Spectral Asymptotics of Pauli Operators and Orthogonal Polynomials in Complex Domains

We consider the spectrum of a two-dimensional Pauli operator with a compactly supported electric potential and a variable magnetic field with a positive mean value. The rate of accumulation of eigenvalues to zero is described in terms of the logarithmic capacity of the support of the electric potential. A connection between these eigenvalues and orthogonal polynomials in complex domains is established. On leave of absence from King's College London, U.K.     

Density of Complex Zeros of a System of Real Random Polynomials

We study the density of complex zeros of a system of real random SO(m+1) polynomials in m variables. We show that the density of complex zeros of this random polynomial system with real coefficients rapidly approaches the density of complex zeros in the complex coefficients case. We also show that the behavior the scaled density of complex zeros near ℝ ^m of the system of real random polynomials is different in the m≥2 case than in the m=1 case: the density approaches infinity instead of tending linearly to zero.     

On Absolute Moments of Characteristic Polynomials of a Certain Class of Complex Random Matrices

The integer moments of the spectral determinant | det (zI − W) |² of complex random matrices W are obtained in terms of the characteristic polynomial of the positive-semidefinite matrix WW ^† for the class of matrices W = AU, where A is a given matrix and U is random unitary. This work is motivated by studies of complex eigenvalues of random matrices and potential applications of the obtained results are discussed in this context. The research in Nottingham was supported by EPSRC grant EP/C515056/1: “Random Matrices and Polynomials: a tool to understand complexity”. Part of this work was carried out during the Newton Institute programme on Random Matrix Approaches in Number Theory.     

Relative Entropy and Identification of Gibbs Measures in Dynamical Systems

In this work we explore the idea of using the relative entropy of ergodic measures for the identification of Gibbs measures in dynamical systems. The question we face is how to estimate the thermodynamic potential (together with a grammar) from a sample produced by the corresponding Gibbs state.     

Relative Entropy of Correct Proximal Policy Optimization Algorithms with Modified Penalty Factor in Complex Environment

In the field of reinforcement learning, we propose a Correct Proximal Policy Optimization (CPPO) algorithm based on the modified penalty factor β and relative entropy in order to solve the robustness and stationarity of traditional algorithms. Firstly, In the process of reinforcement learning, this paper establishes a strategy evaluation mechanism through the policy distribution function. Secondly, the state space function is quantified by introducing entropy, whereby the approximation policy is used to approximate the real policy distribution, and the kernel function estimation and calculation of relative entropy is used to fit the reward function based on complex problem. Finally, through the comparative analysis on the classic test cases, we demonstrated that our proposed algorithm is effective, has a faster convergence speed and better performance than the traditional PPO algorithm, and the measure of the relative entropy can show the differences. In addition, it can more efficiently use the information of complex environment to learn policies. At the same time, not only can our paper explain the rationality of the policy distribution theory, the proposed framework can also balance between iteration steps, computational complexity and convergence speed, and we also introduced an effective measure of performance using the relative entropy concept.     

An Explicit Family of Probability Measures for Passive Scalar Diffusion in a Random Flow

We explore the evolution of the probability density function (PDF) for an initially deterministic passive scalar diffusing in the presence of a uni-directional, white-noise Gaussian velocity field. For a spatially Gaussian initial profile we derive an exact spatio-temporal PDF for the scalar field renormalized by its spatial maximum. We use this problem as a test-bed for validating a numerical reconstruction procedure for the PDF via an inverse Laplace transform and orthogonal polynomial expansion. With the full PDF for a single Gaussian initial profile available, the orthogonal polynomial reconstruction procedure is carefully benchmarked, with special attentions to the singularities and the convergence criteria developed from the asymptotic study of the expansion coefficients, to motivate the use of different expansion schemes. Lastly, Monte-Carlo simulations stringently tested by the exact formulas for PDF’s and moments offer complete pictures of the spatio-temporal evolution of the scalar PDF’s for different initial data. Through these analyses, we identify how the random advection smooths the scalar PDF from an initial Dirac mass, to a measure with algebraic singularities at the extrema. Furthermore, the Péclet number is shown to be decisive in establishing the transition in the singularity structure of the PDF, from only one algebraic singularity at unit scalar values (small Péclet), to two algebraic singularities at both unit and zero scalar values (large Péclet).     

General Asymptotics of the Density of a Restricted Coalescing Random Walk System

These results explore the asymptotic behavior of the density of a system of coalescing random walks where particles begin from only a subspace of the integer lattice and are allowed to walk anywhere on the lattice. They generalize results by Bramson and Griffeath from 1980.⁽¹⁾ Since the probability that a given site is occupied depends on how far that site is from the originating subspace, the density of the system at a given time must be re-defined. However, the general idea is still that if the density is larger than we expect at a given time, more coalescing events will occur, and the density will correct itself over time.     

Exact Asymptotics of the Freezing Transition of a Logarithmically Correlated Random Energy Model

We consider a logarithmically correlated random energy model, namely a model for directed polymers on a Cayley tree, which was introduced by Derrida and Spohn. We prove asymptotic properties of a generating function of the partition function of the model by studying a discrete time analogy of the KPP-equation—thus translating Bramson’s work on the KPP-equation into a discrete time case. We also discuss connections to extreme value statistics of a branching random walk and a rescaled multiplicative cascade measure beyond the critical point.     

EEG-Based Emotion Recognition by Exploiting Fused Network Entropy Measures of Complex Networks across Subjects

It is well known that there may be significant individual differences in physiological signal patterns for emotional responses. Emotion recognition based on electroencephalogram (EEG) signals is still a challenging task in the context of developing an individual-independent recognition method. In our paper, from the perspective of spatial topology and temporal information of brain emotional patterns in an EEG, we exploit complex networks to characterize EEG signals to effectively extract EEG information for emotion recognition. First, we exploit visibility graphs to construct complex networks from EEG signals. Then, two kinds of network entropy measures (nodal degree entropy and clustering coefficient entropy) are calculated. By applying the AUC method, the effective features are input into the SVM classifier to perform emotion recognition across subjects. The experiment results showed that, for the EEG signals of 62 channels, the features of 18 channels selected by AUC were significant (p < 0.005). For the classification of positive and negative emotions, the average recognition rate was 87.26%; for the classification of positive, negative, and neutral emotions, the average recognition rate was 68.44%. Our method improves mean accuracy by an average of 2.28% compared with other existing methods. Our results fully demonstrate that a more accurate recognition of emotional EEG signals can be achieved relative to the available relevant studies, indicating that our method can provide more generalizability in practical use.     

A Lower Bound for the Wehrl Entropy of Quantum Spin with Sharp High-Spin Asymptotics

A lower bound for the Wehrl entropy of a single quantum spin is derived. The high-spin asymptotics of this bound coincides with Liebs conjecture up to, but not including, terms of first and higher order in the inverse spin quantum number. The result presented here may be seen as complementary to the verification of the conjecture in cases of lowest spin by Schupp [Commun. Math. Phys. 207, 481 (1999)]. The present result for the Wehrl-entropy is obtained from interpolating a sharp norm bound that also implies a sharp lower bound for the so-called Rényi-Wehrl entropy with certain indices that are evenly spaced by half of the inverse spin quantum number.     

Relative Entropy of Entanglement of a Kind of Two-Qubit Entangled States

We strictly prove that some block diagonalizable two-qubit entangled state with six non-zero elements reaches its quantum relative entropy entanglement by a separable state having the same matrix structure. The entangled state comprises local filtering result state as a special case.     

Entropy Production for a Class of Inverse SRB Measures

We study the entropy production for inverse SRB measures for a class of hyperbolic folded repellers presenting both expanding and contracting directions. We prove that for most such maps we obtain strictly negative entropy production of the respective inverse SRB measures. Moreover we provide concrete examples of hyperbolic folded repellers where this happens.     

Random Versus Deterministic Exponents in a Rich Family of Diffeomorphisms

We study, both numerically and theoretically, the relationship between the random Lyapunov exponent of a family of area preserving diffeomorphisms of the 2-sphere and the mean of the Lyapunov exponents of the individual members. The motivation for this study is the hope that a rich enough family of diffeomorphisms will always have members with positive Lyapunov exponents, that is to say, positive entropy. At question is what sort of notion of richness would make such a conclusion valid. One type of richness of a family—invariance under the left action of SO(n+1)—occurs naturally in the context of volume preserving diffeomorphisms of the n-sphere. Based on some positive results for families linear maps obtained by Dedieu and Shub, we investigate the exponents of such a family on the 2-sphere. Again motivated by the linear case, we investigate whether there is in fact a lower bound for the mean of the Lyapunov exponents in terms of the random exponents (with respect to the push-forward of Haar measure on SO(3)) in such a family. The family that we study contains a twist map with stretching parameter . In the family , we find strong numerical evidence for the existence of such a lower bound on mean Lyapunov exponents, when the values of the stretching parameter are not too small. Even moderate values of like 10 are enough to have an average of the metric entropy larger than that of the random map. For small the estimated average entropy seems positive but is definitely much less than the one of the random map. The numerical evidence is in favor of the existence of exponentially small lower and upper bounds (in the present example, with an analytic family). Finally, the effect of a small randomization of fixed size of the individual elements of the family is considered. Now the mean of the local random exponents of the family is indeed asymptotic to the random exponent of the entire family as tends to infinity.     

On the Correlation Functions of the Characteristic Polynomials of Non-Hermitian Random Matrices with Independent Entries

Journal of Statistical Physics - The paper is concerned with the asymptotic behavior of the correlation functions of the characteristic polynomials of non-Hermitian random matrices with independent...     

Stochastic Hydrodynamics of Complex Fluids: Discretisation and Entropy Production

Many complex fluids can be described by continuum hydrodynamic field equations, to which noise must be added in order to capture thermal fluctuations. In almost all cases, the resulting coarse-grained stochastic partial differential equations carry a short-scale cutoff, which is also reflected in numerical discretisation schemes. We draw together our recent findings concerning the construction of such schemes and the interpretation of their continuum limits, focusing, for simplicity, on models with a purely diffusive scalar field, such as ‘Model B’ which describes phase separation in binary fluid mixtures. We address the requirement that the steady-state entropy production rate (EPR) must vanish for any stochastic hydrodynamic model in a thermal equilibrium. Only if this is achieved can the given discretisation scheme be relied upon to correctly calculate the nonvanishing EPR for ‘active field theories’ in which new terms are deliberately added to the fluctuating hydrodynamic equations that break detailed balance. To compute the correct probabilities of forward and time-reversed paths (whose ratio determines the EPR), we must make a careful treatment of so-called ‘spurious drift’ and other closely related terms that depend on the discretisation scheme. We show that such subtleties can arise not only in the temporal discretisation (as is well documented for stochastic ODEs with multiplicative noise) but also from spatial discretisation, even when noise is additive, as most active field theories assume. We then review how such noise can become multiplicative via off-diagonal couplings to additional fields that thermodynamically encode the underlying chemical processes responsible for activity. In this case, the spurious drift terms need careful accounting, not just to evaluate correctly the EPR but also to numerically implement the Langevin dynamics itself.     

Design and Test of an Integrated Random Number Generator with All-Digital Entropy Source

In the cybersecurity field, the generation of random numbers is extremely important because they are employed in different applications such as the generation/derivation of cryptographic keys, nonces, and initialization vectors. The more unpredictable the random sequence, the higher its quality and the lower the probability of recovering the value of those random numbers for an adversary. Cryptographically Secure Pseudo-Random Number Generators (CSPRNGs) are random number generators (RNGs) with specific properties and whose output sequence has such a degree of randomness that it cannot be distinguished from an ideal random sequence. In this work, we designed an all-digital RNG, which includes a Deterministic Random Bit Generator (DRBG) that meets the security requirements for cryptographic applications as CSPRNG, plus an entropy source that showed high portability and a high level of entropy. The proposed design has been intensively tested against both NIST and BSI suites to assess its entropy and randomness, and it is ready to be integrated into the European Processor Initiative (EPI) chip.     

Belief Entropy Tree and Random Forest: Learning from Data with Continuous Attributes and Evidential Labels

As well-known machine learning methods, decision trees are widely applied in classification and recognition areas. In this paper, with the uncertainty of labels handled by belief functions, a new decision tree method based on belief entropy is proposed and then extended to random forest. With the Gaussian mixture model, this tree method is able to deal with continuous attribute values directly, without pretreatment of discretization. Specifically, the tree method adopts belief entropy, a kind of uncertainty measurement based on the basic belief assignment, as a new attribute selection tool. To improve the classification performance, we constructed a random forest based on the basic trees and discuss different prediction combination strategies. Some numerical experiments on UCI machine learning data set were conducted, which indicate the good classification accuracy of the proposed method in different situations, especially on data with huge uncertainty.     

Asymptotics of a Fredholm Determinant Corresponding to the First Bulk Critical Universality Class in Random Matrix Models

We study the determinant det(I?K _PII) of an integrable Fredholm operator K _PII acting on the interval (?s, s) whose kernel is constructed out of the Ψ-function associated with the Hastings–McLeod solution of the second Painlevé equation. This Fredholm determinant describes the critical behavior of the eigenvalue gap probabilities of a random Hermitian matrix chosen from the unitary ensemble in the bulk double scaling limit near a quadratic zero of the limiting mean eigenvalue density. Using the Riemann–Hilbert method, we evaluate the large s-asymptotics of det(I?K _PII) .     

Influence of Coating Layer on Acoustic Wave Propagation in a Random Complex Medium with Resonant Scatterers

We investigate the influence of coating layer on acoustic wave propagation in a dispersed random medium consisting of coa.ted fibers.In the strong-scattering regime, the characteristics of wave scattering resonances are found to evolve regularly with the properties of the coating layer.By theoretical calculation,frequency gaps are found in acoustic excitation spectra in a random medium.The scattering cross section results present the evolution of scattering resonances with the properties of the coating layer,which offers a good explanation for the change of the frequency gaps.The velocity of the propagation quasi-mode is also shown to depend on the filling fraction of the coating layer.We use the generalized coherent potential-approximation approach to solve acoustic wave dispersion relations in a complicated random medium consisting of coating-structure scatterers.It is shown that our model reveals subtle changes in the behavior of the acoustic wave propagating quasi-modes.