Gaussian Term Research Articles

Homogenization for singularly perturbed stochastic wave equations with Hölder continuous coefficients

This work is concerned with the homogenization problem for singularly perturbed stochastic wave equations. Under the assumption that the coefficients are only Hölder continuous, we prove the weak convergence of the original system to a limit equation with an extra Gaussian term by using the technique of Poisson equation in Hilbert space. The optimal convergence rate is also obtained.

Solvable limit of ETH matrix model for double-scaled SYK

We study the two-matrix model for double-scaled SYK model, called ETH matrix model introduced by Jafferis et al. [arXiv:2209.02131]. If we set the parameters qA, qB of this model to zero, the potential of this two-matrix model is given by the Gaussian terms and the q-commutator squared interaction. We find that this model is solvable in the large N limit and we explicitly construct the planar one- and two-point function of resolvents in terms of elliptic functions.

Computing Arbitrage-Free Yields in Multi-Factor Gaussian Shadow Rate Term Structure Models

This paper develops an approximation to arbitrage-free bond yields in Gaussian shadow rate term structure models. In this class of models, yields are constrained to be above an effective lower bound, thus rendering standard bond pricing methods inapplicable. I propose approximating the nonlinear relationship between yields and state variables using moments of the censored normal distribution. In an empirical application, this approximation technique is accurate to within a fraction of a basis point. As I show, minimizing the yield approximation error is crucial for model estimation as even seemingly small errors can lead to economically meaningful inference biases.

Primordial black holes from a curvaton scenario with strongly non-Gaussian perturbations

We investigate the production of primordial black holes (PBHs) in a mixed inflaton-curvaton scenario with a quadratic curvaton potential, assuming the curvaton is in de Sitter equilibrium during inflation with 〈χ〉 = 0. In this setup, the curvature perturbation sourced by the curvaton is strongly non-Gaussian, containing no leading Gaussian term. We show that for m 2/H 2 ≳ 0.3, the curvaton contribution to the spectrum of primordial perturbations on CMB scales can be kept negligible but on small scales the curvaton can source PBHs. In particular, PBHs in the asteroid mass range 10-16 M ⊙ ≲ M ≲ 10-10 M ⊙ with an abundance reaching F PBH = 1 can be produced when the inflationary Hubble scale H ≳ 1012 GeV and the curvaton decay occurs in the window from slightly before the electroweak transition to around the QCD transition.

Spontaneous symmetry breaking in the BFSS model: analytical results using the Gaussian expansion method

We apply the Gaussian expansion method to the BFSS matrix model in the high temperature limit. When the (Euclidean) BFSS action is expanded about a Gaussian ansatz, it is shown that the SO(9) symmetry is spontaneously broken, analogous to what happens in the IKKT model. The analysis of the free energy, using the set of gap equations which determines the width of the Gaussian terms, is sufficient to show that this symmetry breaking happens only when the fermionic terms are included and is absent in the bosonic case.

Parallel Kalman filter group integrated particle filter method for the train nonlinear operational status high-precision estimation under non-Gaussian environment

For the problem of multi-mode state estimation in actual train operation, this paper proposes a nonlinear non-gaussian high-precision parallel Kalman filter group (NN-HEKFG) integrated Particle Filter. A multi-model Gaussian decomposition of the probability density function for state equations and measurement equations is performed, and each local state model is represented by a multi-dimensional high-order polynomial to establish the expanded dimensional state model. Then, by updating the mean and variance of the local state expanded dimensional model and in turn solving the particle filtering posterior probability density distribution function, the global estimation results are obtained. In reducing the number of Gaussian terms, a new parameter reduction criterion is established, which can effectively carry out the re-identification of parameters such as weights and means, so as to avoid the problem of parameter explosion. The superiority of NN-HEKFG over particle filters and Gaussian sum filters and its effectiveness for train running state estimation are verified by simulating the multi-model running state of trains.

Unconventional Monetary Policies and the Yield Curve: Estimating Non-Affine Term Structure Models with Unspanned Macro Risk by Factor Extraction

Abstract We show how the Joslin, Singleton, and Zhu (2011) factor extraction approach to estimating the Gaussian term structure model can be modified to handle the interest rate lower bound without the approximations used in other approaches. This drastically reduces the computation time and produces more robust estimates of the lower bound parameter and the shadow rate. It makes feasible the extensive specification search necessary to allow for unspanned factors as in Joslin, Priebsch, and Singleton (2014), allowing the term structure model to be used to better assess the effects of policy on the term premium and market expectations. (JEL G12, C13, E43)

Eigenmode analysis of the multiple temperature model: spectrum properties, hierarchical structures, and temperature inversion

Recent developments of ultrafast laser pulse techniques enable us to study the subpicosecond scale dynamics out of thermal equilibrium. Multiple temperature models (MTMs) are frequently used to describe such dynamics where the total system is divided into subsystems each of which is in local thermal equilibrium. Typical examples include the electron-lattice two temperature model and electron-spin-phonon three temperature model. We present the exact analytical solutions of linear MTM, based on the Fourier series expansion and the Green’s function method. We then discuss their properties for the case of the two and three temperature models. We show that the general solution of MTM is expressed as a linear combinations of a spatially uniform, single-temperature stationary mode and the other non-oscillatory, decaying “eigenmodes” characterized by different wave vectors and well-defined mode lifetimes. The eigenmode picture enables us to explore the hierarchical structure of models with respect to space, time and the coupling parameter. Excitation by source term is included by the Green’s function method. As an example, we derive an analytical solution for a Gaussian type source term. We report a phenomenon “temperature inversion” where the lattice temperature exceeds electron’s temperature for ns time scale. Furthermore, we show how physical requirements such as energy conservation and equilibration are realized in the general linear MTM in terms of the eigenmode picture.

Analysis of collaborative urban public crisis governance in complex system: A multi-agent stochastic evolutionary game approach

As the world economy is leaping forward, countries have been exposed to the constant threat of public crises. Accelerated urbanization has currently led to a proliferation of urban public crisis events. Conventional governance methods become ineffective under the effect of the highly uncertain and dynamic nature of urban public crisis events. Accordingly, this study primarily aims to facilitate the construction of a collaborative multi-agent alliance in urban public crisis governance. Thus, evolutionary game theory is combined with system dynamics, Gaussian white noise stochastic disturbance terms are introduced to model the complex characteristics of urban public crisis events, and evolutionary game models and stochastic evolutionary game models are built for three types of groups (i.e., local governments, enterprises, and the public) to study the evolutionary process of group strategy choice behavior with or without central government supervision. The result of this study suggests that under the effect of the complexity factor, the central government's dynamic reward and punishment mechanism plays a role of in reaching a facilitative and stable alliance for collaborative urban public crisis governance. Furthermore, contributions, unresolved issues and vital challenges are identified. In this study, a solid basis is laid for future research and widespread use of collaborative urban public crisis governance, which provides a reference for stakeholders in the field of urban public crisis governance.

Direct computation of nth-order correlations of the solution of a non-linear stochastic equation

SummaryA system of nonlinear stochastic equations excited by a Gaussian random term leading to a statistically stationary solution is considered. The Carleman linearization is used to handle the nonlinearity and the statistical characterization of the solution is formulated in terms of a sequence of correlations of increasing order. Truncated in a finite but arbitrary order, the problem leads to a linear system of equations yielding directly the correlations. It is demonstrated in two examples that, if the excitation is not too strong, the solution converges as a function of the truncation order and provides an alternative to the Monte Carlo approach consisting in ensemble averaging a large number of time-dependent random solutions. For a low-dimensional system, it is shown that replacing the tensor indexing by a numbering accounting for redundancies makes it possible to keep the total problem dimension within reasonable limits even if relatively high-order correlations are accounted for. A model reduction based on the singular value decomposition of the second-order correlation matrix is tested with success for a case of a partial differential equation (Burgers’ equation), showing that the method can be potentially applied even to high-dimensional systems originating in partial differential equations.

Cosmic shear in harmonic space from the Dark Energy Survey Year 1 Data: compatibility with configuration space results

ABSTRACT We perform a cosmic shear analysis in harmonic space using the first year of data collected by the Dark Energy Survey (DES-Y1). We measure the cosmic weak lensing shear power spectra using the metacalibration catalogue and perform a likelihood analysis within the framework of CosmoSIS. We set scale cuts based on baryonic effects contamination and model redshift and shear calibration uncertainties as well as intrinsic alignments. We adopt as fiducial covariance matrix an analytical computation accounting for the mask geometry in the Gaussian term, including non-Gaussian contributions. A suite of 1200 lognormal simulations is used to validate the harmonic space pipeline and the covariance matrix. We perform a series of stress tests to gauge the robustness of the harmonic space analysis. Finally, we use the DES-Y1 pipeline in configuration space to perform a similar likelihood analysis and compare both results, demonstrating their compatibility in estimating the cosmological parameters S8, σ8, and Ωm. We use the DES-Y1 metacalibration shape catalogue, with photometric redshifts estimates in the range of 0.2−1.3, divided in four tomographic bins finding σ8(Ωm/0.3)0.5 = 0.766 ± 0.033 at 68 per cent CL. The methods implemented and validated in this paper will allow us to perform a consistent harmonic space analysis in the upcoming DES data.

Mutual information analysis of mutation, nonlinearity, and triple interactions in proteins.

Mutations are the cause of several diseases as well as the underlying force of evolution. A thorough understanding of their biophysical consequences is essential. We present a computational framework for evaluating different levels of mutual information (MI) and its dependence on mutation. We used molecular dynamics trajectories of the third PDZ domain and its different mutations. Nonlinear MI between all residue pairs are calculated by tensor Hermite polynomials up to the fifth order and compared with results from multivariate Gaussian distribution of joint probabilities. We show that MI is written as the sum of a Gaussian and a nonlinear component. Results for the PDZ domain show that the Gaussian term gives a sufficiently accurate representation of MI when compared with nonlinear terms up to the fifth order. Changes in MI between residue pairs show the characteristic patterns resulting from specific mutations. Emergence of new peaks in the MI versus residue index plots of mutated PDZ shows how mutation may change allosteric pathways. Triple correlations are characterized by evaluating MI between triplets of residues. We observed that certain triplets are strongly affected by mutation. Susceptibility of residues to perturbation is obtained by MI and discussed in terms of linear response theory.

Gaussian dispersion analysis in the time domain: Efficient conversion with Padé approximants

We present an approach for adapting the Gaussian dispersion analysis (GDA) of optical materials to time-domain simulations. Within a GDA model, the imaginary part of a measured dielectric function is presented as a sum of Gaussian absorption terms. Such a simple model is valid for materials where inhomogeneous broadening is substantially larger than the homogeneous linewidth. The GDA model is the essential broadband approximation for the dielectric function of many glasses, polymers, and other natural and artificial materials with disorder. However, efficient implementation of this model in time-domain full-wave electromagnetic solvers has never been fully achieved. We start with a causal form of an isolated oscillator with Gaussian-type absorption — Causal Dawson-Gauss oscillator. Then, we derive explicit analytical formulas to implement the Gaussian oscillator in a finite-difference time-domain (FDTD) solver with minimal use of memory and floating point operations. The derivation and FDTD implementation employ our generalized dispersive material (GDM) model — a universal, modular approach to describing optical dispersion with Padé approximants. We share the FDTD prototype codes that include automated generation of the approximants and a universal FDTD dispersion implementation that employs various second-order accurate numerical schemes. The codes can be used with non-commercial solvers and commercial software for time-domain simulations of light propagation in dispersive media, which are experimentally characterized with GDA models. Program summaryProgram Title:MADISCPC Library link to program files:https://doi.org/10.17632/x69b4krsy8.1Licensing provisions: GPLv3Programming language: MATLABNature of problem: The problem of efficient time-domain simulation of the Gaussian absorption is essential for wideband modeling of the optical response from materials with inhomogeneous spectral broadening, such as glasses, polymers, and other natural and artificial composite materials with structural or phase disorder.Solution method: A Coupled Oscillator (CO) approximation to the Gaussian absorption in both the frequency and time domains is derived to solve this problem. The time-domain CO approximation is coupled to the finite-difference time-domain (FDTD) solver for the Maxwell equations in the MADIS (MAterial DIspersion Simulator) package. Verification of the Maxwell solvers' accuracy and stability is performed with the FDTD solver coupled to a compact universal implementation of the CO model, employing second-order schemes (either ADE or RC). Code prototypes of these efficient CO schemes can be ported to other methods and platforms for implementing the Gaussian absorption in open-source codes or commercial software.Additional comments including restrictions and unusual features: Patent pending. Restrict any commercial use, including for profit reproduction.

Multi-Extended Target Tracking Algorithm Based on VBEM-CPHD

Considering that the extended targets tracking problem of the measurement is a glint noise with an unknown inverse covariance, a new algorithm from a multi-extended target tracking based on the Variational Bayesian Expectation Maximization (VBEM) is proposed. To improve the variational Bayesian technique, a modeled Student’s t distribution is proposed based on multiple Gaussian mixture terms in order to replace the probability hypothesis density (PHD) intensity. The extended target was modeled with a Student’s t distribution with the glint noise, using Gauss-Gamma distribution combining the variational Bayesian technique to obtain an approximate distribution and applying the expectation-maximization algorithm for iterative estimation. The two experiments are compared and analyzed with the VBEM-CPHD algorithm and the traditional extended target tracking algorithm. Experiment 1 estimated the trajectory of the target, compared algorithms of VBEM-CPHD and GM-CPHD with OSPA distance, varied glint noise with the three different measurement noise standard deviations. Experiment 2 completed the examination of the tracking performance and stability of the proposed method and performed to compare the VBEM-CPHD algorithm proposed with the VB-based GM-CPHD (GM-VBCPHD) algorithm under an unknown measurement noise covariance. The experimental results indicate that the algorithm of VBEM-CPHD has high tracking accuracy, good adaptability, and a strong antijamming ability for multiple extended targets under glint noise conditions.

Symplectic tomographic probability distribution of crystallized Schrödinger cat states

Within the framework of the probability representation of quantum mechanics, we study a superposition of generic Gaussian states associated to symmetries of a regular polygon of n sides; in other words, the cyclic groups (containing the rotational symmetries) and dihedral groups (containing the rotational and inversion symmetries). We obtain the Wigner functions and tomographic probability distributions (symplectic and optical tomograms) determining the density matrices of the states explicitly as the sums of Gaussian terms. The obtained Wigner functions demonstrate nonclassical behavior, i.e., contain negative values, while the tomograms show a series of maxima and minima different for each state, where the number of the critical points reflects the order of the group defining the states. We discuss general properties of such a generalization of normal probability distributions.

Image segmentation using active contour model driven by RSF and difference of Gaussian energy

As we all know, the region scalable fitting method is sensitive to initialisations and suffers from bad results in images with complex scene. In our article, we put forward a new framework by integrating region scalable fitting (RSF) term and difference of Gaussian (DOG) term for segmenting images. We first propose a DOG function which can enhance the contrast at the edges of objects. Then, the RSF energy term is introduced to drive the curve closer to the edge. In the next step, the regularisation term is established which can avoid of the process of reinitialisation. Compared with traditional classical methods, the proposed technique is more flexibility with initialisation and has better segmentation performance.

Prognostics of State-dependent Fractional Degradation Processes with Stochastic Disturbance

Predicting the remaining useful life (RUL) of in-service commercial plants is closely linked with safety production and maintenance cost optimization. It is noteworthy that real-life degradation processes could be affected by their past states coupled with unknown disturbances. To make a comprehensive consideration of multisource dependencies and uncertainties, we develop a generalized non-stationary, nonlinear, and non-Markovian degradation model consisting of a Gaussian disturbed drift term and a sub-fractional Brownian motion (sub-FBM) based diffusion term. Both parts are state-dependent, and reflect two different types of memory effects. The main parameters and the approximate probability density function (PDF) of RUL can be solved on foundation of the Markovian transformation theories. A case study finally illustrates the effectiveness of the proposed scheme.

Free Space Realization of the Symmetrical Tunable Auto‐Focusing Lommel Gaussian Vortex Beam

AbstractAn auto‐focusing beam with adjustable symmetry is proposed numerically and experimentally. The initial field of the auto‐focusing Lommel Gaussian vortex beam (ALoGVB) consists of the Lommel function and the Gaussian term. When the asymmetry factor ε of the beam approaches zero, the ALoGVB will degenerate into a circular Bessel Gaussian vortex beam (CBGVB). Compared with the CBGVB, the ALoGVB is more diverse. The shape, symmetry, normalized focusing intensity, and focus distance of the ALoGVB can be controlled by flexibly adjusting the asymmetry factor ε and the topological charge n. In addition, the effect of off‐axis vortices on the propagating characteristics of the beam is also discussed. The experiment is in good agreement with the numerical simulation. The results could be beneficial for potential applications in optical trapping or micromachining.

Exploring a Simple Stochastic Mathematical Model Including Fear with a Linear Functional Response

This paper concentrates on a simple population model incorporating fear. Firstly, positivity and steady state analysis are performed, where the theoretical investigations show that change in the level of fear in prey population does not effect the local stability of the system around each equilibria (either stable or unstable). For the deterministic model, the numerical simulations are plotted for the density of prey species as a function of various system parameters. The stability analysis of the coexisting state shows that only transcritical bifurcation, where the steady states intersect, is observed. Secondly, the model is analysed with Gaussian noise term incorporated in the prey’s death rate. The model comprising noise term turns the system into stochastic differential equations and irregular noise related oscillations are observed in the densities of both species.

Diffusion approximation for multi-scale stochastic reaction-diffusion equations

In this paper, we study the diffusion approximation for singularly perturbed stochastic reaction-diffusion equation with a fast oscillating term. The asymptotic limit for the original system is obtained, where an extra Gaussian term appears. Such a term is explicitly given in terms of the solution of Poisson equation in Hilbert space. Moreover, we also obtain the rate of convergence, and the convergence rate is shown not to depend on the regularity of the coefficients of the original system with respect to the fast variable, which coincides with the intuition since the fast component has been totally homogenized out in the limit equation.