Gaussian Functional Approximation Research Articles

An analytical model to calculate the primary and secondary acoustic forces acting on microbubbles due to a short pulsed laser excitation

This work presents a comprehensive analytical approach to describe photoacoustic waves resulting from a pulsed laser excitation. The spatial part of the laser is modeled by a zeroth-order Bessel function of the first kind, which has a wide range of applications in optics and medical physics, such as optical trapping and nonlinear effects resulting from the interaction of a pulsed laser with tissue in photoacoustic imaging. The temporal part of the laser is described by a Gaussian function, which is a pretty realistic approximation since the interaction of the laser with the medium is instantaneous. The photoacoustic wave equation is solved analytically using the Fourier transform and the Greens’ function methods. The solution of the photoacoustic wave equation depends explicitly on position, time, pulse duration, and beam-width of the pulsed laser. The effects of these dependencies on the photoacoustic wave are investigated. Later, the primary and secondary radiation forces acting on microbubbles Albunex and Quantison are calculated using the magnitude of the photoacoustic pressure wave. The primary and secondary radiation forces decrease considerably with the distance from the photoacoustic absorber. These forces increase as the beam-width increases while they decrease as the pulse duration gets longer. The primary radiation forces on the microbubbles are on the order of nanonewtons. The force at this scale can be used to manipulate microbubbles. The secondary radiation force between identical microbubbles is in the range of piconewtons. Hence, this force can be used to determine the viscoelastic properties of microbubbles even though it is very small compared to the primary radiation force. The radiation forces determined by this work are also compared with those calculated by another study describing the laser’s spatial profile with a Gaussian function. The forces obtained by this work are larger than the forces determined by the Gaussian function approximation at the positions near the source. The forces obtained by the two approaches show similar behaviors, and they decrease remarkably with the distance from the source. Thus, the model presented in this work can be used to study the nonlinear mechanism in photoacoustics, such as enhancing image contrast and determining the tissue temperature. It can also be helpful for the applications of microbubbles in medical imaging and drug delivery as carriers.

Particle filter algorithm based on geometric center and likelihood estimation

The improved particle filter (PF) based on the geometric center and likelihood estimation is proposed in order to solve the problem of particle dilution and degradation. In the resampling stage, the geometric center is used to resample the particles. The particles are filtered according to the distance between the particles and the geometric center, and then the particles are resampled. The resampled particles are composed of newborn particles and non-resampled particles. The former can help alleviate the degradation problem, while the latter can keep the diversity of the particle set. In order to ensure effectiveness of the PF, the positioning error threshold of the particle filter is introduced. In the phase of particle weighting calculation, in view of the problem of low accuracy and divergence of PF state estimation caused by non-stationary and non-Gaussian noise, it adopts non-Gaussian noise parameter estimation based on likelihood to approximately estimate the measurement noise instead of the Gaussian density function. The proposed model is applied to particle weight calculation to avoid particle degradation caused by Gaussian density function approximation. The simulation results show that, after the improved algorithm, the root-mean-square error is reduced to 0.085, the variance is reduced to 0.014, and the running time is shortened by 14.8% compared with the polynomial resampling algorithm, which can effectively alleviate particle degradation and dilution in the traditional PF algorithm, and the positioning accuracy is also improved.

A Novel Method for Solving Ordinary Differential Equations with Artificial Neural Networks

This research work investigates the use of Artificial Neural Network (ANN) based on models for solving first and second order linear constant coefficient ordinary differential equations with initial conditions. In particular, we employ a feed-forward Multilayer Perceptron Neural Network (MLPNN), but bypass the standard back-propagation algorithm for updating the intrinsic weights. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the initial or boundary conditions and contains no adjustable parameters. The second part involves a feed-forward neural network to be trained to satisfy the differential equation. Numerous works have appeared in recent times regarding the solution of differential equations using ANN, however majority of these employed a single hidden layer perceptron model, incorporating a back-propagation algorithm for weight updation. For the homogeneous case, we assume a solution in exponential form and compute a polynomial approximation using statistical regression. From here we pick the unknown coefficients as the weights from input layer to hidden layer of the associated neural network trial solution. To get the weights from hidden layer to the output layer, we form algebraic equations incorporating the default sign of the differential equations. We then apply the Gaussian Radial Basis function (GRBF) approximation model to achieve our objective. The weights obtained in this manner need not be adjusted. We proceed to develop a Neural Network algorithm using MathCAD software, which enables us to slightly adjust the intrinsic biases. We compare the convergence and the accuracy of our results with analytic solutions, as well as well-known numerical methods and obtain satisfactory results for our example ODE problems.

Landau damping and frequency-shift of (0, 0, 2) scissors mode in a disc-shaped Bose-Einstein condensate

应用哈特里-福克-博戈留波夫平均场理论近似和基于托马斯-费米近似的解析方法, 研究碟形玻色-爱因斯坦凝聚体中(0, 0, 2)剪刀模的朗道阻尼和频移, 计算阻尼系数和频移大小以及它们的温度依赖. 计算中, 在集体激发本征频移微扰关系中考虑元激发弛豫及其弛豫之间的正交关系以获得阻尼和频移的计算公式, 把凝聚体基态波函数取为高斯分布函数的一级近似以消除托马斯-费米近似中三模耦合矩阵元的发散. 采用与相关实验研究相同的粒子数、囚禁频率和各向异性参量, 理论计算结果与相关实验测量结果相符合. 由于理论的复杂性和计算的困难性, 在大多数基于平均场理论的单分量和两分量玻色-爱因斯坦凝聚集体激发阻尼和频移的研究中采用半经典近似, 把准粒子激发能谱看成是连续的来积分计算各个准粒子跃迁对阻尼和频移的贡献, 而本文和本文前期工作按分立的准粒子激发频谱计算阻尼或频移, 并在研究过程中提出了考虑元激发弛豫及弛豫之间正交关系的改进方法, 希望这种方法对今后的工作有一定参考价值.

Data-driven risk assessment and multicriteria optimization of UAV operations

This paper introduces a novel data-driven risk metric and assessment method for UAVs operating in environments typically encountered in civilian applications. A truly “data-driven risk measure” is derived through a probabilistic formulation that not only accounts for the intrinsically stochastic nature of the considered environmental factors (such as weather and signal strength), but also incorporates extrinsic prediction uncertainties originating from the geographical sparsity of data collection sources. We present a data-driven modeling of the stochastic environmental factors using Gaussian process-based function approximations. Notably, the proposed mathematical definition of the risk metric is based on the probabilistic predictions of such a Gaussian process model and introduced through a path-integral formulation. The problem of minimizing operational risk for multiple UAVs in partially unknown environments is then defined in a multicriteria optimization framework to address the trade-off between the path-integral risk measure and classical path-efficiency (distance). We show that such approach can be embedded into current standard risk assessment methods which could be easily integrated into UAVs traffic management initiatives. We analyze the results through a number of simulations, including realistic scenarios.

Landau damping and frequency-shift of a quadrupole mode in a disc-shaped rubidium Bose—Einstein condensate

The damping and frequency-shift in Landau mechanism of a quadrupole mode in a disc-shaped rubidium Bose—Einstein condensate are investigated by using the Hartree—Fock—Bogoliubov approximation. The practical relaxations of the elementary excitations and the orthometric relation among them are taken into account to obtain advisable calculation formula for damping as well as frequency-shift. The first approximation of Gaussian distribution function is employed for the ground-state wavefunction to suitably eliminate the divergence of the analytic three-mode coupling matrix elements. According to these methods, both Landau damping rate and frequency-shift of the quadrupole mode are analytically calculated. In addition, all the theoretical results agree with the experimental ones.

Gaussian functional approximation to ’t Hooft’s extension of the linearΣmodel

We apply a self-consistent relativistic mean-field variational ``Gaussian functional'' (or optimized one-loop perturbation theory, or $\mathrm{Hartree}+\mathrm{RPA}$) approximation to the extended ${N}_{f}=2$ linear $\ensuremath{\sigma}$ model with spontaneously and explicitly broken chiral $\mathrm{S}{\mathrm{U}}_{\mathrm{R}}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{S}{\mathrm{U}}_{\mathrm{L}}(2)\ifmmode\times\else\texttimes\fi{}{\mathrm{U}}_{\mathrm{A}}(1)\ensuremath{\equiv}\mathrm{O}(4)\ifmmode\times\else\texttimes\fi{}\mathrm{O}(2)$ symmetry. We set up the self-consistency, or gap equations that dress up the bare fields with ``cactus tree'' loop diagrams, and the Bethe-Salpeter equations that provide further dressing with one-loop irreducible diagrams. In a previous publication [V. Dmitra\ifmmode \check{s}\else \v{s}\fi{}inovi\ifmmode \acute{c}\else \'{c}\fi{} and I. Nakamura, J. Math. Phys. (N.Y.) 44, 2839 (2003).] we have already shown the ability of this approximation to create composite (i.e., bound and/or resonance) states. With explicit $\mathrm{S}{\mathrm{U}}_{\mathrm{R}}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{S}{\mathrm{U}}_{\mathrm{L}}(2)\ifmmode\times\else\texttimes\fi{}{\mathrm{U}}_{\mathrm{A}}(1)$ chiral symmetry breaking first we consider how the ${\mathrm{U}}_{\mathrm{A}}(1)$ symmetry induced scalar-pseudoscalar meson mass relation that is known to hold in fermionic chiral models is modified by the bosonic gap equations. Then we solve the gap and Bethe-Salpeter equations numerically and discuss the solutions' properties and the particle content of the theory. We show that in the strong-coupling regime two, sometimes even three solutions to the $\ensuremath{\eta}$ meson channel Bethe-Salpeter equation may coexist.

THE EFFECTIVE MASS OF STRONG-COUPLING MAGNETOPOLARONS IN A PARABOLIC QUANTUM DOT

Within the framework of the Landau–Pekar variational method we have investigated the effective mass of strong-coupling magnetopolarons in a parabolic quantum dot. The effective mass as functions of the magnetic field strength and the confinement length of the quantum dot are obtained in the Gaussian function approximation. It is shown that the effective mass increases with the increasing magnetic field strength and increases with the decrease in the size of the quantum dot (QD).

Performance analysis of a fiber Bragg grating filter-based strain/temperature sensing system based on a modified Gaussian function approximation method

This paper analyzes the performance of a fiber Bragg grating (FBG) filter-based strain and/or temperature sensing system based on a modified Gaussian function (MGF) approximation method. Instead of using a conventional Gaussian function, we propose the MGF, which can capture the characteristics of the sidelobes of the reflected spectrum, to model the FBG sensor and filter. We experimentally demonstrate that, by considering the contributions of the sidelobes with the MGF approximation method, behaviors of the FBG filter-based FBG displacement and/or temperature sensing system can be predicted more accurately. The predicted behaviors include the saturation, the sensitivity, the sensing range, and the optimal initial Bragg wavelengths of the FBG sensing system.

Probability of Power Depletion in SRS Cross-Talk and Optimum Detection Threshold for Minimum BER in a WDM Receiver

In this paper, the probability of power depletion due to stimulated Raman scattering (SRS) is derived for an arbitrary number of interferers to estimate bit error rate and power penalty of a wavelength division multiplexed (WDM) receiver. Gaussian function approximation can be used only when the numbers of interferers is very large. The effects of number of interfering channels, channel separation and receiver noise on the bit error rate (BER) are studied. Minimum BER varies non-monotonically with channel separation. Optimum detection thresholds for minimum BER in the WDM receiver in presence of SRS crosstalk are investigated and summarized in tabular form.

DEVELOPMENT OF A LORENTZIAN-FUNCTION APPROXIMATION UTILIZING IN THE CHARGED- PARTICLE-INDUCED NONRESONANT REACTION RATE

A development has been made for the charged-particle-induced nonresonant reaction-rate equations. The forms of reaction-rate equations for nonresonant and resonant reactions have been united in a frame of Lorentzian-Function Approximation (LFA) mathematically. In the frame of LFA, the nonresonant reaction taken place within the Gamow window can be considered, in form, as a "resonance" reaction with a full width at half maximum (FWHM, Γ nr ) equal to the 1/e width (Δ) in a well-known Gaussian-Function Approximation (GFA).

On Development of Fuzzy Controller: The Case of Gaussian and Triangular Membership Functions

In recent years, the use of Fuzzy set theory has been popularised for handling overlap domains in control engineering but this has mostly been within the context of triangular membership functions. In actual practice however, such domains are hardly triangular and in fact for most engineering applications the membership functions are usually Gaussian and sometimes cosine. In an earlier paper, we derived explicit Fourier series expressions for systematic and dynamic computation of grade of membership in the overlap and non-overlap regions of triangular Fuzzy sets. In another paper, we extended the methodology to cover cases of cosine, exponential and Gaussian Fuzzy sets by presenting explicit Fourier series representation for encoding fuzziness in the overlap and non-overlap domains of Fuzzy sets. This current paper presents the development of a “Fuzzy Controller” device, which incorporates the formal mathematical representation for computing grade of membership of Gaussian and triangular Fuzzy sets. It is shown that triangular approximation of Gaussian membership function in Fuzzy control can lead to wrong linguistic classification which may have adverse effects on operational and control decisions. The development of the Fuzzy controller demonstrates that the proposed technique can indeed be incorporated in engineering systems for dynamic and systematic computation of grade of membership in the overlap and non-overlap regions of Fuzzy sets; and thus provides a basis for the design of embedded Fuzzy controller for mission critical applications.

Pseudoscalar mesons in theSU(3)linear sigma model with Gaussian functional approximation

We study the $SU(3)$ linear sigma model for the pseudoscalar mesons in the Gaussian functional approximation (GFA). We use the $SU(3)$ linear sigma model Lagrangian with nonet scalar and pseudoscalar mesons, including symmetry breaking terms. In the GFA, we take the Gaussian Ansatz for the ground state wave function and apply the variational method to minimize the ground state energy. We derive the gap equations for the dressed meson masses, which are actually just variational parameters in the GFA method. We use the Bethe-Salpeter equation for meson-meson scattering, which provides the masses of the physical nonet mesons. We construct the projection operators for the flavor $SU(3)$ in order to work out the scattering $T$ matrix in an efficient way. In this paper, we discuss the properties of the Nambu-Goldstone bosons in various limits of the chiral ${U}_{L}(3)\ifmmode\times\else\texttimes\fi{}{U}_{R}(3)$ symmetry.

A Biologically Inspired Compound-Eye Detector Array—Part I: Modeling and Fundamental Limits

This is the first part of a two-part paper. In this paper, we propose a detector array for detecting and localizing sources that emit particles including photons, neutrons, or charged particles. The array consists of multiple ldquoeyelets.rdquo Each eyelet has a conical module with a lens on its top and an inner subarray containing multiple particle detectors. The array configuration is inspired by and generalizes the biological compound eye: it is spherically shaped and has a larger number of detectors in each individual eyelet. Potential applications of this biomimetic array include artificial vision in medicine (e.g., artificial eyes for the blind) or robotics (e.g., for industry or space missions), astronomy and astrophysics, security (e.g., for radioactive materials), and particle communications. In this part, we assume Poisson distribution for each detector's measurement within the observation time window. Then we construct a general parametric model for the detection rate of the Poisson-distributed measurements illustrated by a circular Gaussian lens-shaping function (LSF) approximation, which is commonly used in optical and biological disciplines. To illustrate how this ldquoprototyperdquo model fits practical cases, we apply it to an example of localizing a candle from 20 miles away and estimating the parameters under this circumstance. In addition, we also discuss the hardware setup and performance measure of the proposed array, as well as its fundamental constraints. Part I forms the theoretical basis for Part II, in which we analyze the performance of the array, both analytically and numerically.

A Lorentzian-Function Approximation developed in calculating the charged-particle-induced nonresonant reaction rate

A Lorentzian-Function Approximation (LFA) has been developed in calculating the nonresonant reaction rate of charged-particle-induced reactions. The nonresonant reaction rate and the effective S -factor have been represented in terms of LFA. In the frame of LFA, the nonresonant reaction taken place within the Gamow window can be considered as a “resonance reaction” with a width of Γ which is equal to that of 1/e width (Δ in a well-known Gaussian-Function Approximation (GFA).

Linear [formula omitted] model in the Gaussian wave functional approximation II: analyticity of the S-matrix and the effective potential/action

We show an explicit connection between the solution to the equations of motion in the Gaussian functional approximation [I. Nakamura, V. Dmitrašinović, Prog. Theor. Phys. 106 (2001) 1195] and the minimum of the (Gaussian) effective potential/action of the linear Σ model, as well as with the N/D method in dispersion theory. The resulting equations contain analytic functions with branch cuts in the complex mass squared plane. Therefore the minimum of the effective action may lie in the complex mass squared plane. Many solutions to these equations can be found on the second, third, etc. Riemann sheets of the equation, though their physical interpretation is not clear. Our results and the established properties of the S-matrix in general, and of the N/D solutions in particular, guide us to the correct choice of the Riemann sheet. We count the number of states and find only one in each spin-parity and isospin channel with quantum numbers corresponding to the fields in the Lagrangian, i.e., to Castillejo–Dalitz–Dyson (CDD) poles. We examine the numerical solutions in both the strong and weak coupling regimes and calculate the Källén–Lehmann spectral densities and then use them for physical interpretation.

Linear Model in the Gaussian Functional Approximation

We apply a self-consistent relativistic mean-field variational “Gaussian functional” (or Hartree) approximation to the linear σ model with spontaneously and explicitly broken chiral O(4) symmetry. We set up the self-consistency, or “gap” and the Bethe-Salpeter equations. We check and confirm the chiral Ward-Takahashi identities, among them the Nambu-Goldstone theorem and the (partial) axial current conservation [CAC], both in and away from the chiral limit. With explicit chiral symmetry breaking, we confirm the Dashen relation for the pion mass and partial CAC. We solve numerically the gap and Bethe-Salpeter equations, discuss the solutions’ properties and the particle content of the theory.

Valuing American Put Options Using Gaussian Quadrature

This article develops an efficient and accurate method for numerical evaluation of the integral equation which defines the American put option value function. Numerical integration using Gaussian quadrature and function approximation using Chebyshev polynomials are combined to evaluate recursive expectations and produce an approximation of the option value function in two dimensions, across stock prices and over time to maturity. A set of such solutions results in a multidimensional approximation that is extremely accurate and very quick to compute. The method is an effective alternative to finite difference methods, the binomial model, and various analytic approximations. Article published by Oxford University Press on behalf of the Society for Financial Studies in its journal, The Review of Financial Studies.

Particle content of the Gaussian functional approximation to the scalar φ4 theory

The particle content of the Gaussian functional approximation to the φ 4 theory with a spontaneously broken O(2) symmetry is investigated in view of apparent existence of two kinds of particles with quantum numbers of the Goldstone boson: a massive “elementary” and a massless Goldstone “composite” particle. We calculate the Källén-Lehmann spectral functions of the two-point Green functions in the Gaussian approximation. The massive particle is absent from the spectral function in the “pion” channel. Similar results hold in the “sigma” channel, our treatment of which is novel.

Analysis of texture and mechanical anisotropy of Al-Mg alloy sheets for autobody applications

The effect of texture on mechanical anisotropy was investigated in three commercial Al-Mg alloy sheets for autobody applications. The textures were identified using ODF (oriented distribution function) analyses of XRD pole figures. The volume fractions of existing texture components were calculated based on the Gaussian function approximation. The tensile tests were performed in different directions to the rolling direction of the sheet in order to analyze the mechanical anisotropy. The Cube and CubeND recrystallization textures were identified as dominant texture components of the Al-Mg alloy sheets. The experimental R-values were in good agreement with the prediction based on the CMTP (continuum mechanics of textured polycrystals). The mechanical anisotropy originated from CubeND texture component was weaker than that originated from Cube texture component. The anisotropy in R-value decreased with increasing the amount of random spread components and the AR value was measured as 0.01 when the random spread component was 70%. The biaxial stretchability obtained from Erichen test was enhanced with decreasing the mechanical anisotropy in Al-Mg alloy sheet.