Gaussian Part Research Articles

Simplified calculation method of non-Gaussian wind pressure peak factor for large-span spatial roof structures

Based on the wind tunnel test data of a long-span spatial roof structure, the correlation between higher-order statistics and peak factors was obtained according to the time and frequency domain characteristics of wind pressure-time history, as well as the coincidence condition between wind pressure power spectra of measurement points and wind speed power spectra. A simplified peak factor calculation formula was proposed based on the moment-based Hermite polynomial model. The Gaussian part of the peak factor was calculated through the wind speed spectrum based on the zero-crossing theory, and the parameters related to higher-order statistics were considered as the non-Gaussian part of the peak factor. The simplified method was verified through the instantaneous extreme value method based on a coal shed structure with available long-term wind pressure-time history. It revealed that the results of the proposed simplified method correlate well with that of the instantaneous extreme value method, indicating its accuracy and potential value for engineering practice.

Construction of orthogonal-MaxPro Latin hypercube designs

Orthogonal Latin hypercube designs (LHDs) and maximum projection (MaxPro) LHDs are widely used in computer experiments. They are efficient for estimating the trend part and the Gaussian process part of the universal Kriging (i.e., the Gaussian process) model, respectively, especially when only some of the factors are active. Yet, the orthogonality and the MaxPro criteria often do not agree with each other. In this work, we propose a new class of optimal designs, called orthogonal-MaxPro LHDs, optimizing a well-defined multi-objective criterion combining the correlation and the MaxPro metrics. An efficient parallel algorithm via level permutations and expansions is developed, whose efficiency is guaranteed by theories. Numerical results are presented to show that the construction is fast and the obtained designs are attractive, especially for large computer experiments.

Computational extreme value statistics of strength for WC-Co composites at small specimens

The mechanical behavior of ceramic–metal (WC-Co) composites at small specimen level is subject to significant uncertainty due to the random assembly of (1) its constituent phases, (2) the crystal orientation of the WC grains, (3) the intrinsic residual stresses, and (4) the variable level of confinement of the metallic binder, within the contiguous network of much stiffer WC grains. In the existing literature, a Weibull distribution of strength is typically assumed at large specimen level. However, it is unknown whether this assumption can be carried over to small samples. Thus, the aim of this study is to shed light into the randomness in the mechanical strength of WC-Co composites, by using Monte Carlo simulations employing Latin Hypercube Sampling. The marginal distribution of the model parameters is assumed to be a grafted zero-threshold Weibull-Gaussian probability distribution, applied individually to the ceramic particles and the metallic binder using autocorrelation fields. The deterministic models are the recently calibrated microplane models M7WC and MPJ2 for the ceramic phase (WC) and the metallic matrix (Co) respectively. Test data in the form of strength histograms, gathered from literature on tensile tests conducted on nanowires of WC-Co composites, are used to calibrate the Weibull and Gaussian parameters, the grafting point locations and the autocorrelation lengths. The deterministic models M7WC and MPJ2 are assumed to be predictive in this study, as they were already calibrated in a previously published work by the authors. The finite element meshes used in the Monte Carlo simulations are generated using real tomography images, which easily captures the Gaussian part of the probability density function (pdf) without any random variables. This approach is shown to account successfully for the randomness observed in the strength of these ceramic–metal composites. Predictions are made for the strength distribution of differently sized nanowire specimens in tension.

Nonlocal Fractional Quantum Field Theory and Converging Perturbation Series

The main purpose of this paper is to derive a new perturbation theory (PT) that has converging series. Such series arise in the nonlocal scalar quantum field theory (QFT) with fractional power potential. We construct a PT for the generating functional (GF) of complete Green functions (including disconnected parts of functions) Zj as well as for the GF of connected Green functions Gj=lnZj in powers of coupling constant g. It has infrared (IR)-finite terms. We prove that the obtained series, which has the form of a grand canonical partition function (GCPF), is dominated by a convergent series—in other words, has majorant, which allows for expansion beyond the weak coupling g limit. Vacuum energy density in second order in g is calculated and researched for different types of Gaussian part S0[ϕ] of the action S[ϕ]. Further in the paper, using the polynomial expansion, the general calculable series for Gj is derived. We provide, compare and research simplifications in cases of second-degree polynomial and hard-sphere gas (HSG) approximations. The developed formalism allows us to research the physical properties of the considered system across the entire range of coupling constant g, in particular, the vacuum energy density.

Optimality of Threshold Strategies for Spectrally Negative Lévy Processes and a Positive Terminal Value at Creeping Ruin

This article investigates a dividend optimization problem with a positive creeping-associated terminal value at ruin for spectrally negative Lévy processes. We consider an insurance company whose surplus process evolves according to a spectrally negative Lévy process with a Gaussian part and a finite Lévy measure. Its objective function relates to dividend payments until ruin and a creeping-associated terminal value at ruin. The positive creeping-associated terminal value represents the salvage value or the creeping reward when creeping happens. Owing to formulas from fluctuation theory, the objective considered is represented explicitly. Under certain restrictions on the terminal value and the surplus process, we show that the threshold strategy should be the optimal one over an admissible class with bounded dividend rates.

Second spectrum of charge carrier density fluctuations in graphene due to trapping/detrapping processes

We investigate the second spectrum of charge carrier density fluctuations in graphene within the McWorther model, where noise is induced by electron traps in the substrate. Within this simple picture, we obtain a closed-form expression including both Gaussian and non-Gaussian fluctuations. We show that a very extended distribution of switching rates of the electron traps in the substrate leads to a carrier density power spectrum with a non-trivial structure on the scale of the measurement bandwidth. This explains the appearance of a 1/f component in the Gaussian part of the second spectrum, which adds up to the expected frequency-independent term. Finally, we find that the non-Gaussian part of the second spectrum can become quantitatively relevant by approaching extremely low temperatures.

On the Convergence from Boltzmann to Navier–Stokes–Fourier for General Initial Data

In this work, we prove the convergence of strong solutions of the Boltzman equation, for initial data having polynomial decay in the velocity variable, towards those of the incompressible Navier–Stokes–Fourier system. We show in particular that the solutions of the rescaled Boltzmann equation do not blow up before their hydrodynamic limit does. This is made possible by adapting the strategy from [M. Briant, S. Merino-Aceituno, and C. Mahout, Anal. Appl. (Singap.), 17 (2019), pp. 85–116] of writing the solution to the Boltzmann equation as the sum of a part with polynomial decay and a second one with Gaussian decay. The Gaussian part is treated with an approach reminiscent of the one from [I. Gallagher and I. Tristani, Ann. Henri Lebesgue, 3 (2020), pp. 561–614].

Improved Spin-Wave Estimate for Wilson Loops in U(1) Lattice Gauge Theory

Abstract In this paper, we obtain bounds on the Wilson loop expectations in 4D $U(1)$ lattice gauge theory, which quantify the effect of topological defects. In the case of a Villain interaction, by extending the non-perturbative technique introduced in [24], we obtain the following estimate for a large loop $\gamma $ at low temperatures: $ |\langle W_\gamma \rangle _{\beta }|\leq \exp \Big (-\frac {C_{GFF}} {2\beta }(1+C \beta e^{- 2\pi ^2 \beta } )(|\gamma |+o(|\gamma |)) \Big )\,.$ Our result is in line with recent works [4, 9, 13, 15] which analyze the case where the gauge group is discrete. In the present case where the gauge group is continuous and Abelian, the fluctuations of the gauge field decouple into a Gaussian part, related to the so-called free electromagnetic wave [11, 23], and a gas of topological defects. As such, our work gives new quantitative bounds on the fluctuations of the latter which complement the works by Guth and Frölich-Spencer [17, 27]. Finally, we improve, also in a non-perturbative way, the correction term from $e^{-2\pi ^2\beta }$ to $e^{-\pi ^2\beta }$ in the case of the free-energy of the system. This provides a matching lower-bound with the prediction of Guth [27] based on renormalization group techniques.

Where is the distribution tail threshold? A tale on tail and copulas in financial risk measurement

Estimating the market risk is conditioned by the fat tail of the distribution of returns. But the tail index depends on the threshold of this distribution fat tail. We propose a methodology based on the decomposition of the series into positive outliers, Gaussian central part and negative outliers and uses the latter to estimate this cutoff point. Additionally, from this decomposition, we estimate extreme dependence correlation matrix which is used in the measurement of portfolio risk. For a sample consisting of six assets (Bitcoin, Gold, Brent, Standard&Poor-500, Nasdaq and Real Estate index), we find that our methodology presents better results, in terms of normality and volatility of the tail index, than the Kolmogorov–Smirnov distance, and its unnecessary capital consumption is lower. Also, in the measurement of the risk of a portfolio, the results of our proposal improve those of a t-Student copula and allow us to estimate the extreme dependence and the corresponding indexes avoiding the implicit restrictions of the elliptic and Archimedean copulas.

The renormalization group transformation in the generalized fermionic hierarchical model

We consider a two-dimensional hierarchical lattice in which the vertices of a square represent an elementary cell. In the generalized hierarchical model, the distance between opposite vertices of a square differs from that between adjacent vertices and is a parameter of the new model. The Gaussian part of the Hamiltonian of the 4-component generalized fermionic hierarchical model is invariant under the block-spin renormalization group transformation. The transformation of the renormalization group in the space of coefficients, which specify the Grassmann-valued density of the free measure, is explicitly calculated as a homogeneous mapping of degree four in the two-dimensional projective space.

Connectedness and Gaussian parts for compact quantum groups

We introduce the Gaussian part of a compact quantum group G, namely the largest quantum subgroup of G supporting all the Gaussian functionals of G. We prove that the Gaussian part is always contained in the Kac part, and characterise Gaussian parts of classical compact groups, duals of classical discrete groups and q-deformations of compact Lie groups. The notion turns out to be related to a new concept of “strong connectedness” and we exhibit several examples of both strongly connected and totally strongly disconnected compact quantum groups.

Fluctuating kinetic theory and fluctuating hydrodynamics of aligning active particles: the dilute limit

Kinetic and hydrodynamic theories are widely employed for describing the collective behavior of active matter systems. At the fluctuating level, these have been obtained from explicit coarse-graining procedures in the limit where each particle interacts weakly with many others, so that the total forces and torques exerted on each of them is of order unity at all times. Such limit is however not relevant for dilute systems that mostly interact via alignment; there, collisions are rare and make the self-propulsion direction to change abruptly. We derive a fluctuating kinetic theory, and the corresponding fluctuating hydrodynamics, for aligning self-propelled particles in the limit of dilute systems. We discover that fluctuations at kinetic level are not Gaussian and depend on the interactions among particles, but that only their Gaussian part survives in the hydrodynamic limit. At variance with fluctuating hydrodynamics for weakly interacting particles, we find that the noise variance at hydrodynamic level depends on the interaction rules among particles and is proportional to the square of the density, reflecting the binary nature of the aligning process. The results of this paper, which are derived for polar self-propelled particles with polar alignment, could be straightforwardly extended to polar particles with nematic alignment or to fully nematic systems.

One-loop correction to the enhanced curvature perturbation with local-type non-Gaussianity for the formation of primordial black holes

As one of the promising candidates of cold dark matter (DM), primordial black holes (PBHs) were formed due to the collapse of over-densed regions generated by the enhanced curvature perturbations during the radiation-dominated era. The enhanced curvature perturbations are expected to be non-Gaussian in some relevant inflation models and hence the higher-order loop corrections to the curvature power spectrum might be non-negligible as well as altering the abundance of PBHs. In this paper, we calculate the one-loop correction to the curvature power spectrum with local-type non-Gaussianities characterizing by $F_{\mathrm{NL}}$ and $G_{\mathrm{NL}}$ standing for the quadratic and cubic non-Gaussian parameters, respectively. Requiring that the one-loop correction be subdominant, we find a perturbativity condition, namely $|2cAF_{\mathrm{NL}}^2+6AG_{\mathrm{NL}}|\ll 1$, where $c$ is a constant coefficient which can be explicitly calculated in the given model and $A$ denotes the variance of Gaussian part of enhanced curvature perturbation, and such a perturbativity condition can provide a stringent constraint on the relevant inflation models for the formation of PBHs.

Nonlinear investigation of Gol’denveizer’s problem of a circular and elliptic elastic torus

The Gol’denveizer problem of a torus can be simply described as follows: the inner and outer equator of the torus shell is loaded by the vertical opposite balanced forces. Gol’denveizer proposed that the membrane theory of shells does not yield a valid solution to this problem. Following this, Audoly and Pomeau used a membrane theory of a toroidal shell and nonlinear boundary-layer to study the linear Gol’denveizer problem and gave a simple solution relating resultant force F and vertical displacement. Sun used the bending theory of shells to solve the linear Gol’denveizer problem, and demonstrated that the solution by Audoly and Pomeau for the linear Gol’denveizer problem is extremely accurate. However, to the best of our knowledge, no one has examined the nonlinear Gol’denveizer problem. In this study, the finite element method is used to observe the nonlinear mechanical property and buckling of the Gol’denveizer problem of a circular and elliptic torus. Investigations reveal that the radius ratios a/R and a/b have great influences on the deformation and force of a circular torus and elliptical torus, respectively, and that the negative Gaussian part of a torus has a stronger bearing than the positive Gaussian part. Additionally, when the vertical force is large enough, shear buckling may occur in the circular and elliptic tori. We propose the buckling failure modes of circular and elliptical torus. As a/b increases, the collapse load of an elliptic torus of the Gol’denveizer problem is enhanced gradually.

Gaussian continuous-variable isotropic state

Inspired by the definition of the non-Gaussian two-parametric continuous variable analogue of an isotropic state introduced by Mi\v{s}ta et al. [Phys. Rev. A, 65, 062315 (2002); arXiv:quant-ph/0112062], we propose to take the Gaussian part of this state as an independent state by itself, which yields a simple, but with respect to the correlation structure interesting example of a two-mode Gaussian analogue of an isotropic state. Unlike conventional isotropic states which are defined as a convex combination of a thermal and an entangled density operator, the Gaussian version studied here is defined by a convex combination of the corresponding covariance matrices and can be understood as entangled pure state with additional Gaussian noise controlled by a mixing probability. Using various entanglement criteria and measures, we study the non-classical correlations contained in this state. Unlike the previously studied non-Gaussian two-parametric isotropic state, the Gaussian state considered here features a finite threshold in the parameter space where entanglement sets in. In particular, it turns out that it exhibits an analogous phenomenology as the finite-dimensional two-qubit isotropic state.

Low-rank Gaussian mixture modeling of space-snapshot representation of microphone array measurements for acoustic imaging in a complex noisy environment

The microphone array can simultaneously obtain the multi-dimensional information (space-time-frequency) of sound sources, which is recognized as a fundamental and powerful tool in acoustic imaging. Acoustic Beamforming is one of the widely used methods in acoustic imaging. However, most of the applications of beamforming are based on the Gaussian noise assumption, which is not always accurate in on-site measurements. For example, shock noise with a skewed probability density function (PDF) may appear on the signal record when the turbulent eddies are not controlled. Thus, in this paper, the conventional Gaussian noise model is extended to a Gaussian mixture noise model, which can approximate any probability distribution of the noise in theory. The space-snapshot representation of microphone array measurements is further modeled as a combination of the low-rank matrix part (measurements from the sound sources) and a Gaussian mixture matrix part (measurement noise). The signal from the sources of interest is finally recovered by the Expectation–maximization algorithm, which iterates between the low-rank approximation of the sound sources and the estimation of the parameter of the Gaussian mixture model. The proposed method is further investigated with simulations and compared with robust principal component analysis (RPCA) and Gaussian-based probabilistic factor analysis (PFA). It is concluded that the proposed method outperforms the state-of-the-art denoising methods.

Doi-Peliti field theory of free active Ornstein-Uhlenbeck particles.

We derive a Doi-Peliti field theory for free active Ornstein-Uhlenbeck particles, or, equivalently, free inertial Brownian particles, and present a way to diagonalize the quadratic part of the action and calculate the propagator. Unlike previous coarse-grained approaches this formulation correctly tracks particle identity and yet can easily be expanded to include potentials and arbitrary reactions.

Sizes and Shapes of Sea Ice Floes Broken by Waves–A Case Study From the East Antarctic Coast

The floe size distribution (FSD) is an important characteristics of sea ice, influencing several physical processes that take place in the oceanic and atmospheric boundary layers under/over sea ice, as well as within sea ice itself. Through complex feedback loops involving those processes, FSD might modify the short-term and seasonal evolution of the sea ice cover, and therefore significant effort is undertaken by the scientific community to better understand FSD-related effects and to include them in sea ice models. An important part of that effort is analyzing the FSD properties and variability in different ice and forcing conditions, based on airborne and satellite imagery. In this work we analyze a very high resolution (pixel size: 0.3 m) satellite image of sea ice from a location off the East Antarctic coast (65.6°S, 101.9°E), acquired on February 16, 2019. Contrary to most previous studies, the ice floes in the image have angular, polygonal shapes and a narrow size distribution. We show that the observed FSD can be represented as a weighted sum of two probability distributions, a Gaussian and a tapered power law, with the Gaussian part clearly dominating in the size range of floes that contribute over 90% to the total sea ice surface area. Based on an analysis of the weather, wave and ice conditions in the period preceding the day in question, we discuss the most probable scenarios that led to the breakup of landfast ice into floes visible in the image. Finally, theoretical arguments backed up by a series of numerical simulations of wave propagation in sea ice performed with a scattering model based on the Matched Eigenfunction Expansion Method are used to show that the observed dominating floe size in the three different regions of the image (18, 13 and 51 m, respectively) agree with those expected as a result of wave-induced breaking of landfast ice.

Entanglement entropy in scalar field theory and ZM gauge theory on Feynman diagrams

Entanglement entropy (EE) in interacting field theories has two important issues: renormalization of UV divergences and non-Gaussianity of the vacuum. In this letter, we investigate them in the framework of the two-particle irreducible formalism. In particular, we consider EE of a half space in an interacting scalar field theory. It is formulated as $\mathbb{Z}_M$ gauge theory on Feynman diagrams: $\mathbb{Z}_M$ fluxes are assigned on plaquettes and summed to obtain EE. Some configurations of fluxes are interpreted as twists of propagators and vertices. The former gives a Gaussian part of EE written in terms of a renormalized 2-point function while the latter reflects non-Gaussianity of the vacuum.

Modeling of the quasi-periodic galloping response type under combined harmonic and random excitation

The paper presents a mathematical Single-Degree-of-Freedom model of a cable with ice accretion vibrating in an air flow. It follows an experimental investigation of several aeroelastic models in a wind tunnel. The analysis of the experimental data is very complicated because a number of structural and stream characteristics mutually interact. Distinction between deterministic and random response components is only possible using a theoretical solution. The vibrational movement of the models used can be expressed by van der Pol-Duffing type equations. The aeroelastic excitation that is due to the air flow is modeled as an additive process consisting of a deterministic periodic part and random components. The paper investigates several particular configuration parameter settings for a non-white Gaussian random part of an excitation process and characterizes corresponding response properties. The Fokker-Planck equation is constructed for the random part of the response, and its semi-analytical solution in exponential form is expressed using a probability potential. Partial amplitudes of a harmonic approximation of the response are determined using the stochastic averaging strategy. The existence of a stationary probability distribution of the response is investigated, and its stochastic stability is assessed. Open problems and further research steps are outlined.