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yang chen

University of Macau, Mathematics, Faculty Member

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I am a mathematical physicist working on random matrix theory and its applications. I left the University of Macau in June 2023.
Supervisors: Advisor.
Phone: +85363296824
Address: 25 Neville Street

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Ioannis Ntzoufras

Athens University of Economics and Business

Yuzuncu Yil University

Anoop Chaturvedi

University of Allahabad

California Polytechnic State University at San Luis Obispo

Sevcan Demir Atalay

Ege University

Fikri gökpınar

Gazi University

José Francisco Pessanha

UERJ - Universidade do Estado do Rio de Janeiro / Rio de Janeiro State University

Adebayo D Agunbiade

Olabisi Onabanjo University, Ago-Iwoye, Nigeria

Srinivasa Rao Gadde

The University of Dodoma, Tanzania

Universidade Federal de Pernambuco

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Papers by yang chen

Robust stability condition of an uncertain networked system with delayed data dropout in both forward and feedback channels

2006 SICE-ICASE International Joint Conference, 2006

This paper provides stability conditions of an uncertain networked control system with both forwa... more This paper provides stability conditions of an uncertain networked control system with both forward control signal loss and feedback output signal loss. It is assumed that both the forward control signal and the output measurement signal are intermittent independently, and the remote plant has a model uncertainty. Also, in this paper, an observer-based controller is used in the remote place because the full states are not accessible. Thus, the main research goal is to analyze the overall stability of this uncertain intermittent networked system. In the second part of this paper, we consider delay effects of both forward signal and feedback signal during the network transfer.

Gaussian Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlevé IV System

Gaussian Unitary Ensembles with Jump Discontinuities, PDEs and the coupled Painleve IV equationsuities, , 2023

We study the Hankel determinant generated by the Gaussian weight with jump discontinuities at t 1... more We study the Hankel determinant generated by the Gaussian weight with jump discontinuities at t 1 , • • • , tm. By making use of a pair of ladder operators satisfied by the associated monic orthogonal polynomials and three supplementary conditions, we show that the logarithmic derivative of the Hankel determinant satisfies a second-order partial differential equation which is reduced to the σ-form of a Painlevé IV equation when m = 1. Moreover, under the assumption that t k − t 1 is fixed for k = 2, • • • , m, by considering the Riemann-Hilbert problem for the orthogonal polynomials, we construct direct relationships between the auxiliary quantities introduced in the ladder operators and solutions of a coupled Painlevé IV system.

The smallest eigenvalue of the Hankel matrices associated with a perturbed Jacobi weight

In this paper, we study the large N behavior of the smallest eigenvalue N of the (N + 1) ⇥ (N + 1... more In this paper, we study the large N behavior of the smallest eigenvalue N of the (N + 1) ⇥ (N + 1) Hankel matrix, H N = (µ j+k) 0 j,kN , generated by the dependent Jacobi weight w(z,) = e z z ↵ (1 z) , z 2 [0, 1], 2 R, ↵ > 1, > 1. Applying the arguments of Szegö, Widom and Wilf, we obtain the asymptotic representation of the orthonormal polynomials P N (z), z 2 C\[0, 1], with the weight w(z,) = e z z ↵ (1 z). Using the polynomials P N (z), we obtain the theoretical expression of N , for large N. We also display the smallest eigenvalue N for su ciently large N, computed numerically.

The Eigenvectors of Single Spiked Complex Wishart Matrices Finite and Asymptotic Analyses 1 1

by yang chen and Pasan Dissanayake

Let W ∈ C n×n be a single-spiked Wishart matrix in the class W ∼ CWn (m, In + θvv †) with m ≥ n, ... more Let W ∈ C n×n be a single-spiked Wishart matrix in the class W ∼ CWn (m, In + θvv †) with m ≥ n, where In is the n × n identity matrix, v ∈ C n×1 is an arbitrary vector with unit Euclidean norm, θ ≥ 0 is a non-random parameter, and (•) † represents the conjugate-transpose operator. Let u1 and un denote the eigenvectors corresponding to the smallest and the largest eigenvalues of W, respectively. This paper investigates the probability density function (p.d.f.) of the random quantity Z (n) = ¬ ¬ v † u ¬ ¬ 2 ∈ (0, 1) for = 1, n. In particular, we derive a finite dimensional closed-form p.d.f. for Z (n) 1 which is amenable to asymptotic analysis as m, n diverges with m − n fixed. It turns out that, in this asymptotic regime, the scaled random variable nZ (n) 1 converges in distribution to χ 2 2 /2(1 + θ), where χ 2 2 denotes a chi-squared random variable with two degrees of freedom. This reveals that u1 can be used to infer information about the spike. On the other hand, the finite dimensional p.d.f. of Z (n) n is expressed as a double integral in which the integrand contains a determinant of a square matrix of dimension (n − 2). Although a simple solution to this double integral seems intractable, for special configurations of n = 2, 3, and 4, we obtain closed-form expressions.

DISTRIBUTIONS OF DEMMEL AND RELATED CONDITION NUMBERS

Consider a random matrix A ∈ C m×n (m ≥ n) containing independent complex Gaussian entries with z... more Consider a random matrix A ∈ C m×n (m ≥ n) containing independent complex Gaussian entries with zero mean and unit variance, and let 0 < λ 1 ≤ λ 2 ≤ • • • ≤ λn < ∞ denote the eigenvalues of A * A, where (•) * represents conjugate-transpose. This paper investigates the distribution of the random variables n j=1 λ j λ k for k = 1 and k = 2. These two variables are related to certain condition number metrics, including the so-called Demmel condition number, which have been shown to arise in a variety of applications. For both cases, we derive new exact expressions for the probability densities and establish the asymptotic behavior as the matrix dimensions grow large. In particular, it is shown that as n and m tend to infinity with their difference fixed, both densities scale on the order of n 3. After suitable transformations, we establish exact expressions for the asymptotic densities, obtaining simple closed-form expressions in some cases. Our results generalize the work of Edelman on the Demmel condition number for the case m = n.

Distribution of the Scaled Condition Number of Single-Spiked Complex Wishart Matrices

Let X ∈ C n×m (m ≥ n) be a random matrix with independent columns each distributed as complex mul... more Let X ∈ C n×m (m ≥ n) be a random matrix with independent columns each distributed as complex multivariate Gaussian with zero mean and single-spiked covariance matrix In + ηuu * , where In is the n × n identity matrix, u ∈ C n×1 is an arbitrary vector with unit Euclidean norm, η ≥ 0 is a nonrandom parameter, and (•) * represents the conjugate-transpose. This paper investigates the distribution of the random quantity θ 2 SC (X) = È n k=1 ιk/ι1, where 0 ≤ ι1 ≤ ι2 ≤. .. ≤ ιn < ∞ are the ordered eigenvalues of XX * (i.e., single-spiked Wishart matrix). This random quantity is intimately related to the so called scaled condition number or the Demmel condition number (i.e., θSC(X)) and the minimum eigenvalue of the fixed trace Wishart-Laguerre ensemble (i.e., θ −2 SC (X)). In particular, we use an orthogonal polynomial approach to derive an exact expression for the probability density function of θ 2 SC (X) which is amenable to asymptotic analysis as matrix dimensions grow large. Our asymptotic results reveal that, as m, n → ∞ such that m − n is fixed and when η scales on the order of 1/n, θ 2 SC (X) scales on the order of n 3. In this respect we establish simple closed-form expressions for the limiting distributions. It turns out that, as m, n → ∞ such that n/m → c ∈ (0, 1), properly centered θ 2 SC (X) fluctuates on the scale m 1 3 .

Certificate of Emeritus (1)

Document , 2023

I am now Professor Emeritus at the University of Macau.

Asymptotic Properties of Some Freud Polynomials

Preprint, 2023

We study the asymptotic properties of monic orthogonal polynomials (OPs) with respect to some Fre... more We study the asymptotic properties of monic orthogonal polynomials (OPs) with respect to some Freud weights when the degree of the polynomial tends to infinity, including the asymptotics of the recurrence coefficients, the nontrivial leading coefficients of the monic OPs, the associated Hankel determinants and the squares of L 2-norm of the monic OPs. These results are derived from the combination of the ladder operator approach, Dyson's Coulomb fluid approach and some recent results in the literature.

Contents lists available at ScienceDirect

We study the Hankel determinant generated by the Laguerre weight with jump discontinuities at t k... more We study the Hankel determinant generated by the Laguerre weight with jump discontinuities at t k , k = 1,. .. , m. By employing the ladder operator approach to establish Riccati equations, we show that σ n (t 1 ,. .. , t m), the logarithmic derivative of the n-dimensional Hankel determinant, satisfies an m-variable generalization of the σ-form of a Painlevé V equation. Through investigating the Riemann-Hilbert problem for the associated orthogonal polynomials and via the Lax pair, we express σ n in terms of solutions of a coupled Painlevé V system. We also build relations between the auxiliary quantities introduced in the above two methods, which provides connections between the Riccati equations and the Lax pair. In addition, when each t k tends to the hard edge of the spectrum and n goes to infinity, the scaled σ n is shown to satisfy a generalized σ-form of a Painlevé III equation.

Painlevé V and the Hankel determinant for a singularly perturbed Jacobi weight

Nuclear Physics B, 2020

We study the Hankel determinant generated by a singularly perturbed Jacobi weight w(x, t) := (1 −... more We study the Hankel determinant generated by a singularly perturbed Jacobi weight w(x, t) := (1 − x 2) α e − t x 2 , x ∈ [−1, 1], α > 0, t ≥ 0. If t = 0, it is reduced to the classical symmetric Jacobi weight. For t > 0, the factor e − t x 2 induces an infinitely strong zero at the origin. This Hankel determinant is related to the Wigner time-delay distribution in chaotic cavities. In the finite n dimensional case, we obtain two auxiliary quantities R n (t) and r n (t) by using the ladder operator approach. We show that the Hankel determinant has an integral representation in terms of R n (t), where R n (t) is closely related to a particular Painlevé V transcendent. Furthermore, we derive a second-order nonlinear differential equation and also a second-order difference equation for the logarithmic derivative of the Hankel determinant. This quantity can be expressed in terms of the Jimbo-Miwa-Okamoto σ-function of a particular Painlevé V. Then we consider the asymptotics of the Hankel determinant under a suitable double scaling, i.e. n → ∞ and t → 0 such that s = 2n 2 t is fixed. Based on previous results by using the Coulomb fluid method, we obtain the large s and small s asymptotic behaviors of the scaled Hankel determinant, including the constant term in the asymptotic expansion.

Single-user MIMO system, Painlevé transcendents, and double scaling

Journal of Mathematical Physics, 2017

In this paper we study a particular Painlevé V (denoted P V) that arises from Multi-Input-Multi-O... more In this paper we study a particular Painlevé V (denoted P V) that arises from Multi-Input-Multi-Output (MIMO) wireless communication systems. Such a P V appears through its intimate relation with the Hankel determinant that describes the moment generating function (MGF) of the Shannon capacity. This originates through the multiplication of the Laguerre weight or the Gamma density x α e −x , x > 0, for α > −1 by (1 + x/t) λ with t > 0 a scaling parameter. Here the λ parameter "generates" the Shannon capacity; see Yang Chen and Matthew McKay, IEEE Trans. IT, 58 (2012) 4594-4634. It was found that the MGF has an integral representation as a functional of y(t) and y (t) , where y(t) satisfies the "classical form" of P V. In this paper, we consider the situation where n, the number of transmit antennas, (or the size of the random matrix), tends to infinity, and the signal-to-noise ratio (SNR) P tends to infinity, such that s = 4n 2 /P is finite. Under such double scaling the MGF, effectively an infinite determinant, has an integral representation in terms of a "lesser" P III. We also consider the situations where α = k + 1/2, k ∈ N, and α ∈ {0, 1, 2,. .. } λ ∈ {1, 2,. .. }, linking the relevant quantity to a solution of the two dimensional sine-Gordon equation in radial coordinates and a certain discrete Painlevé-II. From the large n asymptotic of the orthogonal polynomials, that appears naturally, we obtain the double scaled MGF for small and large s , together with the constant term in the large s expansion. With the aid of these, we derive a number of cumulants and find that the capacity distribution function is non-Gaussian.

Gap Probability Distribution of the Jacobi Unitary Ensemble: An Elementary Treatment, from Finitento Double Scaling

Studies in Applied Mathematics, 2017

In this paper, we study the gap probability problem of the (symmetric) Jacobi unitary ensemble of... more In this paper, we study the gap probability problem of the (symmetric) Jacobi unitary ensemble of Hermitian random matrices, namely the probability that the interval (−a, a) (0 < a < 1) is free of eigenvalues. Using the ladder operator technique for orthogonal polynomials and the associated supplementary conditions, we derive three quantities instrumental in the gap probability, denoted by H n (a), R n (a) and r n (a). We find that each one satisfies a second order differential equation. We show that after a double scaling, the large second order differential equation in the variable a with n as parameter satisfied by H n (a), can be reduced to the Jimbo-Miwa-Okamoto σ form of the Painlevé V equation.

Singular linear statistics of the Laguerre unitary ensemble and Painlevé. III. Double scaling analysis

Journal of Mathematical Physics, 2015

We continue with the study of the Hankel determinant, defined by, D n (t, α) = det ∞ 0 x j+k w(x;... more We continue with the study of the Hankel determinant, defined by, D n (t, α) = det ∞ 0 x j+k w(x; t, α)dx n−1 j,k=0 , generated by a singularly perturbed Laguerre weight, w(x; t, α) = x α e −x e −t/x , x ∈ R + , α > 0, t > 0, obtained through a deformation of the Laguerre weight function, w(x; 0, α) = x α e −x , x ∈ R + , α > 0, via the multiplicative factor e −t/x. An earlier investigation was made on the finite n aspect of such determinants, which appeared in [20]. It was found that the logarithm of the Hankel determinant has an integral representation in terms of a particular Painlevé III(P III , for short) and its t derivatives. In this paper we show that, under a double scaling, where n , the order of the Hankel matrix tends to ∞, and t , tends to 0 + , the scaled-and therefore, in some sense, infinite dimensional-Hankel determinant, has an integral representation in terms of a C potential. The second order non-linear ode satisfied by C, after a change of variable, is another P III transcendent, albeit with fewer number of parameters. Expansions of the double scaled determinant for small and large parameter are obtained.

Fixed-Time Consensus Tracking for Second-Order Multiagent Systems Under Disturbance

IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2019

This article focuses on the topic of fixed-time consensus for second-order multiagent systems (MA... more This article focuses on the topic of fixed-time consensus for second-order multiagent systems (MASs) with disturbances. Based on an integration of the nominal control part and discontinuous integral sliding control part or continuous supertwisting-like control part, some control protocols are presented to achieve consensus tracking in the fixed time. A distributed integral sliding mode (ISM) and a fixed-time convergent command filter are introduced. The performance of nominal dynamics is dominated by the nominal control part which ensures the fixedtime convergence in the ISM surface. The discontinuous sliding mode or continuous super-twisting-like part related to the ISM is utilized to compensate disturbances within fixed convergence time. In this article, we have designed the continuous fixed-time consensus tracking controllers which can eliminate the chattering phenomenon and simultaneously guarantee the convergence precision. Independent of any initial state values, the restrictive bound of the convergence time is estimated. Some fair comparisons are performed to demonstrate the merits of the proposed strategies.

New results on the phase diagram of the (TMTSF)2FSO3 salt

by Vladan Celebonovic and yang chen

Synthetic Metals, 1991

We present transport measurements under pressure on the (TMTSF)2FSO 3 salt from P= l bar to 8kbar... more We present transport measurements under pressure on the (TMTSF)2FSO 3 salt from P= l bar to 8kbar using a helium gas pressure cell, in order to clarify the pressure-temperature phase diagram of this compound. The interest in this TMTSF salt comes from the existence of non-centrosymmetric tetrahedral anion with a permanent dipole moment related to the presence of the fluorine atom. Two types of disorder can be involved in the metal to insulator transitions at low temperature: one due to the orientation of the tetrahedra and the second to the position of the fluorine atom inside the anion. For all studied pressures a serniconducting phase occurs at T = 90K via a very sharp transition. At lower temperatures, the activation energy decreases as the pressure is increased. At last, the resistivity saturates at P=8kbar from T=80K down to 10K where the metallic state is restored. This new study shows that phase diagrams derived with the use of the clamped pressure techniques should be revised.

Orthogonal polynomials with discontinuous weights

Journal of Physics A: Mathematical and General, 2005

In this paper we present a brief description of a ladder operator formalism applied to orthogonal... more In this paper we present a brief description of a ladder operator formalism applied to orthogonal polynomials with discontinuous weights. The two coefficient functions, A n (z) and B n (z), appearing in the ladder operators satisfy the two fundamental compatibility conditions previously derived for smooth weights. If the weight is a product of an absolutely continuous reference weight w 0 and a standard jump function, then A n (z) and B n (z) have apparent simple poles at these jumps. We exemplify the approach by taking w 0 to be the Hermite weight. For this simpler case we derive, without using the compatibility conditions, a pair of difference equations satisfied by the diagonal and off-diagonal recurrence coefficients for a fixed location of the jump. We also derive a pair of Toda evolution equations for the recurrence coefficients which, when combined with the difference equations, yields a particular Painlevé IV.

Orthogonal polynomials, asymptotics, and Heun equations

Journal of Mathematical Physics

The Painlevé equations arise from the study of Hankel determinants generated by moment matrices, ... more The Painlevé equations arise from the study of Hankel determinants generated by moment matrices, whose weights are expressed as the product of "classical" weights multiplied by suitable "deformation factors," usually dependent on a "time variable" t. From ladder operators [see A. Magnus, J. Comput. Appl. Math. 57(1-2), 215-237 (1995)], one finds second order linear ordinary differential equations for associated orthogonal polynomials with coefficients being rational functions. The Painlevé and related functions appear as the residues of these rational functions. We will be interested in the situation when n, the order of the Hankel matrix and also the degree of the polynomials Pn(x) orthogonal with respect to the deformed weights, gets large. We show that the second order linear differential equations satisfied by Pn(x) are particular cases of Heun equations when n is large. In some sense, monic orthogonal polynomials generated by deformed weights mentioned below are solutions of a variety of Heun equations. Heun equations are of considerable importance in mathematical physics, and in the special cases, they degenerate to the hypergeometric and confluent hypergeometric equations. In this paper, we look at three types of weights: the Jacobi type, the Laguerre type, and the weights deformed by the indicator function of χ (a,b) (x) and the step function θ(x). In particular, we consider the following Jacobi type weights: (1.1) x α (1 − x) β e −tx , x ∈ [0, 1], α, β, t > 0; (1.2) x α (1 − x) β e −t/x , x ∈ (0, 1], α, β, t > 0; (1.3) (1 − x 2) α (1 − k 2 x 2) β , x ∈ [−1, 1], α, β > 0, k 2 ∈ (0, 1); the Laguerre type weights: (2.1) x α (x + t) λ e −x , x ∈ [0, ∞), t, α, λ > 0; (2.2) x α e −x−t/x , x ∈ (0, ∞), α, t > 0; and another type of deformation when the classical weights are multiplied by χ (a,b) (x) or θ(x): (3.1) e −x 2 (1 − χ (−a,a) (x)), x ∈ R, a > 0; (3.2) (1 − x 2) α (1 − χ (−a,a) (x)), x ∈ [−1, 1], a ∈ (0, 1), α > 0; (3.3) x α e −x (A + Bθ(x − t)), x ∈ [0, ∞), α, t > 0, A ≥ 0, A + B ≥ 0. The weights mentioned above were studied in a series of papers related to the deformation of "classical" weights.

Smallest Eigenvalue of Large Hankel Matrices at Critical Point: Comparing Conjecture With Parallelised Computation

We propose a novel parallel numerical algorithm for calculating the smallest eigenvalues of highl... more We propose a novel parallel numerical algorithm for calculating the smallest eigenvalues of highly ill-conditioned matrices. It is based on the LDLT decomposition and involves finding a k × k sub-matrix of the inverse of the original N × N Hankel matrix H −1 N. The computation involves extremely high precision arithmetic, message passing interface, and shared memory parallelisation. We demonstrate that this approach achieves good scalability on a high performance computing cluster (HPCC) which constitute a major improvement of the earlier approaches. We use this method to study a family of Hankel matrices generated by the weight w(x) = e −x β , supported on [0, ∞) and β > 0. Such weight generates Hankel determinant, a fundamental object in random matrix theory. In the situation where β > 1/2, the smallest eigenvalue tend to 0, exponentially fast. If β < 1/2, the situation where the classical moment problem is indeterminate, the smallest eigenval-ue is bounded from below by ...

Orthogonal Polynomials, Asymptotics and Heun Equations

The Painlevé equations arise from the study of Hankel determinants generated by moment matrices, ... more The Painlevé equations arise from the study of Hankel determinants generated by moment matrices, whose weights are expressed as the product of "classical" weights multiplied by suitable "deformation factors", usually dependent on a "time variable" t. From ladder operators [12-14, 30] one finds second order linear ordinary differential equations for associated orthogonal polynomials with coefficients being rational functions. The Painlevé and related functions appear as the residues of these rational functions. We will be interested in the situation when n, the order of the Hankel matrix and also the degree of the polynomials P n (x) orthogonal with respect to the deformed weights, gets large. We show that the second order linear differential equations satisfied by P n (x) are particular cases of Heun equations when n is large. In some sense, monic orthogonal polynomials generated by deformed weights mentioned below are solutions of a variety of Heun equ...

Quantum criticality and the Tomonaga-Luttinger liquid in one-dimensional Bose gases

Physical Review Letters

We experimentally investigate the quantum criticality and Tomonaga-Luttinger liquid (TLL) behavio... more We experimentally investigate the quantum criticality and Tomonaga-Luttinger liquid (TLL) behavior within one-dimensional (1D) ultracold atomic gases. Based on the measured density profiles at different temperatures, the universal scaling laws of thermodynamic quantities are observed. The quantum critical regime and the relevant crossover temperatures are determined through the double-peak structure of the specific heat. In the TLL regime, we obtain the Luttinger parameter by probing sound propagation. Furthermore, a characteristic power-law behavior emerges in the measured momentum distributions of the 1D ultracold gas, confirming the existence of the TLL.

Robust stability condition of an uncertain networked system with delayed data dropout in both forward and feedback channels

2006 SICE-ICASE International Joint Conference, 2006

This paper provides stability conditions of an uncertain networked control system with both forwa... more This paper provides stability conditions of an uncertain networked control system with both forward control signal loss and feedback output signal loss. It is assumed that both the forward control signal and the output measurement signal are intermittent independently, and the remote plant has a model uncertainty. Also, in this paper, an observer-based controller is used in the remote place because the full states are not accessible. Thus, the main research goal is to analyze the overall stability of this uncertain intermittent networked system. In the second part of this paper, we consider delay effects of both forward signal and feedback signal during the network transfer.

Gaussian Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlevé IV System

Gaussian Unitary Ensembles with Jump Discontinuities, PDEs and the coupled Painleve IV equationsuities, , 2023

We study the Hankel determinant generated by the Gaussian weight with jump discontinuities at t 1... more We study the Hankel determinant generated by the Gaussian weight with jump discontinuities at t 1 , • • • , tm. By making use of a pair of ladder operators satisfied by the associated monic orthogonal polynomials and three supplementary conditions, we show that the logarithmic derivative of the Hankel determinant satisfies a second-order partial differential equation which is reduced to the σ-form of a Painlevé IV equation when m = 1. Moreover, under the assumption that t k − t 1 is fixed for k = 2, • • • , m, by considering the Riemann-Hilbert problem for the orthogonal polynomials, we construct direct relationships between the auxiliary quantities introduced in the ladder operators and solutions of a coupled Painlevé IV system.

The smallest eigenvalue of the Hankel matrices associated with a perturbed Jacobi weight

In this paper, we study the large N behavior of the smallest eigenvalue N of the (N + 1) ⇥ (N + 1... more In this paper, we study the large N behavior of the smallest eigenvalue N of the (N + 1) ⇥ (N + 1) Hankel matrix, H N = (µ j+k) 0 j,kN , generated by the dependent Jacobi weight w(z,) = e z z ↵ (1 z) , z 2 [0, 1], 2 R, ↵ > 1, > 1. Applying the arguments of Szegö, Widom and Wilf, we obtain the asymptotic representation of the orthonormal polynomials P N (z), z 2 C\[0, 1], with the weight w(z,) = e z z ↵ (1 z). Using the polynomials P N (z), we obtain the theoretical expression of N , for large N. We also display the smallest eigenvalue N for su ciently large N, computed numerically.

The Eigenvectors of Single Spiked Complex Wishart Matrices Finite and Asymptotic Analyses 1 1

by yang chen and Pasan Dissanayake

Let W ∈ C n×n be a single-spiked Wishart matrix in the class W ∼ CWn (m, In + θvv †) with m ≥ n, ... more Let W ∈ C n×n be a single-spiked Wishart matrix in the class W ∼ CWn (m, In + θvv †) with m ≥ n, where In is the n × n identity matrix, v ∈ C n×1 is an arbitrary vector with unit Euclidean norm, θ ≥ 0 is a non-random parameter, and (•) † represents the conjugate-transpose operator. Let u1 and un denote the eigenvectors corresponding to the smallest and the largest eigenvalues of W, respectively. This paper investigates the probability density function (p.d.f.) of the random quantity Z (n) = ¬ ¬ v † u ¬ ¬ 2 ∈ (0, 1) for = 1, n. In particular, we derive a finite dimensional closed-form p.d.f. for Z (n) 1 which is amenable to asymptotic analysis as m, n diverges with m − n fixed. It turns out that, in this asymptotic regime, the scaled random variable nZ (n) 1 converges in distribution to χ 2 2 /2(1 + θ), where χ 2 2 denotes a chi-squared random variable with two degrees of freedom. This reveals that u1 can be used to infer information about the spike. On the other hand, the finite dimensional p.d.f. of Z (n) n is expressed as a double integral in which the integrand contains a determinant of a square matrix of dimension (n − 2). Although a simple solution to this double integral seems intractable, for special configurations of n = 2, 3, and 4, we obtain closed-form expressions.

DISTRIBUTIONS OF DEMMEL AND RELATED CONDITION NUMBERS

Consider a random matrix A ∈ C m×n (m ≥ n) containing independent complex Gaussian entries with z... more Consider a random matrix A ∈ C m×n (m ≥ n) containing independent complex Gaussian entries with zero mean and unit variance, and let 0 < λ 1 ≤ λ 2 ≤ • • • ≤ λn < ∞ denote the eigenvalues of A * A, where (•) * represents conjugate-transpose. This paper investigates the distribution of the random variables n j=1 λ j λ k for k = 1 and k = 2. These two variables are related to certain condition number metrics, including the so-called Demmel condition number, which have been shown to arise in a variety of applications. For both cases, we derive new exact expressions for the probability densities and establish the asymptotic behavior as the matrix dimensions grow large. In particular, it is shown that as n and m tend to infinity with their difference fixed, both densities scale on the order of n 3. After suitable transformations, we establish exact expressions for the asymptotic densities, obtaining simple closed-form expressions in some cases. Our results generalize the work of Edelman on the Demmel condition number for the case m = n.

Distribution of the Scaled Condition Number of Single-Spiked Complex Wishart Matrices

Let X ∈ C n×m (m ≥ n) be a random matrix with independent columns each distributed as complex mul... more Let X ∈ C n×m (m ≥ n) be a random matrix with independent columns each distributed as complex multivariate Gaussian with zero mean and single-spiked covariance matrix In + ηuu * , where In is the n × n identity matrix, u ∈ C n×1 is an arbitrary vector with unit Euclidean norm, η ≥ 0 is a nonrandom parameter, and (•) * represents the conjugate-transpose. This paper investigates the distribution of the random quantity θ 2 SC (X) = È n k=1 ιk/ι1, where 0 ≤ ι1 ≤ ι2 ≤. .. ≤ ιn < ∞ are the ordered eigenvalues of XX * (i.e., single-spiked Wishart matrix). This random quantity is intimately related to the so called scaled condition number or the Demmel condition number (i.e., θSC(X)) and the minimum eigenvalue of the fixed trace Wishart-Laguerre ensemble (i.e., θ −2 SC (X)). In particular, we use an orthogonal polynomial approach to derive an exact expression for the probability density function of θ 2 SC (X) which is amenable to asymptotic analysis as matrix dimensions grow large. Our asymptotic results reveal that, as m, n → ∞ such that m − n is fixed and when η scales on the order of 1/n, θ 2 SC (X) scales on the order of n 3. In this respect we establish simple closed-form expressions for the limiting distributions. It turns out that, as m, n → ∞ such that n/m → c ∈ (0, 1), properly centered θ 2 SC (X) fluctuates on the scale m 1 3 .

Certificate of Emeritus (1)

Document , 2023

I am now Professor Emeritus at the University of Macau.

Asymptotic Properties of Some Freud Polynomials

Preprint, 2023

We study the asymptotic properties of monic orthogonal polynomials (OPs) with respect to some Fre... more We study the asymptotic properties of monic orthogonal polynomials (OPs) with respect to some Freud weights when the degree of the polynomial tends to infinity, including the asymptotics of the recurrence coefficients, the nontrivial leading coefficients of the monic OPs, the associated Hankel determinants and the squares of L 2-norm of the monic OPs. These results are derived from the combination of the ladder operator approach, Dyson's Coulomb fluid approach and some recent results in the literature.

Contents lists available at ScienceDirect

We study the Hankel determinant generated by the Laguerre weight with jump discontinuities at t k... more We study the Hankel determinant generated by the Laguerre weight with jump discontinuities at t k , k = 1,. .. , m. By employing the ladder operator approach to establish Riccati equations, we show that σ n (t 1 ,. .. , t m), the logarithmic derivative of the n-dimensional Hankel determinant, satisfies an m-variable generalization of the σ-form of a Painlevé V equation. Through investigating the Riemann-Hilbert problem for the associated orthogonal polynomials and via the Lax pair, we express σ n in terms of solutions of a coupled Painlevé V system. We also build relations between the auxiliary quantities introduced in the above two methods, which provides connections between the Riccati equations and the Lax pair. In addition, when each t k tends to the hard edge of the spectrum and n goes to infinity, the scaled σ n is shown to satisfy a generalized σ-form of a Painlevé III equation.

Painlevé V and the Hankel determinant for a singularly perturbed Jacobi weight

Nuclear Physics B, 2020

We study the Hankel determinant generated by a singularly perturbed Jacobi weight w(x, t) := (1 −... more We study the Hankel determinant generated by a singularly perturbed Jacobi weight w(x, t) := (1 − x 2) α e − t x 2 , x ∈ [−1, 1], α > 0, t ≥ 0. If t = 0, it is reduced to the classical symmetric Jacobi weight. For t > 0, the factor e − t x 2 induces an infinitely strong zero at the origin. This Hankel determinant is related to the Wigner time-delay distribution in chaotic cavities. In the finite n dimensional case, we obtain two auxiliary quantities R n (t) and r n (t) by using the ladder operator approach. We show that the Hankel determinant has an integral representation in terms of R n (t), where R n (t) is closely related to a particular Painlevé V transcendent. Furthermore, we derive a second-order nonlinear differential equation and also a second-order difference equation for the logarithmic derivative of the Hankel determinant. This quantity can be expressed in terms of the Jimbo-Miwa-Okamoto σ-function of a particular Painlevé V. Then we consider the asymptotics of the Hankel determinant under a suitable double scaling, i.e. n → ∞ and t → 0 such that s = 2n 2 t is fixed. Based on previous results by using the Coulomb fluid method, we obtain the large s and small s asymptotic behaviors of the scaled Hankel determinant, including the constant term in the asymptotic expansion.

Single-user MIMO system, Painlevé transcendents, and double scaling

Journal of Mathematical Physics, 2017

In this paper we study a particular Painlevé V (denoted P V) that arises from Multi-Input-Multi-O... more In this paper we study a particular Painlevé V (denoted P V) that arises from Multi-Input-Multi-Output (MIMO) wireless communication systems. Such a P V appears through its intimate relation with the Hankel determinant that describes the moment generating function (MGF) of the Shannon capacity. This originates through the multiplication of the Laguerre weight or the Gamma density x α e −x , x > 0, for α > −1 by (1 + x/t) λ with t > 0 a scaling parameter. Here the λ parameter "generates" the Shannon capacity; see Yang Chen and Matthew McKay, IEEE Trans. IT, 58 (2012) 4594-4634. It was found that the MGF has an integral representation as a functional of y(t) and y (t) , where y(t) satisfies the "classical form" of P V. In this paper, we consider the situation where n, the number of transmit antennas, (or the size of the random matrix), tends to infinity, and the signal-to-noise ratio (SNR) P tends to infinity, such that s = 4n 2 /P is finite. Under such double scaling the MGF, effectively an infinite determinant, has an integral representation in terms of a "lesser" P III. We also consider the situations where α = k + 1/2, k ∈ N, and α ∈ {0, 1, 2,. .. } λ ∈ {1, 2,. .. }, linking the relevant quantity to a solution of the two dimensional sine-Gordon equation in radial coordinates and a certain discrete Painlevé-II. From the large n asymptotic of the orthogonal polynomials, that appears naturally, we obtain the double scaled MGF for small and large s , together with the constant term in the large s expansion. With the aid of these, we derive a number of cumulants and find that the capacity distribution function is non-Gaussian.

Gap Probability Distribution of the Jacobi Unitary Ensemble: An Elementary Treatment, from Finitento Double Scaling

Studies in Applied Mathematics, 2017

In this paper, we study the gap probability problem of the (symmetric) Jacobi unitary ensemble of... more In this paper, we study the gap probability problem of the (symmetric) Jacobi unitary ensemble of Hermitian random matrices, namely the probability that the interval (−a, a) (0 < a < 1) is free of eigenvalues. Using the ladder operator technique for orthogonal polynomials and the associated supplementary conditions, we derive three quantities instrumental in the gap probability, denoted by H n (a), R n (a) and r n (a). We find that each one satisfies a second order differential equation. We show that after a double scaling, the large second order differential equation in the variable a with n as parameter satisfied by H n (a), can be reduced to the Jimbo-Miwa-Okamoto σ form of the Painlevé V equation.

Singular linear statistics of the Laguerre unitary ensemble and Painlevé. III. Double scaling analysis

Journal of Mathematical Physics, 2015

We continue with the study of the Hankel determinant, defined by, D n (t, α) = det ∞ 0 x j+k w(x;... more We continue with the study of the Hankel determinant, defined by, D n (t, α) = det ∞ 0 x j+k w(x; t, α)dx n−1 j,k=0 , generated by a singularly perturbed Laguerre weight, w(x; t, α) = x α e −x e −t/x , x ∈ R + , α > 0, t > 0, obtained through a deformation of the Laguerre weight function, w(x; 0, α) = x α e −x , x ∈ R + , α > 0, via the multiplicative factor e −t/x. An earlier investigation was made on the finite n aspect of such determinants, which appeared in [20]. It was found that the logarithm of the Hankel determinant has an integral representation in terms of a particular Painlevé III(P III , for short) and its t derivatives. In this paper we show that, under a double scaling, where n , the order of the Hankel matrix tends to ∞, and t , tends to 0 + , the scaled-and therefore, in some sense, infinite dimensional-Hankel determinant, has an integral representation in terms of a C potential. The second order non-linear ode satisfied by C, after a change of variable, is another P III transcendent, albeit with fewer number of parameters. Expansions of the double scaled determinant for small and large parameter are obtained.

Fixed-Time Consensus Tracking for Second-Order Multiagent Systems Under Disturbance

IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2019

This article focuses on the topic of fixed-time consensus for second-order multiagent systems (MA... more This article focuses on the topic of fixed-time consensus for second-order multiagent systems (MASs) with disturbances. Based on an integration of the nominal control part and discontinuous integral sliding control part or continuous supertwisting-like control part, some control protocols are presented to achieve consensus tracking in the fixed time. A distributed integral sliding mode (ISM) and a fixed-time convergent command filter are introduced. The performance of nominal dynamics is dominated by the nominal control part which ensures the fixedtime convergence in the ISM surface. The discontinuous sliding mode or continuous super-twisting-like part related to the ISM is utilized to compensate disturbances within fixed convergence time. In this article, we have designed the continuous fixed-time consensus tracking controllers which can eliminate the chattering phenomenon and simultaneously guarantee the convergence precision. Independent of any initial state values, the restrictive bound of the convergence time is estimated. Some fair comparisons are performed to demonstrate the merits of the proposed strategies.

New results on the phase diagram of the (TMTSF)2FSO3 salt

by Vladan Celebonovic and yang chen

Synthetic Metals, 1991

We present transport measurements under pressure on the (TMTSF)2FSO 3 salt from P= l bar to 8kbar... more We present transport measurements under pressure on the (TMTSF)2FSO 3 salt from P= l bar to 8kbar using a helium gas pressure cell, in order to clarify the pressure-temperature phase diagram of this compound. The interest in this TMTSF salt comes from the existence of non-centrosymmetric tetrahedral anion with a permanent dipole moment related to the presence of the fluorine atom. Two types of disorder can be involved in the metal to insulator transitions at low temperature: one due to the orientation of the tetrahedra and the second to the position of the fluorine atom inside the anion. For all studied pressures a serniconducting phase occurs at T = 90K via a very sharp transition. At lower temperatures, the activation energy decreases as the pressure is increased. At last, the resistivity saturates at P=8kbar from T=80K down to 10K where the metallic state is restored. This new study shows that phase diagrams derived with the use of the clamped pressure techniques should be revised.

Orthogonal polynomials with discontinuous weights

Journal of Physics A: Mathematical and General, 2005

In this paper we present a brief description of a ladder operator formalism applied to orthogonal... more In this paper we present a brief description of a ladder operator formalism applied to orthogonal polynomials with discontinuous weights. The two coefficient functions, A n (z) and B n (z), appearing in the ladder operators satisfy the two fundamental compatibility conditions previously derived for smooth weights. If the weight is a product of an absolutely continuous reference weight w 0 and a standard jump function, then A n (z) and B n (z) have apparent simple poles at these jumps. We exemplify the approach by taking w 0 to be the Hermite weight. For this simpler case we derive, without using the compatibility conditions, a pair of difference equations satisfied by the diagonal and off-diagonal recurrence coefficients for a fixed location of the jump. We also derive a pair of Toda evolution equations for the recurrence coefficients which, when combined with the difference equations, yields a particular Painlevé IV.

Orthogonal polynomials, asymptotics, and Heun equations

Journal of Mathematical Physics

The Painlevé equations arise from the study of Hankel determinants generated by moment matrices, ... more The Painlevé equations arise from the study of Hankel determinants generated by moment matrices, whose weights are expressed as the product of "classical" weights multiplied by suitable "deformation factors," usually dependent on a "time variable" t. From ladder operators [see A. Magnus, J. Comput. Appl. Math. 57(1-2), 215-237 (1995)], one finds second order linear ordinary differential equations for associated orthogonal polynomials with coefficients being rational functions. The Painlevé and related functions appear as the residues of these rational functions. We will be interested in the situation when n, the order of the Hankel matrix and also the degree of the polynomials Pn(x) orthogonal with respect to the deformed weights, gets large. We show that the second order linear differential equations satisfied by Pn(x) are particular cases of Heun equations when n is large. In some sense, monic orthogonal polynomials generated by deformed weights mentioned below are solutions of a variety of Heun equations. Heun equations are of considerable importance in mathematical physics, and in the special cases, they degenerate to the hypergeometric and confluent hypergeometric equations. In this paper, we look at three types of weights: the Jacobi type, the Laguerre type, and the weights deformed by the indicator function of χ (a,b) (x) and the step function θ(x). In particular, we consider the following Jacobi type weights: (1.1) x α (1 − x) β e −tx , x ∈ [0, 1], α, β, t > 0; (1.2) x α (1 − x) β e −t/x , x ∈ (0, 1], α, β, t > 0; (1.3) (1 − x 2) α (1 − k 2 x 2) β , x ∈ [−1, 1], α, β > 0, k 2 ∈ (0, 1); the Laguerre type weights: (2.1) x α (x + t) λ e −x , x ∈ [0, ∞), t, α, λ > 0; (2.2) x α e −x−t/x , x ∈ (0, ∞), α, t > 0; and another type of deformation when the classical weights are multiplied by χ (a,b) (x) or θ(x): (3.1) e −x 2 (1 − χ (−a,a) (x)), x ∈ R, a > 0; (3.2) (1 − x 2) α (1 − χ (−a,a) (x)), x ∈ [−1, 1], a ∈ (0, 1), α > 0; (3.3) x α e −x (A + Bθ(x − t)), x ∈ [0, ∞), α, t > 0, A ≥ 0, A + B ≥ 0. The weights mentioned above were studied in a series of papers related to the deformation of "classical" weights.

Smallest Eigenvalue of Large Hankel Matrices at Critical Point: Comparing Conjecture With Parallelised Computation

We propose a novel parallel numerical algorithm for calculating the smallest eigenvalues of highl... more We propose a novel parallel numerical algorithm for calculating the smallest eigenvalues of highly ill-conditioned matrices. It is based on the LDLT decomposition and involves finding a k × k sub-matrix of the inverse of the original N × N Hankel matrix H −1 N. The computation involves extremely high precision arithmetic, message passing interface, and shared memory parallelisation. We demonstrate that this approach achieves good scalability on a high performance computing cluster (HPCC) which constitute a major improvement of the earlier approaches. We use this method to study a family of Hankel matrices generated by the weight w(x) = e −x β , supported on [0, ∞) and β > 0. Such weight generates Hankel determinant, a fundamental object in random matrix theory. In the situation where β > 1/2, the smallest eigenvalue tend to 0, exponentially fast. If β < 1/2, the situation where the classical moment problem is indeterminate, the smallest eigenval-ue is bounded from below by ...

Orthogonal Polynomials, Asymptotics and Heun Equations

The Painlevé equations arise from the study of Hankel determinants generated by moment matrices, ... more The Painlevé equations arise from the study of Hankel determinants generated by moment matrices, whose weights are expressed as the product of "classical" weights multiplied by suitable "deformation factors", usually dependent on a "time variable" t. From ladder operators [12-14, 30] one finds second order linear ordinary differential equations for associated orthogonal polynomials with coefficients being rational functions. The Painlevé and related functions appear as the residues of these rational functions. We will be interested in the situation when n, the order of the Hankel matrix and also the degree of the polynomials P n (x) orthogonal with respect to the deformed weights, gets large. We show that the second order linear differential equations satisfied by P n (x) are particular cases of Heun equations when n is large. In some sense, monic orthogonal polynomials generated by deformed weights mentioned below are solutions of a variety of Heun equ...

Quantum criticality and the Tomonaga-Luttinger liquid in one-dimensional Bose gases

Physical Review Letters

We experimentally investigate the quantum criticality and Tomonaga-Luttinger liquid (TLL) behavio... more We experimentally investigate the quantum criticality and Tomonaga-Luttinger liquid (TLL) behavior within one-dimensional (1D) ultracold atomic gases. Based on the measured density profiles at different temperatures, the universal scaling laws of thermodynamic quantities are observed. The quantum critical regime and the relevant crossover temperatures are determined through the double-peak structure of the specific heat. In the TLL regime, we obtain the Luttinger parameter by probing sound propagation. Furthermore, a characteristic power-law behavior emerges in the measured momentum distributions of the 1D ultracold gas, confirming the existence of the TLL.

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