we consider polynomials orthogonal with respect to a varying perturbed Laguerre weight $e^{-n(z-\log z+t/z)}$ for $t<0$ and $z$ on certain contours in the complex plane. When the parameters $n$, $t$ and the degree $k$ are fixed, the Hankel determinant for the singular complex weight is shown to be the isomonodromy $τ$-function of the Painlevé III equation. When the degree $k=n$, $n$ is large and $t$ is close to a critical value, inspired by the study of the Wigner time delay in quantum transport, we show that the double scaling asymptotic behaviors of the recurrence coefficients and the Hankel determinant are described in terms of a Boutroux tronquée solution to the Painlevé I equation. Our approach is based on the Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems.

math.CA↗

arXiv2015

Uniform Asymptotics of Orthogonal Polynomials Arising from Coherent States

In this paper, we study a family of orthogonal polynomials $\{ϕ_n(z)\}$ arising from nonlinear coherent states in quantum optics. Based on the three-term recurrence relation only, we obtain a uniform asymptotic expansion of $ϕ_n(z)$ as the polynomial degree $n$ tends to infinity. Our asymptotic results suggest that the weight function associated with the polynomials has an unusual singularity, which has never appeared for orthogonal polynomials in the Askey scheme. Our main technique is the Wang and Wong's difference equation method. In addition, the limiting zero distribution of the polynomials $ϕ_n(z)$ is provided.

math.CA↗

arXiv2014

Painlevé III asymptotics of Hankel determinants for a singularly perturbed Laguerre weight

In this paper, we consider the Hankel determinants associated with the singularly perturbed Laguerre weight $w(x)=x^αe^{-x-t/x}$, $x\in (0, \infty)$, $t>0$ and $α>0$. When the matrix size $n\to\infty$, we obtain an asymptotic formula for the Hankel determinants, valid uniformly for $t\in (0, d]$, $d>0$ fixed. A particular Painlevé III transcendent is involved in the approximation, as well as in the large-$n$ asymptotics of the leading coefficients and recurrence coefficients for the corresponding perturbed Laguerre polynomials. The derivation is based on the asymptotic results in an earlier paper of the authors, obtained by using the Deift-Zhou nonlinear steepest descent method.

math.CA↗

arXiv2014

Critical edge behavior and the Bessel to Airy transition in the singularly perturbed Laguerre unitary ensemble

In this paper, we study the singularly perturbed Laguerre unitary ensemble $$ \frac{1}{Z_n} (\det M)^αe^{- \textrm{tr}\, V_t(M)}dM, \qquad α>0, $$ with $V_t(x) = x + t/x$, $x\in (0,+\infty)$ and $t>0$. Due to the effect of $t/x$ for varying $t$, the eigenvalue correlation kernel has a new limit instead of the usual Bessel kernel at the hard edge 0. This limiting kernel involves $ψ$-functions associated with a special solution to a new third-order nonlinear differential equation, which is then shown equivalent to a particular Painlevé III equation. The transition of this limiting kernel to the Bessel and Airy kernels is also studied when the parameter $t$ changes in a finite interval $(0, d]$. Our approach is based on Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems.

math-ph↗