Closed-form Function Research Articles

Performance Analysis of FSO Systems Over a Lognormal-Rician Turbulence Channel With Generalized Pointing Errors

In this paper, an approximate closed-form probability density function expression for the Lognormal-Rician atmospheric turbulence with generalized pointing errors model is developed by using a series representation. The Kolmogorov-Smirnov (KS) goodness-of-fit statistical tool is employed to verify the accuracy of the proposed approximation. We demonstrate that a more efficient approximation can be achieved for the smaller jitter variances and nonzero boresight displacements. To reveal the importance of the proposed approximation, the approximate analytic expressions for the ergodic capacity and outage performance of single-input single-output free-space optical (FSO) systems are derived. It is observed that the pointing errors significantly degrade the performance of FSO links. By using the obtained asymptotic expressions, the impact of pointing errors on the ergodic capacity is considerably mitigated by selecting the optimum beam width at the transmitter side. We find that the optimum beam width is independent of atmospheric turbulence and the Pearson correlation coefficient in high signal-to-noise ratio conditions. All of the derived analytical results are confirmed by Monte Carlo simulation results.

Value of Information in Feedback Control: Quantification

Although transmission of a data packet containing sensory information in a networked control system improves the quality of regulation, it has indeed a price from the communication perspective. It is, therefore, rational that such a data packet be transmitted only if it is valuable in the sense of a cost-benefit analysis. Yet, the fact is that little is known so far about this valuation of information and its connection with traditional event-triggered communication. In the present article, we study this intrinsic property of networked control systems by formulating a rate-regulation trade-off between the packet rate and the regulation cost with an event trigger and a controller as two distributed decision makers, and show that the valuation of information is conceivable and quantifiable grounded on this trade-off. In particular, we characterize an equilibrium in the rate-regulation trade-off, and quantify the value of information <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\rm{VoI}}_k$</tex-math></inline-formula> there as the variation in a so-called value function with respect to a piece of sensory information that can be communicated to the controller at each time <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> . We prove that, for a multi-dimensional Gauss--Markov process, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\rm{VoI}}_k$</tex-math></inline-formula> is a symmetric function of the discrepancy between the state estimates at the event trigger and the controller, and that a data packet containing sensory information at time <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> should be transmitted to the controller only if <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\rm{VoI}}_k$</tex-math></inline-formula> is nonnegative. Moreover, we discuss that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\rm{VoI}}_k$</tex-math></inline-formula> can be computed with arbitrary accuracy, and that it can be approximated by a closed-form quadratic function with a performance guarantee.

A Closed-Form Transfer Function for Sample and Hold Master Slave Sampling Filter

A switched capacitor circuit frequently appears in literature in form of sample and hold or master-slave sampling filter. We call this block Sample and Hold Master-Slave Sampling Filter (SH-MSSF). Considering the time varying nature of this circuit it is difficult to predict its behavior, though it looks like a very simple circuit. So there exists no formula describing its function in all of its applications. In this brief, we present a closed-form transfer function, explaining the operation of SH-MSSF, based on state-space analysis. Then by interpolating an infinite number of such transfer functions a continuous-time formula is derived that can exactly model the behavior of this (Linear Periodically Time-Varying) LPTV circuit. The advantage and applicability of such a formula are shown by applying it to three different well-known applications as case studies. The validity of that compact model has been confirmed by comparing it with the simulation results of Cadence Periodic AC (PAC) throughout this brief.

Biological Effectiveness of Ion Beam for In Vitro Cell Irradiations.

Despite numerous ion beam irradiation of cell experiments performed over the past five decades, the relationship between the biological effectiveness of ion beams and the physical characteristics of the ion beam remains unclear. Using 1,118 sets of in vitro cell survival experiments with ion beam irradiation, compiled by the Particle Irradiation Data Ensemble (PIDE) project, the relationship between cell survival and the fluence and linear energy transfer (LET) of the ion beam was established. Unlike previous studies, the closed-form analytical function is independent of photon irradiation and takes a universal form across all ion and cell species. A new understanding of the biological effectiveness of ion beams is crucial for predicting tumor response and toxicities in ion beam radiation therapy, along with radiation protection for high-LET ion beams with low fluence.

Approximate Bayesian inference for agent-based models in economics: a case study

Abstract Estimation of agent-based models in economics and finance confronts researchers with a number of challenges. Typically, the complex structures of such models do not allow to derive closed-form likelihood functions so that either numerical approximations to the likelihood or moment-based estimators have to be used for parameter inference. However, all these approaches suffer from extremely high computational demands as they typically work with simulations (of the agent-based model) embedded in (Monte Carlo) simulations conducted for the purpose of parameter identification. One approach that is very generally applicable and that has the potential of alleviating the computational burden is Approximate Bayesian Computation (ABC). While popular in other areas of agent-based modelling, it seems not to have been used so far in economics and finance. This paper provides an introduction to this methodology and demonstrates its potential with the example of a well-studied model of speculative dynamics. As it turns out, ABC appears to make more efficient use of moment-based information than frequentist SMM (Simulated Method of Moments), and it can be used for sample sizes of an order far beyond the reach of numerical likelihood methods.

Use of physical testing data for the accurate prediction of the ultimate compressive strength of steel stiffened panels

ABSTRACT Physical testing data can be used to predict the ultimate compressive strength of steel stiffened panels. Moreover, useful empirical formulae have been developed by fitting curves to data from relevant testing databases. A representative example is the Paik–Thayamballi formula, which is based on physical testing data available until 1997 and is a closed-form function of plate and column slenderness ratios. Since 1997, high-precision data-acquisition equipment and large-scale physical models have been used to generate databases contained advanced testing data of modern materials such as AH32 high-tensile steel made by the thermo-mechanical control process (TMCP) technology together with modern fabrication technologies, e.g., flux-cored arc welding technique, under a strict control of welding parameters, e.g., current, voltage, speed and heat input, to achieve a required weld leg length. It is therefore important to determine if these advanced testing data are compatible with the established empirical formulae. This paper describes benchmark studies which were conducted to determine such compatibility, with a focus on the Paik–Thayamballi formula, and summarises key findings and insights obtained from the present study.

Corrections to “Closed-Form Spectral-Domain Green's Functions for Infinitesimal Current Source in Multilayer Soil” [Dec 20 2814-2822

In [1], in Section V, equations (59) and (60) are incorrect. The correct equations are given as follows: \begin{align*} &\bar{\Theta }_i^{\varphi v} = \frac{{{{\bar{\alpha }}_i}}}{{{{\bar{\alpha }}_s}}} \cdot \frac{1}{{{{\bar{\alpha }}_s}}} \cdot \frac{{\partial \,\bar{\Theta }_i^{Avv}}}{{\partial d}} \tag{59}\\ &\bar{X}_i^{\varphi v} = - \frac{{{{\bar{\alpha }}_i}}}{{{{\bar{\alpha }}_s}}} \cdot \frac{1}{{{{\bar{\alpha }}_s}}} \cdot \frac{{\partial \bar{X}_i^{Avv}}}{{\partial d}}. \tag{60} \end{align*}

Internal impedance analysis of rectangular conductors at high frequencies

This paper presents the internal impedance analysis of rectangular conductors using analytical calculations. A novel 2D analytical formula is presented using a variational method. And its impedance calculation is compared with other analytical formulas with the format of sum of series for internal impedances calculations of rectangular conductors at high frequencies. It is shown that the novel analytical formula for internal impedance calculation has excellent accuracy despite its simple and closed form function. Dirichlet boundary condition is used by applying zero magnetic vector potential on the edges of rectangular conductor for the analytical calculations of internal impedance. The internal inductance and resistance are calculated at different frequencies between 1 MHz and 1000 MHz. The accuracy and simplicity of different analytical formulas for the internal impedance of rectangular conductors and equivalent circular conductors are compared.

Easy and computer-time-saving implementation of the van der Pauw method including anisotropy and probe positioning correction factors using approximate closed-form analytical functions.

Measurements in a van der Pauw configuration often require the use of complicated mathematical expressions or implicit relations, which may be computer-time-consuming or tedious to implement in a given software. Thus, a closed-form expression is often desirable. We propose to approximate these relations by closed-form analytical functions using only basic operators that can be easily implemented. We present explicitly the functions and the numerical values of their parameters for the cases of standard van der Pauw measurement, anisotropic resistivities, and the probe position correction factor. The discrepancy with the exact values is smaller than 10-5, which is sufficient for practical purposes.

Performance of Vertical Underwater Wireless Optical Communications With Cascaded Layered Modeling

This paper investigates the performance of vertical underwater wireless optical communication (UWOC) systems in the presence of air bubbles and temperature gradients. We consider a generalized UWOC channel model that contains <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> layers, each having the same distribution but with different parameters to consider the vertically inhomogeneous nature of the underwater environment. To capture the effects of air bubbles and temperature gradients on channel statistics, we model each layer with a mixture exponential-generalized Gamma distribution. A closed-form probability density function (PDF) is presented for the end-to-end signal-to-noise ratio. Based on the PDF, we derive the average bit-error-rate (BER) and the ergodic capacity. Moreover, we provide asymptotic BER and ergodic capacity results in simple forms. The performance and behavior of vertical UWOC systems are thoroughly analyzed, and all the derived expressions are verified via Monte Carlo simulations.

A Review of Flow Characterization of Metallic Materials in the Cold Forming Temperature Range and Its Major Issues.

We focus on the importance of accurately describing the flow behaviors of metallic materials to be cold formed; we refer to several valuable examples. We review the typical experimental methods by which flow curves are obtained, in addition to several combined experimental-numerical methods. The characteristics of four fundamental flow models including the Ludwik, Voce, Hollomon, and Swift models are explored in detail. We classify all flow models in the literature into three groups, including the Ludwik and Voce families, and blends thereof. We review the experimental and numerical methods used to optimize the flow curves. Representative flow models are compared via tensile testing, with a focus on the necking point and pre- or post-necking strain hardening. Several closed-form function models employed for the non-isothermal analyses of cold metal forming are also examined. The traditional bilinear C-m model and derivatives thereof are used to describe the complicated flow behaviors of metallic materials at cold forming temperatures, particularly in terms of their applications to metal forming simulations and process optimization.

The BerG generalized autoregressive moving average model for count time series

In this work, we present a new generalized autoregressive moving average model (GARMA), based on the Bernoulli-geometric (BerG) distribution, for modeling the conditional mean of count time series. The proposed model is able to deal with the equi, under or over-dispersed data. Our main contribution is to suggest a GARMA model with a response variable following a BerG distribution, which also accommodates zero inflated (or deflated) data. The proposed model combines the dispersion flexibility with the inclusion of covariates and lagged terms to model the conditional mean response, inducing an autocorrelation structure (usually relevant in time series). We exhibit the conditional maximum likelihood estimation, the hypothesis testing inference, the diagnostic analysis, and the out-of-sample forecasting procedure. Using the closed-form quantile function of the BerG distribution, the confidence intervals for out-of-sample forecasts are easily obtained. In particular, we provide the closed-form expressions for the conditional score vector and conditional Fisher information matrix. Moreover, we developed a computational study which confirmed that the maximum likelihood estimators are consistent for all dispersion scenarios. And finally, we illustrate the applicability of the postulated model by exploring two real data applications.

Some analytical results on bivariate stable distributions with an application in operational risk

The multivariate stable distributions are widely applicable as they can accommodate both skewness and heavy tails. Although one-dimensional stable distributions are well known, there are many open questions in the multivariate regime, since the tractability of the multivariate Gaussian universe, does not extend to non-Gaussian multivariate stable distributions. In this work, we provide the Laplace transform of bivariate stable distributions and its certain cut in the first quadrant. Given the lack of a closed-form likelihood function, we propose to estimate the parameters by means of Approximate Maximum Likelihood, a simulation-based method with desirable asymptotic properties. Simulation experiments and an application to truncated operational losses illustrate the applicability of the model.

An extended separation-of-variable method for free vibration of rectangular Reddy plates

The Reddy plate theory is widely used in the vibration analysis of thick plates, but there are few closed-form analytical solutions for free vibration of rectangular Reddy plates with arbitrary homogenous boundary conditions. This work is to develop the extended separation-of-variable (SOV) method for the free vibration analysis of rectangular Reddy plates. In the present method, the mode functions are in an explicit and closed form, and the natural frequencies corresponding to two-direction eigenfunctions are mathematically assumed to be independent of each other. According to two pairs of boundary conditions, the closed-form mode functions and frequency equations are achieved. The present method is numerically compared with other approximate and analytical methods, and some valuable observations are presented. The present solutions are exact for the plates with at least two opposite sides simply supported, and agree well with the results in literatures for other boundary conditions.

Anomalous transport of colloids in heterogeneous porous media: A multi-scale statistical theory

HypothesisTransport of suspended colloids in heterogeneous porous media is a multi-scale process that exhibits anomalous behavior and cannot be described by the Fickian dispersion theory. Although many studies have documented colloids’ transport at different length scales, a theoretical basis that links pore- to core-scale observations remains lacking. It is hypothesized that a recently proposed pore-scale statistical kinetic theory is able to capture the results observed experimentally. ExperimentsWe implement a multi-scale approach via conducting core-flooding experiments of colloidal particles in a sandstone sample, simulating particles flowing through a sub-volume of the rock’s digital twin, and developing a core-scale statistical theory for particles’ residence times via upscaling the pore-scale kinetic theory. Experimental and computational results for solute transport are used as benchmark. FindingsBased on good agreement across the scales achieved in our investigation, we show that the macroscopically observed anomalous transport is particle-type dependent and stems from particles’ microscopic dispersion and deposition in heterogeneous flow fields. In particular, we reveal that residence-time distributions (i.e., breakthrough curve) obey a closed-form function that encompasses particles’ microscopic dynamics, which allows investigations of a whole transition from pre-asymptotic to asymptotic behavior. The physical insights attained could be useful for interpreting experimental data and designing colloidal tracers.

Validation Solutions to the Full-sky Radio Interferometry Measurement Equation for Diffuse Emission

Abstract Low-frequency radio observatories are reaching unprecedented levels of sensitivity in an effort to detect the 21 cm signal from the Cosmic Dawn. High precision is needed because the expected signal is overwhelmed by foreground contamination, largely from so-called diffuse emission—a nonlocalized glow comprising Galactic synchrotron emission and radio galaxies. The impact of this diffuse emission on observations may be better understood through detailed simulations, which evaluate the Radio Interferometry Measurement Equation (RIME) for a given instrument and sky model. Evaluating the RIME involves carrying out an integral over the full sky, which is naturally discretized for point sources but must be approximated for diffuse emission. The choice of integration scheme can introduce errors that must be understood and isolated from the instrumental effects under study. In this paper, we present several analytically defined patterns of unpolarized diffuse sky emission for which the RIME integral is manageable, yielding closed-form or series visibility functions. We demonstrate the usefulness of these RIME solutions for validation by comparing them to simulated data and show that the remaining differences behave as expected with varied sky resolution and baseline orientation and length.

A Computationally Efficient Optimal Wigner Distribution in LCT Domains for Detecting Noisy LFM Signals

Recently, Wigner distribution (WD) associated with linear canonical transforms (LCTs) is quickly becoming a promising technique for detecting linear frequency-modulated (LFM) signals corrupted with noises by establishing output signal-to-noise ratio (SNR) inequality model or optimization model. Particularly, the closed-form instantaneous cross-correlation function type of WD (CICFWD), a unified linear canonical Wigner distribution, has shown to be competitive in detecting noisy LFM signals under an extremely low SNR. However, the CICFWD has up to nine LCT free parameters so that it requires a heavy computational load. To improve the efficiency of real-time processing, this paper focuses on the instantaneous cross-correlation function type of WD (ICFWD), which has only six LCT free parameters but is not a special case of the CICFWD. The main advantage of ICFWD is that it could be expected to reduce the computational complexity while maintaining detection performance. This paper first proposes an optimization model to the ICFWD’s output SNR with respect to deterministic signals embedded in additive zero-mean noises. It then deduces the model’s solution to a single component LFM signal added with white noise, leading to the optimal selection strategy on LCT free parameters. Simulation results demonstrate that the ICFWD improves almost a doubling of computing speed in comparison with the CICFWD while sharing the same level of detection performance. To be specific, the computing time of ICFWD in sampling frequencies 5 Hz, 10 Hz, 15 Hz, and 20 Hz is about 0.048 s, 0.111 s, 0.226 s, and 0.392 s, respectively, while 0.075 s, 0.233 s, 0.478 s, and 0.821 s for the computing time of CICFWD; the ICFWD and CICFWD have nearly the same output SNR higher than that of the WD.

An alternative constitutive model for elastic particle-reinforced hyperelastic matrix composites with explicitly expressed Eshelby tensor

Herein, an alternative theoretical model is developed based on a two-phase Mori–Tanaka method within the framework of finite deformation continuum theory, in order to characterize the nonlinear mechanical behavior of particle-reinforced hyperelastic matrix composites (PRHMCs). The equivalent inclusion problems of a hyperelastic medium (HEM) and a reference elastic medium (REM) are analyzed, both of which have the same configuration and elastic property. Based on the equivalence between incremental stress fields in the HEM and REM, the Eshelby tensor for large deformation cases can be explicitly expressed as a closed-form function depending on the elastic modulus and tangent modulus of the hyperelastic matrix and the classical Eshelby tensor for linearly elastic composites. The Eshelby tensor for hyperelastic composites can be easily determined, without requiring time-consuming numerical ellipsoidal integrals in existing meso-mechanical models for PRHMCs. Incremental constitutive relations depending on the material tangent modulus are adopted to describe the mechanical behaviors of hyperelastic matrix and elastic particles. An incrementally effective constitutive relation of PRHMCs is finally achieved with a homogenization procedure, which is used to well reproduce the stress-strain responses of several types of particle-reinforced hyperelastic matrix composites in uniaxially and biaxially tensile experiments. These results demonstrate the feasibility and convenience of the present model not only in predicting the mechanical behavior of particle-reinforced elastomers, but also in guiding the design of advanced flexible composites with desirable mechanical performances.

The Polarimetric Detection Optimization Filter and its Statistical Test for Ship Detection

Ship detection via synthetic aperture radar (SAR) has been demonstrated to be very useful as polarimetric information helps discriminate between targets and sea clutter. Among the available polarimetric detectors, optimal polarimetric detection (OPD) theoretically provides the best detection performance under the assumption that the fully developed speckle hypothesis stands. This study proposes a polarimetric detection optimization filter (PDOF). The target clutter ratio (TCR) over the speckle variation was maximized using a matrix transform to derive the PDOF. The objective function based on a matrix transform instead of a vector transform is optimized to obtain synthetic effects by combining a polarimetric whitening filter (PWF) and a polarimetric matched filter (PMF). Subspace form of the PDOF (SPDOF) is also proposed, which gives performance comparable to the PDOF. Assuming a Wishart distribution, the exact and approximate expressions of the closed-form probability density function (PDF) of the PDOF are derived. The probability of false alarm (PFA) was derived in a closed-form expression, which allows obtaining the PDOF threshold analytically. Moreover, the gamma model is extended to a generalized gamma distribution ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{G}\Gamma \text{D}$ </tex-math></inline-formula> ) to adapt complicated resolutions and sea states. Experiments with simulated and real data validate the correctness and effectiveness of the results. The PDOF detector achieves the best performance in most virtual and real-world environments, especially in cases where the target statistics and clutter are not Wishart-distributed.

Resolution Enhancement for Large-Scale Real Beam Mapping Based on Adaptive Low-Rank Approximation

Recently, a variety of super-resolution (SR) methods have been devoted to enhancing the angular resolution of real beam mapping (RBM) imagery in modern microwave remote sensing applications. When addressing large-scale datasets, however, they suffer from notably high computational complexity due to high-dimensional matrix inversion, multiplication, or singular value decomposition (SVD). To overcome this limitation, this article presents a low-complexity SR strategy based on adaptive low-rank approximation (LRA). Our underlying idea is first to construct a random matrix sketching to sample the raw echo measurements and restore the surface map of reflectivity in a low-dimensional linear space. The resulting low-complexity strategy enables substantial computational complexity reduction for a group of SR methods, at the cost of introducing a manually adjusted LRA parameter. Using the Fourier transform-based antenna analysis method, we further reveal that the LRA parameter that ensures support resolution improvement can be determined by a closed-form function of the aperture length, the wavelength, and the field of view, allowing for adaptively and efficiently selecting the optimal LRA parameter that well balances the tradeoff between LRA error and computational efficiency. We use both simulated and real datasets to demonstrate that the proposed LRA-based SR strategy can provide significant speedup without performance loss.