Closed-form Expressions Research Articles

A convexity-constrained parameterization of the random effects generalized partial credit model.

An alternative closed-form expression for the marginal joint probability distribution of item scores under the random effects generalized partial credit model is presented. The closed-form expression involves a cumulant generating function and is therefore subjected to convexity constraints. As a consequence, complicated moment inequalities are taken into account in maximum likelihood estimation of the parameters of the model, so that the estimation solution is always proper. Another important favorable consequence is that the likelihood function has a single local extreme point, the global maximum. Furthermore, attention is paid to expected a posteriori person parameter estimation, generalizations of the model, and testing the goodness-of-fit of the model. Procedures proposed are demonstrated in an illustrative example.

Precision tests of bulk entanglement entropy

We consider linear superpositions of single particle excitations in a scalar field theory on AdS3 and evaluate their contribution to the bulk entanglement entropy across the Ryu-Takayanagi surface. We compare the entanglement entropy of these excitations obtained using the Faulkner-Lewkowycz-Maldacena formula to the entanglement entropy of linear superposition of global descendants of a conformal primary in a large c CFT obtained using the replica trick. We show that the closed form expressions for the entanglement entropy in the small interval expansion both in gravity and the CFT precisely agree. The agreement serves as a non-trivial check of the FLM formula for the quantum corrections to holographic entanglement entropy as well as the methods developed in the CFT to evaluate entanglement entropy of descendants. Our checks includes an example in which the state is time dependent and spatially in-homogenous as well another example involving a coherent state with a Bañados geometry as its holographic dual.

Gordon Decomposition of the Magnetizability of a Dirac One-Electron Atom in an Arbitrary Discrete Energy State

We present a Gordon decomposition of the magnetizability of a Dirac one-electron atom in an arbitrary discrete energy eigenstate, with a pointlike, spinless, and motionless nucleus of charge Ze. The external magnetic field, by which the atomic state is perturbed, is assumed to be weak, static, and uniform. Using the Sturmian expansion of the generalized Dirac–Coulomb Green function proposed by Szmytkowski in 1997, we derive a closed-form expressions for the diamagnetic (χd) and paramagnetic (χp) contributions to χ. Our calculations are purely analytical; the received formula for χp contains the generalized hypergeometric functions 3F2 of the unit argument, while χd is of an elementary form. For the atomic ground state, both results reduce to the formulas obtained earlier by other author. This work is a prequel to our recent article, where the numerical values of χd and χp for some excited states of selected hydrogenlike ions with 1⩽Z⩽137 were obtained with the use of the general formulas derived here.

Application of pole allocation to optimize passive viscous dampers represented by the Maxwell model

The Maxwell model is used to understand the realistic effectiveness of vibration reduction in the building structures. The model consists of a dashpot and spring in series. The dashpot and spring represent a viscous damper and joint between the damper and the structural frame, respectively. However, few studies have investigated the optimal damper capacity and the corresponding spring stiffness in the practical parameter ranges. Additionally, the optimal damper derived using the fixed-point theory has no relationships with that derived via pole allocation. Therefore, in this study, to examine the effects of the joint spring, the pole allocation method was used to design a structural system wherein the Maxwell model was incorporated into a single-degree-of-freedom damped model. We introduced a closed-form expression, which explicitly described the relationships between the target structural damping ratio, damper capacity, and joint spring. The Maxwell model constrained the control effectiveness using damper parameters and simultaneously suggested an optimal and realistic joint spring for the damper. Pole allocation was also applied to a multi-degree-of-freedom (M-DOF) damped structural system employing multiple Maxwell models. The newly proposed optimized joint spring could easily control the additional damping effect on the M-DOF system. The Maxwell model can be optimized almost manually. The scheme is more useful in the preliminary design stage than the previous numerical optimizations. This study extended the mathematical equation governing the building vibrations to the Maxwell model. The extended equation served as a unified description for considering structural passive control.

Partial stochastic resetting with refractory periods

Abstract The effect of refractory periods in partial resetting processes is studied. Under Poissonian partial resets, a state variable jumps to a value closer to the origin by a fixed fraction at constant rate, $x\to a x$. Following each reset, a stationary refractory period of arbitrary duration takes place. We derive an exact closed-form expression for the propagator in Fourier-Laplace space, which shows rich dynamical features such as connections not only to other resetting schemes but also to intermittent motion. For diffusive processes, we use the propagator to derive exact expressions for time dependent moments of $x$ at all orders. At late times the system reaches a non-equilibrium steady state which takes the form of a mixture distribution that splits the system into two subpopulations; trajectories that at any given time in the stationary regime find themselves in the freely evolving phase, and those that are in the refractory phase. In contrast to conventional resetting, partial resets give rise to non-trivial steady states even for the refractory subpopulation. Moments and cumulants associated with the steady state density are studied, and we show that a universal optimum for the kurtosis can be found as a function of mean refractory time, determined solely by the strength of the resetting and the mean inter-reset time. The presented results could be of relevance to growth-collapse processes with periods of inactivity following a collapse.

High-dimensional asymptotics of denoising autoencoders*

Abstract We address the problem of denoising data from a Gaussian mixture using a two-layer non-linear autoencoder with tied weights and a skip connection. We consider the high-dimensional limit where the number of training samples and the input dimension jointly tend to infinity while the number of hidden units remains bounded. We provide closed-form expressions for the denoising mean-squared test error. Building on this result, we quantitatively characterize the advantage of the considered architecture over the autoencoder without the skip connection that relates closely to principal component analysis. We further show that our results accurately capture the learning curves on a range of real data sets.

Coverage Probability Analysis of a Double IRS Assisted SISO Communication System

Intelligent reflecting surface (IRS) improves wireless communication efficiency and coverage probability. Existing studies focus on IRS deployment but overlook optimal fading channels. This research aims to identify and analyze the most effective channel fading estimation for the double IRS system model. The coverage probability using two-IRS is derived under different fading channels like Rayleigh and Rician to find the optimal solution in a Single-Input-Single-Output (SISO) system. Additionally, a closed-form expression for the coverage probability using statistical channel state information is derived while assuming no direct path between the transmitter and receiver. The analytical results are validated through simulations. It offers valuable insights into the modeling and analysis of double IRS-assisted wireless communication systems.

Estimating Social Effects with Randomized and Observational Network Data

Abstract This paper introduces an innovative approach to identifying and estimating the parameters of interest in the widely recognized linear-in-means regression model under conditions where the initial randomization of peers determines the observed network. We assert that peers who are initially randomized do not produce social effects. However, after randomization, agents can endogenously develop significant connections that potentially generate peer influences. We present a moment condition that compiles local heterogeneous identifying information for all agents within the population. Under the assumption of ψ-dependence in the endogenous network space, we propose a Generalized Method of Moments (GMM) estimator, which is proven to be consistent, asymptotically normally distributed, and straightforward to implement using commonly available statistical software due to its closed-form expression. Monte Carlo simulations demonstrate the GMM estimator’s strong small-sample performance. An empirical analysis utilizing data from Hong Kong high school students reveals substantial positive spillover effects on math test scores among study partners in our sample, provided that their seatmates were exogenously assigned by their teachers.

Radiative-conductive heat transfer dynamics in dissipative dispersive anisotropic media

Abstract We develop a self-consistent theoretical formalism to model the dynamics of heat transfer in dissipative, dispersive, anisotropic nanoscale media, such as metamaterials. We employ our envelope dyadic Green’s function method to solve Maxwell’s macroscopic equations for the propagation of fluctuating electromagnetic fields in these media. We assume that the photonic radiative heat transfer mechanism in these media is complemented by dynamic phononic mechanisms of heat storage and conduction, accounting for effects of local heat generation. By employing the Poynting theorem and the fluctuation-dissipation theorem, we derive novel closed-form expressions for the radiative heat flux and the coupling term of photonic and phononic subsystems, which contains the heating rate and the radiative heat power contributions. We apply our formalism to the paraxial heat transfer in uniaxial media and present relevant closed-form expressions. By considering a Gaussian transverse temperature profile, we also obtain and solve a system of integro-differential heat diffusion equations to model the paraxial heat transfer in uniaxial reciprocal media. By applying the developed analytical model to radiative-conductive heat tranfer in nanolayered media constructed by layers of silica and germanium, we compute the temperature profiles for the three first orders of expansion and the total temperature profile as well. The results of this research can be of interest in areas of science and technology related to thermophotovoltaics, energy harvesting, radiative cooling, and thermal management at micro- and nanoscale.

Efficient Sampling From the Watson Distribution in Arbitrary Dimensions

In this paper, we present two efficient methods for sampling from the Watson distribution in arbitrary dimensions. The first method adapts the rejection sampling algorithm from Kent et al. (2018), originally designed for Bingham distributions, using angular central Gaussian envelopes. For the Watson distribution, we derive a closed-form expression for the parameters that maximize sampling efficiency, which is further investigated and bounded by asymptotic results. This approach avoids the curse of dimensionality through a smart matrix inversion, enabling fast runtimes even in high dimensions. The second method, based on Saw (1978), employs adaptive rejection sampling from a projected distribution. This algorithm is also effective in all dimensions and offers rapid sampling capabilities. Finally, our simulation study compares the two main methods, revealing that each excels under different conditions: the first method is more efficient for small samples or large dimensions, while the second performs better with larger samples and more concentrated distributions. Both algorithms are available in the R package watson on CRAN. Supplementary materials for this article are available online.

Exploring the fundamental properties and MGF of a 3-component mixture of power distributions: a simulation study

This paper estimates the parameters of the 3-component mixture of power distributions. The expressions of fundamental properties of the said distribution are presented, and for judgment, the numerical results of mean, median, variance, mode and skewness are evaluated. The closed-form expressions of the reliability properties, the statistical properties and necessary inequality measures are discussed. The trend of reliability measures is shown with the help of numerical results and graphs by taking different values of parameters and proportion parameters. The mathematical expressions of certain related statistical functions are derived from the reliability functions, the moment generating function, the characteristics function, the probability generating function, and the factorial moment generating function. The expressions of essential entropies, Shannon’s entropy, the B-entropy and Renyi’s entropy, are derived. The mathematical expressions of inequality indexes, the Gini index, the Lorenz curve, the Bonferroni curve, the Zinga index, the Atkinson index and the generalized entropy index are also analysed. The RF, the HRF, the CHR, the RHR, the MRL, and the MWT functions are derived in reliability analysis. The numerical values and graphs for different parameters and proportion components are discussed to judge the behaviour of reliability properties.

Bivariate BMM-based hybrid domain image watermark detector

Robustness, imperceptibility, and watermark capacity are three indispensable and contradictory properties for any image watermarking systems. It is a challenging work to achieve the balance among the three important properties. In this paper, by using the bivariate Beta mixture model (Bivariate BMM), we present a statistical image watermark scheme in nonsubsampled Contourlet transform (NSCT)-fractional order Jacobi Fourier moments (FJFMs) amplitude hybrid domain. The whole watermarking algorithm includes two parts: watermark embedding and detection. NSCT is firstly performed on host image to obtain the frequency subbands, and the NSCT subbands are divided into non overlapping blocks. Then, the significant NSCT domain blocks are selected using local binary patterns (LBP). Meanwhile, for each selected NSCT coefficient block, FJFMs are calculated to obtain the NSCT-FJFMs amplitude. Finally, watermark signals are inserted into the amplitude hybrid domain of NSCT-FJFMs. In order to detect accurately watermark signal, the statistical characteristics of NSCT-FJFMs magnitudes are analyzed in detail. Then, NSCT-FJFMs magnitudes are described statistically by Bivariate BMM, which can simultaneously capture the marginal distribution and strong dependencies of NSCT-FJFMs magnitudes. Also, Bivariate BMM parameters are estimated accurately by the rough-enhanced-Bayes mixture estimation & expectation–maximization (REBMIX&EM) approach. Finally, a statistical watermark detector based on the locally optimum (LO) decision rule and Bivariate BMM is developed in NSCT-FJFMs magnitude hybrid domain. Also, the closed-form expressions on the LO statistic is derived and the receiver operating characteristic (ROC) about our detector is analyzed. Extensive experimental results show the superiority of the proposed image watermark detector over some state-of-the-art statistical watermarking methods.

Performance analysis of mean level CFAR detectors in homogeneous gamma-distributed radar clutter

ABSTRACT In this paper, a novel and exact mathematical formulation of three mean level constant false alarm rate (CFAR) detectors has been developed considering a homogeneous gamma-distributed (GM) clutter. These detectors include the cell averaging (CA), the greatest-of (GO), and the smallest-of (SO)-CFAR detection schemes. To do so, we derived closed-form expressions for the probability density function (pdf), and the cumulative distribution function (cdf) of the sum of an exponential fluctuating target submerged in gamma-distributed sea clutter. Then, we developed exact and explicit expressions for the probability of false alarm (P fa ) and probability of detection (P d ) of the optimal fixed threshold detector and the three considered CFAR detectors. The correctness of our expressions is then examined and validated via numerical simulations. Assuming a homogeneous GM background, the detection performance of these three detectors is carried out, discussed, and compared with that of the optimal detector. Experimental results with Sentinel-1 synthetic aperture radar (SAR) data demonstrate the effectiveness of the proposed closed-form expressions and show that the considered CFAR schemes are efficient for ship detection in sea clutter backgrounds.

Optimal scheduling of battery energy storage system operations under load uncertainty

This paper investigates the optimal scheduling of battery energy storage system operations considering energy load uncertainty. We develop a novel two-stage distributionally robust optimization model to determine an optimal battery usage schedule that minimizes the worst-case energy costs considering peak load costs. The model leverages deep-learning-based probabilistic forecasting in the construction of the ambiguity set. Specifically, we develop a Deep Autoregressive Recurrent Networks model to generate a probabilistic forecast of energy loads over a time horizon. The output of the forecasting model is then used to construct a marginal-moment ambiguity set for the distributionally robust optimization model. To solve the proposed model, we establish a closed-form characterization of the optimal second-stage objective function value. Leveraging this closed-form expression and using second-order conic duality, we derive an exact single-level mixed integer second-order conic reformulation of the problem. Extensive computational experiments, conducted on a real dataset, demonstrate the value of our proposed model and the effectiveness of the resulting battery schedule. The results show that the proposed model outperforms several benchmarks, including two-stage stochastic programming. Furthermore, the accuracy of the load forecast significantly impacts the effectiveness of the optimal battery schedule in eliminating peak loads by achieving up to 18% reduction in the maximum energy load.

Analysis of the multi-term fractional Bateman equations in radioactive decay by means of Mikusiński algebraic calculus

Fractional Bateman equations model the concentrations of nuclear species through a radioactive decay chain, by considering non-Markovian behavior. Cruz-López and Espinosa-Paredes (Nuclear Engineering and Technology, 2022) solved the mathematical problem with a single memory coefficient, based on the Laplace-transform technique. Cruz-López, Espinosa-Paredes and François (Computer Physics Communications, 2022) extended the analysis to several fractional orders, with Laplace transforms and series expansions checked by induction. In this paper, we solve such a case of distinct memory effects, via the more general method of Mikusiński operational calculus. We do not distinguish between equal and different decay rates. Multivariate Mittag-Leffler functions appear in the closed-form expression.

BER Analysis of Underlay Cooperative Cognitive Radio-based NOMA System

Background: The integration of Cooperative Communications (CC), Cognitive Radio (CR) technology, and Non-Orthogonal Multiple Access (NOMA) techniques, termed Cooperative Cognitive Radio used NOMA (CCR-NOMA) systems, has emerged as a promising solution to address spectrum scarcity and connectivity challenges anticipated in sixth generation (6G) networks. Aim: This studyaims to investigate the Bit Error Rate (BER) performance of underlay CCRNOMA systems. Methods: We derive precise closed-form expressions for BER at distant users under perfect and imperfect Channel State Information (CSI) conditions. These mathematical formulations are validated through Monte Carlo simulations. Results: Our results indicate that the near user CU! exhibits superior performance compared to the far user CU!. Additionally, distant users utilizing the CCR-OMA protocol demonstrate better BER performance than those employing the CCR-NOMA protocol. Conclusion: The presence of imperfect CSI adversely affects BER performance. Moreover, the derived closed-form expressions for BER in the investigated system align well with Monte Carlo simulations. These findings provide valuable insights for optimizing the performance of CCR-NOMA systems in real-world scenarios.

Cognitive radio inspired RSMA-MEC under hardware impairments: Performance analysis and optimization

In this paper, we investigate a cognitive radio (CR) inspired rate splitting multiple access (RSMA) aided mobile edge computing (MEC) network, where hardware impairments (HIs) in all transceivers are considered. Given that allowing the secondary user (SU) to share the resource with the primary user (PU) for data offloading does not degrade the PU’s offloading performance, we derive closed-form expressions for the rate-splitting parameter and power allocation coefficient at the SU to maximize its offloading rate. Using the derived parameters, we then derive the successful computation probability (SCP), which is defined as the probability that both the PU and SU can successfully compute their task bits within a given latency budget, into the closed form. To further enhance the performance of the CR inspired RSMA-MEC network with HIs, a SCP maximization-based problem is formulated to jointly optimize the task offloading ratio and task offloading time of both the SU and PU. Leveraging the convex theory, we obtain the optimal solutions in the closed form. Simulation results confirm the following two insights. First, the presence of HIs leads to a decreasing on the SCP and the SCP approaches to a constant which is less than 1 with the increase of the transmit power. Second, with the optimal task offloading ratio and time, the CR inspired RSMA-MEC achieves the highest SCP compared to the existing schemes.

Publisher Correction: On closed-form expressions for the Fisher–Rao distance

Publisher Correction: On closed-form expressions for the Fisher–Rao distance

Improving the security of NOMA cognitive networks

Aiming at the scenario where the eavesdropper try to eavesdrop on the confidential messages of secondary trusted users, two physical-layer network coding (PNC) strategies, namely dual power superposition (PS-PS) encoding strategy and bit-level exclusive-or power superposition (XOR-PS) encoding strategy, are proposed to improve the security performance of cognitive non-orthogonal multiple access (NOMA) networks with the help of the data transmitted by the primary user. Considering the overlay spectrum sharing mechanism and the imperfect successive interference cancellation (SIC) technology, we first derive the closed-form accurate expressions of the outage probability and intercept probability of the secondary network with these two schemes over Rayleigh fading, and also give the asymptotic outage probability in the high signal-to-noise ratio (SNR) regime. Then, the correctness of the theoretical results is verified by simulation. For comparison, we also use the scheme without the assistance of the primary user as the benchmark scheme, which is represented by NPS. Finally, the simulation results show that the security of the proposed PS-PS and XOR-PS schemes is much better than that of the NPS scheme in the case of without reducing the outage performance.

Sound transmission through an unbaffled plate

Commonly occurring complex structures can often be modelled as simple finite plates mounted without baffles. Thus, the objective here is to study the sound transmission loss of a simply-supported rectangular finite unbaffled plate impinged upon by an acoustic plane wave. The total pressure jump across the plate surface is expressed as the sum of the pressure jump due to pure diffraction, where the plate normal displacement is zero and the pressure jump due to plate radiation. These pressure jumps are computed using the definition of the free-space Green's function in the wavenumber domain and the Euler equation. Then, the coupled equation of motion is solved for the plate displacement. The method of stationary phase is used to derive the closed-form expression for the improper double integral that appears in the far-field acoustic pressure. Finally, the far-field transmitted power is numerically computed and the sound transmission loss of the unbaffled plate is compared with that of the baffled one.