Closed-form Expressions Research Articles

Fast fracture in toughened glass when impacted randomly by Ice

Modelling the triggering of fracture at a pre-existing flaw is an evolving method of predicting ultimate failure in glass. This fracture mechanics approach of modelling has been shown to give more reliable predictions than a calibrated probabilistic distribution model (as is commonly adopted) when dealing with hail impact which is highly transient in nature. The dynamic stress intensity factor controlling fast crack growth is sensitive to the complex stress state surrounding the critical flaw. Finite element simulations of localised stresses in 3D could incur high computation cost which is compounded by the need to repeat computations until convergence and to simulate multiple strikes in emulating a storm scenario. In this study, closed-form expressions were developed to waive away the need of any simulations. With hail impact, boundary conditions of the glass panel need not be factored into the modelling, as the highly transient stresses are wave controlled. Input parameters are the thickness and level of prestress in glass; offset of position of strike from the known crack and its depth; and the size, velocity, and temperature of the ice impactor. Results from 40 test scenarios involving 5 offset distances, 3 crack depths, 2 glass thicknesses, and 2 sizes of ice were used to validate the prediction, and to reveal the sensitivity of the outcome of the impact to changes in the impact position relative to the known crack.

Performance analysis of CSI independent cache-aided NOMA ordering

This paper introduces a Non-Orthogonal Multiple Access (NOMA) ordering technique, called the region-based ordering method, which operates independently of channel state information (CSI) for vehicular-to-infrastructure (V2I) networks. Caching has been Integrated with NOMA to significantly enhance outage probability, ergodic sum rate, and latency performance compared to traditional NOMA systems. The performance of the proposed method is evaluated under various cache content availability scenarios and compared to dynamic ordered NOMA, which relies on instantaneous CSI for user ordering. A closed-form analytical expression for outage probability is derived for a Rayleigh fading channel and validated through Monte Carlo simulations. The analysis demonstrates that the proposed method is as effective as the dynamic ordering method. Additionally, cache-aided NOMA significantly improves outage performance and reduces delivery delay compared to conventional NOMA.

The analysis of the non-Kolmogorov turbulence effect on the performance of uplink coherent DPSK laser satellite communication system in the presence of tilt-corrected beam wander

In the present study, a model for analyzing the performance of laser-based satellite uplinks that account for beam wander and atmospheric turbulence is presented. The developed scintillation index model is a continuation of the recent works considering the effects of non-Kolmogorov turbulence on the propagation of Gaussian beams across turbulent media. We specifically concentrate on the specific regime under the tilt-corrected beam wander. The on-axis scintillation index is developed assuming an isotropic non-Kolmogorov spectrum model where the power law exponent (PLE) effective value, α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document}, is taken to be 3 < α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} < 4. Within that regime, effective irradiance models for tilt-corrected beam wander are presented, using the Gamma and Gamma-Gamma distributions. Closed-form expressions of the average bit error rate (BER) are derived to evaluate the performance of the optical communication system considering coherent detection of differential phase shift keying. The obtained numerical results show the impact of the Gaussian beam radius at the transmitter, the PLE effective value, and the zenith angle on the on-axis scintillation index and the average BER of the system under consideration. Due to beam wander, the link performance in terms of scintillation index and average BER depends on the beam radius at the transmitter. With tilt-corrected beam wander the link performance becomes best at an intermediate value of beam radius. With the non-Kolmogorov turbulent medium model, the PLE effective value has a major effect on the beam radius with the best performance. The obtained results guide the choice of the optimum beam radius for best performance for each PLE effective value.

A scalar field theory of 1+1-dimensional laser wakefield accelerators

Abstract A relativistic non-linear scalar field theory is developed from a 2+2-dimensional decomposition of the cold plasma field equations, and the theory is used to investigate a 1+1-dimensional description of a laser wakefield accelerator. The relationship between the properties of a compact laser pulse and its wake is explored. Non-linear solutions are sought describing a regular (i.e. unbroken) wake driven by a prescribed rectangular circularly-polarised laser pulse. An upper bound on the dimensionless amplitude a 0 of the laser pulse is determined as a function of the phase speed v of the wake. The asymptotic behaviour of the upper bound on a 0 as v → c is shown to agree with well-established, but approximate, results obtained using the conventional encoding of the plasma degrees of freedom. Our approach leads to a closed-form expression for the upper bound on a 0 which is exact for all values of the phase speed of the wake, unlike conventional results that are applicable only when v is sufficiently close to c.

Modeling and Performance Analysis of LBT-Based RF-Powered NR-U Network for IoT.

Energy harvesting combined with spectrum sharing offers a promising solution to the growing demand for spectrum while keeping energy costs low. New Radio Unlicensed (NR-U) technology enables telecom operators to utilize unlicensed spectrum in addition to the licensed spectrum already in use. Along with this, the energy demands for the Internet of Things (IoT) can be met through energy harvesting. In this regard, the ubiquity and ease of implementation make the RF-powered NR-U network a sustainable solution for cellular IoT. Using a Markov chain, we model the NR-U network with nodes powered by the base station (BS). We derive closed-form expressions for the normalized saturated throughput of nodes and the BS, along with the mean packet delay at the node. Additionally, we compute the transmit outage probability of the node. These quality of service (QoS) parameters are analyzed for different values of congestion window size, TXOP parameter, maximum energy level, and energy threshold of the node. Additionally, the effect of network density on collision, transmission, and energy harvesting probabilities is observed. We validate our model through simulations.

Time-Critical Unified Rendezvous Guidance for an Unmanned Autonomous Vehicle

Abstract This paper addresses the time-critical rendezvous problem for a pursuing autonomous unmanned vehicle, e.g., an unmanned aerial vehicle (UAV), guided using the concept of true-proportional navigation guidance, which is a variant of proportional-navigation guidance. In existing vehicle routing and flight time-constrained guidance techniques, specific rendezvous guidance commands are designed based on the specific motion of the target. In contrast to that, we propose a unified guidance command for a UAV that guarantees a time-critical rendezvous with a target that moves arbitrarily. We explore the purview of true-proportional navigation guidance and posit that a guidance law thus designed may be a potential candidate for designing time-critical rendezvous strategies against various target motions, even when the pursuer does not necessarily have a speed advantage over the target. We first derive a closed-form expression for the flight duration until rendezvous, over which we exercise control to make the pursuing vehicle rendezvous with the target at any feasible time prescribed a priori. Next, we ensure that the necessary flight time-based error variable converges to zero with an optimal convergence pattern with respect to a suitable cost function. We finally validate the efficacy of the proposed unified guidance command via numerical simulations.

Physical layer security in SWIPT-based cooperative vehicular relaying networks

The future of autonomous transportation systems depends on energy sustainability and secure information exchange from low-power vehicular sensors, hence the increased interest in vehicular sensor charging using simultaneous wireless information and power transfer (SWIPT). This study investigates the physical layer security of a SWIPT-based radio frequency energy harvesting cooperative vehicular relaying network subjected to cascade Nakagami-m and double Nakagami-m (DN) fading channels. In the considered system model, a stationary source communicates with a mobile destination through a power-splitting-based decode-and-forward relay in the presence of a mobile passive eavesdropper. Based on the Gamma-distributed first term of the Laguerre series, new statistical probability density function (PDF) and cumulative distribution function (CDF) expressions for the DN are derived to accurately model the complex cascaded fading scenario. The secrecy performance metrics analyzed are the secrecy outage probability (SOP), the probability of non-zero secrecy capacity (PNZSC), and the intercept probability (IP). In addition, the asymptotic SOP (ASOP) is investigated in the high signal-to-noise ratio (SNR) to enhance the comprehension of the secrecy performance. Based on the derived ASOP, the secrecy diversity order (SDO) of the proposed system is determined and examined. Particularly, we present analytical closed-form expressions for the secrecy performance metrics and provide a detailed understanding of the impact of the system parameters under the cascade fading scenario. Then, a power splitting (PS) optimization problem is formulated to minimize the SOP. The results demonstrate a reduction in the SOP with the proposed PS scheme compared to the equal PS scheme. The obtained analytical findings are validated using Monte Carlo simulations.

The Maxwell-Boltzmann-Exponential distribution with regression model

Abstract This paper proposes a new probability model called as Maxwell-Boltzmann-Exponential (MBE) distribution. The MBE distribution arises as a mixture distribution of the Maxwell-Boltzmann and exponential distributions. The statistical properties of the distributions are studied and obtained in closed-form expressions. Three methodologies are assessed and compared for the estimation of parameters in the MBE distribution. The MBE regression model is defined, with the proposed regression model being an alternative to the gamma regression model for response variables that are extremely right-skewed and bimodal. Two real data sets are used to demonstrate the applicability of the proposed models against the existing models.

Performance Evaluation of UOWC Systems from an Empirical Channel Model Approach for Air Bubble-Induced Scattering.

Underwater optical wireless communication (UOWC) systems provide the potential to establish secure high-data-rate communication links in underwater environments. The uniqueness of oceanic impairments, such as absorption, scattering, oceanic turbulence, and air bubbles demands accurate statistical channel models based on empirical measurements for the development of UOWC systems adapted to different types of water and link conditions. Recently, generalized Gamma and a mixture of two generalized Gamma probability density functions (PDF) were proposed to describe the statistical behavior of small and large air bubbles, respectively, when considering several levels of particle-induced scattering. In this paper, we derive novel closed-form analytic expressions to compute the bit error rate (BER) and outage performance using both proposed PDFs for various scattering conditions. Furthermore, simple asymptotic expressions are obtained to determine the diversity order of each scenario. Monte Carlo simulation results verify the obtained theoretical expressions. Our results also reveal that UOWC systems present lower BER and outage performance under more turbid water cases with respect to the tap water case due to the higher diversity order and despite the significant increases in pathloss at short link distances. Particle-induced scattering provides an inherent mechanism of turbid waters to mitigate air bubble-induced fluctuations and light blockages.

Simple closed-form estimation of a binary latent variable model

Abstract This paper develops a closed-form nonparametric estimator of the conditional distribution function for a binary outcome variable given an unobserved latent variable. This type of function is commonly used in models that involve measurement error and dynamic models with agent-specific unobserved heterogeneity. This paper presents a consistent extremum sieve estimator with the following advantages: i) it has a closed-form expression for all the sieve coefficients; ii) it is computationally straightforward, equivalent to computing eigenvalues and eigenvectors of matrices without the use of iterative optimization algorithms. While as flexible as the sieve maximum likelihood estimator (MLE) previously proposed for this model, our estimator proves computationally simpler. The finite sample properties of the estimator are investigated through a Monte Carlo study, and the developed estimator is applied to a probit model to assess the targeting performance of a social welfare program.

Rotational influence on fermions within negative curvature wormholes

In this research, we examine relativistic fermions within the rotating frame of negative curvature wormholes. Initially, as is typical in our context, we introduce the wormholes by embedding a curved surface into a higher-dimensional flat Minkowski spacetime. Subsequently, we derive the spacetime metric that characterizes the rotating frame of these wormholes. We then investigate analytical solutions of the generalized Dirac equation within this framework. Through exploring a second-order non-perturbative wave equation, we seek exact solutions for fermions within the rotating frame of hyperbolic and elliptic wormholes, also known as negative curvature wormholes. Our analysis provides closed-form energy expressions, and we generalize our findings to Weyl fermions. By considering the impact of the rotating frame and curvature radius of wormholes, we discuss how these factors affect the evolution of fermionic fields, offering valuable insights into their behavior.

A comparative study of numerical methods for reliability assessment based on semi-Markov processes

The semi-Markov process, renowned for its versatile applications, has garnered significant attention in recent years. However, deriving closed-form expressions for computing reliability metrics proves challenging when sojourn times deviate from exponential distributions. This paper investigates three numerical techniques for computing the transition function matrix of the semi-Markov process, including the algebraic method, the truncated method, and the iterative method. It delves into their truncation and discretization errors, as well as their computational complexities. Additionally, the Laplace-based method and the semi-Markov-chain-based method are discussed to contrast their effectiveness with the three devised numerical approaches. Building on the linkage between the Markov renewal equation and system reliability metrics, five computational methods are applied to the evaluation of system reliability and availability. A case study on sequential cyber-attacks is presented to illustrate the applicability of these methods, considering sojourn times that follow exponential, gamma, lognormal, and Weibull distributions respectively. The results reveal that the three proposed numerical methods not only achieve high precision and rapid speed but also address scenarios beyond the capability of the Laplace-based method.

Critical Permeability from Resummation

Special calculation methods are presented for critical indices and amplitudes for the permeability of thin wavy channels dependent on the waviness. The effective permeability and wetted perimeter of the two-dimensional random percolating media are considered as well. A special mathematical framework is developed to characterize the dependencies on porosities, critical points, and indices. Various approximation techniques are applied without involving popular lubrication approximation in any sense. In particular, the Borel summation technique is applied to the effective polynomial approximations with or without optimization. Minimal difference and minimal derivative optimal conditions are adapted to calculations of critical indices and amplitudes for the effective permeability of thin wavy channels. Critical indices, amplitudes, and thresholds are obtained for the effective permeability and wetted perimeter of the two-dimensional percolating random media. Closed-form expressions for all porosities, critical points, and indices are calculated from the polynomial approximations for the first time.

Optimal Investment Strategy for DC Pension Plan with Stochastic Salary and Value at Risk Constraint in Stochastic Volatility Model

This paper studies the optimal asset allocation problem of a defined contribution (DC) pension plan with a stochastic salary and value under a constraint within a stochastic volatility model. It is assumed that the financial market contains a risk-free asset and a risky asset whose price process satisfies the Stein–Stein stochastic volatility model. To comply with regulatory standards and offer a risk management tool, we integrate the dynamic versions of Value-at-Risk (VaR), Conditional Value-at-Risk (CVaR), and worst-case CVaR (wcCVaR) constraints into the DC pension fund management model. The salary is assumed to be stochastic and characterized by geometric Brownian motion. In the dynamic setting, a CVaR/wcCVaR constraint is equivalent to a VaR constraint under a higher confidence level. By using the Lagrange multiplier method and the dynamic programming method to maximize the constant absolute risk aversion (CARA) utility of terminal wealth, we obtain closed-form expressions of optimal investment strategies with and without a VaR constraint. Several numerical examples are provided to illustrate the impact of a dynamic VaR/CVaR/wcCVaR constraint and other parameters on the optimal strategy.

Assessing Nonlinear Polarization in Electrochemical Cells Using AC Impedance Spectroscopy

Electrochemical energy storage and conversion devices are used to power a range of electrical loads including portable electronics, electric vehicles, and utility grids. The energy efficiency of these systems decreases with increasing current load due to energy losses (in the form of heat) associated with different processes. Electrochemical impedance spectroscopy (EIS) is a nondestructive technique commonly used to assess rate-limiting steps in electrochemical devices. Most EIS measurements for energy storage devices are only performed around open-circuit and do not probe system behavior under non-zero biases where these devices operate. Despite its widespread use, this approach does not provide information about the relative overpotentials associated with coupled nonlinear processes (e.g., diffusion limited redox reactions) commonly encountered in these systems. Furthermore, ambiguous terms such as equivalent series resistance and interfacial resistance can lead to confusion and data misinterpretation. For example, Ohm’s law cannot be applied to calculate nonlinear overpotentials using a resistance that was measured from a locally linear response.To determine overpotentials in nonlinear systems using EIS, one must integrate the individual resistive elements of the equivalent circuit as a function of steady state current. Recent experimental work has used this method to decouple ohmic, kinetic, and transport overpotentials in redox flow batteries. In these prior studies, impedance elements were derived, but analytical expressions for the corresponding overpotentials were not presented.While integrated resistance measurements are a powerful experimental tool, a more rigorous mathematical framework describing the approach is needed. The present work bridges this gap by: (i) deriving a general impedance expression for coupled charge transfer and diffusion processes and (ii) integrating the individual resistive elements as a function of steady-state current to obtain global polarization relationships. Analytical expressions for diffusion overpotentials are obtained for special cases, and numerical methods are applied when closed-form expressions do not exist. The work considers three model electrode geometries (planar, cylindrical, and spherical) and two diffusion boundary types (reflective and transmission). Finally, this treatment is extended to macroscopically homogenous porous electrodes which are relevant for describing practical devices.This presentation will highlight how measuring impedance at different steady-state biases provides critical insights on the rate-limiting processes in energy storage devices. While this work focuses on a simple Faradaic charge transfer process, the method can be used to evaluate other nonlinear phenomena in electrochemical power supplies. For example, the resistance of passive films (e.g., solid electrolyte interphase layer) in Li-ion batteries is typically only probed near open-circuit. Similarly, literature on solid-state batteries commonly uses ambiguous terms such as interfacial resistance to describe rate-limiting processes. For these more complex systems, integrating the resistive elements can provide insights on how these phenomena influence device efficiency. Overall, combining these experimental observations with physical models derived using the approach illustrated herein represents a powerful tool to guide component/system design and improve device performance. Acknowledgements This research was conducted at Oak Ridge National Laboratory, managed by UT Battelle, LLC, for the U.S. Department of Energy (DOE). This work is sponsored by the U.S. Department of Energy in the Office of Electricity through the Energy Storage Research Program and the Office of Energy Efficiency and Renewable Energy (EERE) in the Vehicle Technologies Office (VTO) through the Advanced Battery Materials Research (BMR) Program.

All-Analytic Statistical Modeling of Constellations in (Optical) Transmission Systems Driven by High-Speed Electronic Digital to Analog Converters Part I: DAC Mismatch Statistics, Metrics, Symmetries, Error Vector Magnitude

This two-part work develops a comprehensive toolbox for the statistical characterization of nonlinear distortions of DAC-generated signal constellations to be transmitted over communication links, be they electronic (wireline, wireless) or photonic, Mach–Zehnder modulator (MZM)-based optical interconnects in particular. The all-analytic toolbox developed here delivers closed-form expressions for the second-order statistics (means, variances) of all relevant constellation metrics of the DACs’ building blocks and of DAC-driven MZM-based optical transmitters, all the way to the slicer in the optical receivers over a linear channel with coherent detection. The key impairment targeted by the model is the random current mismatch of the ASIC devices implementing the DAC drivers. In particular the (skew-)centrosymmetry of the DAC metrics is formally derived and explored. A key applicative insight is that the conventional INL/DNL (Integral NonLinearity/Differential NonLinearity) constellation metrics, widely adopted in the electronic devices and circuits community, are not quite useful in the context of communication systems, since these metrics are ill-suited to predict communication link statistical performance. To rectify this deficiency of existing electronic DAC metrics, we introduce modified variants of the INL|DNL, namely the integral error vector (IEV) and the differential error vector (DEV) constellation metrics. The new IEV|DEV represent straightforward predictors of relevant communication-minded metrics: error vector magnitude (EVM) treated here in Part I, and Symbol/Bit Error-Rates (SER, BER) treated in the upcoming Part II of this paper.

Structural Analysis of Periodic Trusses and Lattice Materials: States of Self-Stress, Mechanisms, and Mechanical Properties

Abstract In the realm of low relative density, a rigid-jointed lattice material exhibits a mechanical response analogous to that of a pin-jointed periodic truss with an identical micro-architecture. This correspondence simplifies the evaluation of the lattice's structural performance. While established matrix methods in linear algebra are capable of determining the states of self-stress, infinitesimal mechanisms, and mechanical properties of pin-jointed periodic trusses, they lack a systematic approach to deriving closed-form expressions for these properties. This paper presents a straightforward approach to address this gap. Furthermore, we introduce an inventive finite element framework that directly yields self-stress states and infinitesimal mechanisms in periodic trusses subjected to specific uniform macroscopic loading conditions. This framework allows for the determination of mechanical properties for periodic trusses by solving straightforward boundary value problems, such as uniaxial tension/compression or simple shear. The developed finite element framework offers greater simplicity compared to the matrix methods, with its benefits becoming more pronounced for complex micro-architectures. To demonstrate the effectiveness of the two newly developed methods, we conduct structural analyses on five different lattice materials, achieving successful outcomes.

A Spreadsheet-Based Redox Flow Battery Cell Cycling Model Enabled by Closed-Form Approximations

The complex interplay between numerous parasitic processes—voltage losses, crossover, decay—challenges interpretation of cycling characteristics in redox flow batteries (RFBs). Mathematical models offer a means to predict cell performance prior to testing and to interpret experimentally measured cycling data, however most implementations Require extensive domain expertise, programming knowledge, and/or computational resources. Here, we expand on our previously developed zero-dimensional modeling framework by deriving closed-form expressions for key performance metrics. The resulting closed-form model streamlines the computational structure and allows for spreadsheet modeling of cell cycling behavior, which we highlight by developing a simulation package in Microsoft® Excel®. We then apply this model to analyze previously published experimental data from our group and others, highlighting its utility in numerous diagnostic configurations—bulk electrolysis, compositionally unbalanced symmetric cell cycling, and full cell cycling. Given the accessibility of this modeling toolkit, it has potential to be a widely deployable tool for RFB research and education, aiding in data interpretation and performance prediction.

Novel Explicit Models for Assessing the Frictional Resistance of Pipe Piles Subjected to Seismic Effects

This paper introduces novel explicit models to predict the frictional resistance of open and closed-ended pipe piles subjected to seismic loading. This research employs genetic programming (GP) and multiobjective genetic algorithm-based evolutionary polynomial regression (EPR-MOGA) to develop closed-form expressions for estimating pile frictional resistance, utilizing widely used input parameters for enhanced practicality and applicability in engineering practice. The proposed models are developed using only three input variables: the corrected standard penetration test (SPT) blow count (N1)60, the pile slenderness ratio (L/D), and the peak ground acceleration (PGA). This deliberate reduction in input complexity significantly enhances the models' applicability across a wide range of geotechnical scenarios and industries. The accuracy of the developed models was assessed via the coefficient of determination (R2), root mean squared error (RMSE), and mean absolute error (MAE). In the case of the GP model, the evaluation metrics for the testing set for open-ended piles (R2, RMSE, and MAE values) are 0.89, 0.43, and 0.35, respectively, whereas the corresponding values for closed-ended piles are 0.93, 0.38, and 0.3, respectively. On the other hand, the EPR-MOGA approach achieves similarly encouraging results, with performance metrics of 0.92, 0.37, and 0.29 for open-ended piles and 0.91, 0.39, and 0.30 for closed-ended piles.

Joint calibration to SPX and VIX options with signature‐based models

AbstractWe consider a stochastic volatility model where the dynamics of the volatility are described by a linear function of the (time extended) signature of a primary process which is supposed to be a polynomial diffusion. We obtain closed form expressions for the VIX squared, exploiting the fact that the truncated signature of a polynomial diffusion is again a polynomial diffusion. Adding to such a primary process the Brownian motion driving the stock price, allows then to express both the log‐price and the VIX squared as linear functions of the signature of the corresponding augmented process. This feature can then be efficiently used for pricing and calibration purposes. Indeed, as the signature samples can be easily precomputed, the calibration task can be split into an offline sampling and a standard optimization. We also propose a Fourier pricing approach for both VIX and SPX options exploiting that the signature of the augmented primary process is an infinite dimensional affine process. For both the SPX and VIX options we obtain highly accurate calibration results, showing that this model class allows to solve the joint calibration problem without adding jumps or rough volatility.